text:
\section{Introduction}

\begin{figure}[ht!]
	\begin{center}
   	\includegraphics[width=5in]{Fig-1-laser-driven-ion-wake-soliton.jpg}

	\end{center}
\caption{ \small {\bf Laser driven non-linear ion-wake at early time (t = 46$\omega_{pe}^{-1}$ = 0.17 $f_{pi}^{-1}$, where $f_{pi}$ is the plasma ion frequency) in $m_{i}=m_p=1836 ~ m_e$ plasma}. (a) Electron bubble wakefields in cartesian coordinates (fixed-box) with $\frac{\omega_0}{\omega_{pe}}=10$ driven by a matched laser pulse (vector potential $a_0=4$ and frequency $\omega_0$) with $R_B \simeq 4\frac{c}{\omega_{pe}}$. (b) Non-linear ion-wake in the form of a cylindrical ion-soliton of radius $\simeq 4\frac{c}{\omega_{pe}}$ excited behind the bubble electron wake in a proton plasma. (c) Transverse ion-density profile at $z=15~c/\omega_{pe}$.  Notice that the ion density perturbation in this excitation phase is still building up and is a fraction of the background ion density, $\frac{\delta n_i}{n_0} < 1$.}
\label{fig1:ion-wake-laser}
\end{figure}

Plasma ions are generally assumed to be stationary in the theory of ultra-relativistic non-linear plasma electron waves \cite{Akhiezer-Polovin}. Such electron waves are regularly excited as wakefields of high-intensity energy sources such as an ultra-short laser or particle beams and have proved to be promising for accelerating and transporting beams with unprecedented field strengths \cite{wakefield}\cite{cavitation-laser}\cite{cavitation-beam}\cite{Pukhov-laser-bubble}\cite{Lu-bubble-regime}. Important exceptions to the fundamental assumption of stationary ions occur as the intensities of the energy sources become high enough to lead to significant ion motion within a period of the electron wave. Ion motion also invariably becomes important over several periods of the electron wake train further behind the driver as the energy left-over in electron oscillation modes couples to the ion modes.

The motion of ions has significant implications for plasma acceleration as high-intensity conditions occur when the drive beam (or an accelerated witness beam) has fields that lead to ion trajectories that are a considerable fraction of the electron oscillation trajectory \cite{ion-motion-intense-beam}\cite{ion-motion-beam-emittance}. These conditions are predicted to arise in the final-stage of ultra-low emittance future plasma-based collider designs at the TeV energy scale. The subject of this paper however is the ion motion at longer timescales, understanding the long-term ion behavior is important to determine the state of the plasma for succeeding bunches in a high repetition rate future plasma-based collider \cite{long-term-wake}\cite{hot-plasma-wake}. The work presented here shows that a long-lived ion-mode is leftover in the plasma, establishing an upper-limit on the repetition-rate of the plasma-based accelerators.

In this paper, using theoretical analysis and computational modeling, the excitation of a nonlinear ion-wake in the trail of a non-linear ``bubble" plasma electron, is shown. The electron wake may be driven by either an intense laser or particle beam energy source \cite{cavitation-laser-expt}\cite{cavitation-beam-expt}. We show that the time asymmetry of the focusing fields of the bubble leads to the excitation of non-linear ion-acoustic modes in the form of a cylindrical ion-soliton.

The application of the non-linear ion-wake for plasma-based accelerators in ``crunch-in" wakefield regime is explored. The ``crunch-in" regime of plasma wakefields in an ideal hollow-channel was introduced in \cite{positron-IPAC-2015}\cite{crunch-in-regime} and Ch.8 of \cite{Sahai-dissertation}. In this wakefield regime, it was shown that hollow-channel is driven by energy-sources such that the channel-wall electrons collapse to the axis, driving strong wakefields of the order of channel-wall cold-plasma wave-breaking fields. Importantly, it was also shown that the focusing fields excited in this regime have a linear radial dependence of magnitude (with direction favorable for positron transport) and are of the order of accelerating fields \cite{positron-IPAC-2015} (Fig.2 there-in). The excitation of strong focusing fields in this regime is completely opposite to the conventionally established conclusion that relativistic particles have zero focusing fields in hollow-channels \cite{Katsouleas-PRL-1998}. This regime is enabled by the ion-wake channel because it is shown to have an initial radius close to the electron wake transverse size while uniquely its length is as long as the acceleration length. Here we show that the ion-wake channel-wall electrons collapse towards the energy-propagation axis resulting in a non-linear on-axis electron density compression many times the near-void background density. The optimal compression is shown to be only possible if the driving beam properties are matched to the channel radius \cite{positron-IPAC-2015}, a strong dependence on the excitation which is a signature of non-linearity. The choice of appropriate channel radius is enabled by launching the driver at an appropriate time, resulting in excitation at an appropriate channel radius during the expansion of the ion-wake channel.

The ion-wake model shows two distinct phases of the non-linear ion-wake: inertial and thermalized phase. In the inertial or excitation phase the time-asymmetry between the attractive and repulsive radial fields of the bubble on the ions excites them into a soliton-like structure. We show that in this phase the inertial response of the ion rings is dictated by an equilibrium or separatrix radius. The ion rings located within this radius collapse towards the axis whereas rings outside are driven outwards. The outward propagating rings are only driven up to the bubble radius beyond which the force of the bubble radial fields rapidly falls off, resulting in the accumulation of the ion rings at the bubble radius.

At later times, the non-linear radial electron oscillations undergo phase mixing \cite{phase-mixing-longitudinal} leading to coherent electron motion becoming thermalized. The thermalized phase is shown to be a driven non-linear ion-acoustic wave in the form of a cylindrical ion-acoustic soliton. Its characteristics are similar to the solutions of the cylindrical Korteweg - de Vries equation (cKdV) \cite{Maxon-cyl-soliton}\cite{cyl-soliton-observation}\cite{Berezin-Karpman-1964}\cite{KdV-non-linear-ion-waves}. However, the ion-wake shown here differs from a cylindrical-KdV soliton in several aspects: (a) The bubble wake electron oscillations do not thermalize into an isothermal plasma, so the ion-wake soliton is driven (or forced) by the electron temperature gradient; (b) the ion-wake soliton breaks up into N-solitons as it evolves and (c) at early times there is an ion-density spike on the axis which collapses at a later time. The soliton propagates radially outwards leaving behind a flat residue resulting in a near-void ion-wake channel.

Representative PIC simulation results in Fig.\ref{fig1:ion-wake-laser} and Fig.\ref{fig2:ion-wake-beam} illustrate the salient features of the non-linear ion-wake. Figure \ref{fig1:ion-wake-laser} shows the excitation phase at an early time when the bubble wake-train is still executing orderly oscillations and its fields have begun to excite inertial ion motion resulting in a soliton-like ion-wake structure ($\delta n_i/n_0 \simeq 0.2$) as seen in Fig.\ref{fig1:ion-wake-laser}(b),(c). At later times as shown in  Fig.\ref{fig2:ion-wake-beam} the radial oscillations sustaining the bubble train have phase-mixed, converting much of the wave energy into electron thermal energy. The resulting electron thermal pressure drives the ion-soliton ($\delta n_i / n_0 > 1$) outwards. The time evolution of the radial dynamics of the ion rings driven by the time-asymmetric nonlinear electron is shown in a movie in supplementary material \cite{supplementary-movie}. We show below, it is the longitudinal or time asymmetry of the radial electron wakefields that excites the ion soliton which propagates leaving behind a near-void channel shown in the PIC simulations in Fig.\ref{fig1:ion-wake-laser} and Fig.\ref{fig2:ion-wake-beam}.

\begin{figure}[ht!]
	\begin{center}
   	\includegraphics[width=5in]{Fig-2-beam-driven-ion-wake-soliton.jpg}

	\end{center}
\caption{ \small {\bf Electron beam-driven non-linear ion-wake at late time (t = 460 $\omega_{pe}^{-1} = 1.7 f_{pi}^{-1}$) in $m_{i}=m_p=1836 ~ m_e$ plasma}. (a) Beam-driven ion-wake electron density in cylindrical coordinates (fixed-box). The beam parameters are $n_b=5n_0$, $\sigma_r=0.5c/\omega_{pe}$, $\sigma_z = 1.5 c/\omega_{pe}$, $\gamma_b=38,000$, these beam-plasma parameters are quite similar to \cite{cavitation-beam}. (b) Corresponding ion density in cylindrical coordinates (fixed-box). Note the N-soliton formation in the ion-density, $50c/\omega_{pe} \leq z \leq 100 c/\omega_{pe}$. The later times in the time-evolution of the ion-wake is also inferred from density snapshots farther behind the beam. (c) Radial electron and ion density profile at $z=150~c/\omega_{pe}$. A full movie of radial electron and ion density dynamics is presented in supplementary material \cite{supplementary-movie}. }
\label{fig2:ion-wake-beam}
\end{figure}

The paper is organized into the following sections. In section \ref{radial-ion-wave} using the linearized fluid equations for ion dynamics we show the two distinct phases of the ion-wake: the excitation phase and the propagation phase. Using the fact that the thermalizing electron wakefield is non-isothermal with radial electron temperature gradients, we model the non-linear ion-acoustic waves as a driven cylindrical ion soliton. We use an analytical model based on fields of a non-linear plasma wave and simulations to demonstrate the inertial phase of the ion-wake in section \ref{ion-soliton-excitation-phase}. In section \ref{ion-soliton-propagation-phase} the propagation phase of the ion-wake is analytical modeled with simulations verifying the propagation of the cylindrical ion-soliton driven by the radial temperature gradient of the phase-mixed electrons. Finally, in section \ref{ion-soliton-positron-wake} we introduce and analyze the properties of ``crunch-in" wakefield regime in an ion-wake channel, using analytical model and simulations. In appendix \ref{ion-wake-model-considerations} we present considerations and assumptions made to derive the ion-wake model.

\section{Non-linear Ion-wake: \\ as a driven Plasma Ion-wave}
\label{radial-ion-wave}

To develop insight into the ion wake physics, we consider the 1-D simplified dispersion relation of the ion-acoustic plane waves,

\begin{align}
\omega^2 = \frac{c_s^2k^2}{ 1 ~ + ~ (c_s/\omega_{pi})^2 ~ k^2 }
\label{ion-wave-dispersion-relation}
\end{align}

\noindent where, $\omega_{pi}=\omega_{pe}\sqrt{m_e/m_{i}}$ and $c_s = \sqrt{\Upsilon k_BT_{wk}/m_{i}}$ under the collision-less condition, $T^i_{wk}\ll T^e_{wk}$ and $\Upsilon = 1 + 2/f$ is the adiabatic index with $f$ being the degrees of freedom of the ions.

At early times the ion motion is dominated by inertia, thus ions move over the plasma-ion timescales when driven by time varying and asymmetric fields of non-linear electron plasma-wave. As the ion inertia leads to very small spatial displacement scales $k \rightarrow \infty$, the term $k (c_s/\omega_{pi}) \gg 1$ (where $c_s/\omega_{pi} = \lambda_{De} =\sqrt{\frac{k_BT_e}{4\pi e^2 n_0}}$ is the Debye wavelength). Thus, $\omega \simeq \omega_{pi}$ and the ion-soliton density spikes grow over the plasma-ion frequency timescales, $2\pi \omega_{pi}^{-1}$. The radial electron oscillations sustaining the bubble undergo phase-mixing, the electron trajectories lose orderly motion and thermalize. As the electrons thermalize, the ion motion is driven by thermal pressure of electrons.

When the ions gain significant momentum and start oscillating over larger spatial scales in response to the electron dynamics then $k\lambda_{De} \ll 1$. In this thermally driven phase, the acoustic wave propagation becomes dispersion-less with $\omega = k c_s$.

An ion-acoustic wave growing in amplitude undergoes self-steepening, forming a density spike over much smaller spatial scales; $k$ becomes large while dispersion becomes important. The ion-wake modeled here is non-linear, thus the large $k$ dispersion relation retaining the higher-order terms in $k$ in the Taylor series expansion of eq.\ref{ion-wave-dispersion-relation}, is, $\omega = c_s k ~ - ~ \frac{c_s}{2} ~ \lambda_{De}^2 ~ k^3$.

Also, at much later times, the ions undergo heating; $T_i$ increases and modifies the sound speed to $c_s = \sqrt{ k_B (\Upsilon_e T_e + \Upsilon_i T_i) / m_{i}}$.

\subsection{Time-scale separation of Ion-dynamics: \\ a simplified driven linearized ion-fluid model}

\begin{figure}
	\begin{center}
   	\includegraphics[width=5in]{Fig-3-bubble-train.jpg}

	\end{center}
\caption{ \small { \bf Bubble-wake train behind an ultra-relativistic electron beam with bubble: $\beta_g \ll \beta_{\phi} \simeq \beta_{beam}$.}
(a) electron density in 2D cylindrical real-space, (b) corresponding longitudinal electric field profile and (c) corresponding radial-field profile.
Here the beam is located between 170 and 180$\frac{c}{\omega_{pe}}$.
The bubbles just behind the driver in Fig. 3a undergo phase-mixing over several cycles. The intermediate stages of the extent of phase-mixing can be inferred from the bubbles that are closer to the beam.
The beam-plasma parameters are the same as in Fig.\ref{fig2:ion-wake-beam} but the electron-wake is shown at an earlier time $t = 150\omega_{pe}$. }
\label{fig3:bubble-train}
\end{figure}

In this section, we derive the wave equation for the ion-wake in the linear fluid approximation driven by two terms: the electron wakefields and the electron thermal pressure. The linear ion-acoustic wave can be obtained by perturbative expansion of ion density, $n_i$ and ion fluid velocity $v_i$ in the zeroth-order ion fluid continuity equation, $n_0\mathbf{\nabla}\cdot\mathbf{v}^{(1)}_i + \frac{\partial n^{(1)}_i}{\partial t} = 0$. Taking a partial derivative with time, $\mathbf{\nabla}\cdot\frac{\partial \mathbf{v}^{(1)}_i}{\partial t}  + \frac{\partial^2 }{\partial t^2} \frac{n^{(1)}_i}{n_0} = 0$. The ion-fluid equation of motion where the electron temperature ($T_e$) has a spatial gradient and electron wakefields ($\bf{E}_{wk}$) still persist is, $m_i\frac{\partial \mathbf{v}^{(1)}_i}{\partial t} = eZ_i\mathbf{E}_{wk} - \Upsilon k_BT_e\mathbf{\nabla}\frac{n^{(1)}_i}{n_0} - \Upsilon k_B\frac{n^{(1)}_i}{n_0}\mathbf{\nabla}T_e$. The assumption of spatial gradient of electron temperature has been used because electron plasma wave oscillations phase-mix into non-isothermal plasma (this is substantiated through numerical results in the simulations section in Fig.\ref{fig5:elec-temperature-profile}). Upon substituting the equation of motion in the time-derivative of the linearized continuity equation, $\mathbf{\nabla}\cdot\left(\frac{eZ_i}{m_i}\mathbf{E}_{wk} - \frac{\Upsilon k_BT_e}{m_i}\mathbf{\nabla}\frac{n^{(1)}_i}{n_0} - \frac{ \Upsilon k_B}{m_i} \frac{n^{(1)}_i}{n_0}\mathbf{\nabla}T_e \right)  + \frac{\partial^2 }{\partial t^2} \frac{n^{(1)}_i}{n_0} = 0$. Thus, a driven ion-acoustic wave linearized to the first-order in density perturbation has the form,

\begin{align}
\nonumber \left(\frac{\partial^2 }{\partial t^2} - c_s^2\nabla^2\right) & \frac{n^{(1)}_i({\mathbf r},t)}{n_0} \\
& = - \frac{eZ_i}{m_i}\mathbf{\nabla}\cdot\mathbf{E}_{wk}({\mathbf r},t) \biggr\rvert_{\mathrm{wake}} + \frac{ \Upsilon k_B}{m_i}\frac{n^{(1)}_i}{n_0}\nabla^2T_e \big\rvert_{\mathrm{thermal}}
\label{ion-wave-equation-first-order}
\end{align}

In this first-order approximate ion-fluid model the right-hand side of eq.\ref{ion-wave-equation-first-order} shows two separate timescales of the ion-wake.

At earlier times, the first term on the right-hand side dominates. This is the {\it formation or inertial} phase of the ion-wake where the bubble electron oscillations undergo ordered radial motion and the bubble radial electric field excites the inertial response of the ions. The group velocity of the electron bubble wake ($\beta_g\approx 3 v_{th}^2/c^2$, in the 1-D limit, where, $v_{th} \simeq \sqrt{k_BT_e/m_e}$ is the mean electron thermal velocity \cite{Vlasov}) is much smaller than the phase velocity so the bubble fields interact with the background plasma over several oscillations. Fig.\ref{fig3:bubble-train} shows the non-linear electron-wake train (electron density in real space in \ref{fig3:bubble-train}(a)) and its time-asymmetric fields (longitudinal  \ref{fig3:bubble-train}(b) and radial \ref{fig3:bubble-train}(c)) driven by a near speed-of-light energy-source of high-intensity. The fields lead to the formation of the on-axis and $R_B$ ion density spikes. At later times after the phase-mixing between radial oscillators the electrons thermalize and ${\bf E}_{wk}({\bf r},t)\sim 0$. This is the {\it propagation or thermal} phase where the electron thermal pressure gradient drives the cylindrical soliton around $R_B$ radially outwards to many times $R_B$.

Eq.\ref{ion-wave-equation-first-order} is not directly solved as it can be separated over the two different timescales. In the inertial or excitation phase when the plasma is cold ($T_e\simeq 0$, $c_s\simeq 0$), a better description is provided by an ion-ring model driven by the fields of the electron wave. The ion-ring model is developed and verified using PIC simulations in section \ref{ion-soliton-excitation-phase}. The thermally-driven propagation phase is modeled as a driven ion-acoustic soliton and verified by PIC simulations in section \ref{ion-soliton-propagation-phase}.

Before proceeding to the solutions in the two time-scales, we illustrate the time-scale separation using PIC simulation snapshot over tens of electron oscillations behind the driver in Fig. \ref{fig3:bubble-train}. In the Fig. \ref{fig3:bubble-train} (a) the coherent motion in the first four or five oscillations behind the driver is evident; whereas further behind the driver, the electrons begin to de-cohere due to phase-mixing. The falling-off of the electron wakefields to nearly zero as the electrons thermalize is later shown using PIC simulations over a much longer time-scale in sec. \ref{ion-soliton-propagation-phase} (see Fig.\ref{fig6:ion-motion-lineouts}(b) where the radial electric field goes to zero around 200$\omega_{pe}^{-1}$ and the ion-soliton is seen moving outwards radially in \ref{fig6:ion-motion-lineouts}(a)).

\section{Excitation phase: \\ Ion Inertial response to the bubble fields}
\label{ion-soliton-excitation-phase}

Since the characteristic time of ion-motion is much longer than the electron oscillations, the longitudinal field ${\mathbf E}_{wk}\cdot\hat{z}$ averages out over the full bubble electron oscillation. So, the ions gain relatively small net longitudinal momentum. However, atypical radial ion-dynamics arise because the radial fields, ${\mathbf E}_{wk}\cdot\hat{r}$ are asymmetric in time as shown in Fig.\ref{fig4:bubble-ion-dynamics} and do not average to zero, driving an average radial ion-momentum.

\subsection{Ion-ring analytical model: \\ interaction with time-asymmetric bubble radial fields}
\label{excitation-phase-model}

\begin{figure}[ht!]
	\begin{center}
   	\includegraphics[width=5in]{Fig-4-ion-wake-dynamics-bubble.jpg}

	\end{center}
\caption{ \small {\bf Ion dynamics in longitudinally asymmetric phases of the radial forces in an electron bubble}. (a) electron density of a bubble in 2D cylindrical real-space. (b) longitudinal on-axis profile of the electron density (black), longitudinal field (blue), focusing field (red). (c) radial-field profile close to the back of the bubble. This is the focussing ``crunch-in" phase for the ions. (d) the fields at the center of the ion-cavity of the bubble. This is the defocussing ``push-out" phase for the ions. }
\label{fig4:bubble-ion-dynamics}
\end{figure}

The first stage of the ion-wake formation is controlled by the different time-asymmetric phases of ion dynamics inertially responding to the bubble radial field impulses shown in Fig.\ref{fig4:bubble-ion-dynamics} namely, ``suck-in" due to the electron compression in the back of the bubble $F^{back}$ during $\tau_{back}$ shown in Fig.\ref{fig4:bubble-ion-dynamics}(c), and the ``push-out" due to the mutual-ion space-charge Coulomb repulsion force $F^{sc}$ during $\tau_{cav}$ shown in Fig.\ref{fig4:bubble-ion-dynamics}(d). The crunch-in force is spatially-periodic at non-linear plasma wavelength, $\lambda_{Np} \approx 2R_B$ with a duty-cycle $\mathcal{D}=\frac{\tau_{back}}{\tau_{back}+\tau_{cav}}\ll 1$. In addition to the plasma wake, the propagating energy sources themselves impart impulses such as the laser ponderomotive force $F^{pm}\tau_{las}$ ($\tau_{las}$ is laser pulse duration) where $F^{pm}_{e}(r,z) = -\frac{m_ec^2}{2\gamma_e}{\bf \nabla}_{r,z}|{\bf a}(r)|^2$ ($\gamma_e$ is the plasma electron Lorentz factor) and the radial force of the drive beam $F_{b}\tau_{b}$ where $F_{b}(r)=- 2\pi e^2 n_{b} r$. The short driver impulses are neglected (below threshold intensity for direct non-linear ion excitation \cite{ion-motion-intense-beam}\cite{ion-motion-beam-emittance})) because they act on the ions over their sub-wavelength short duration. This is unlike the slowly-propagating wake-plasmon bubbles that undergo continual interaction over many plasma periods. The validity of this assumption is evident from the laser ion-wake in Fig.\ref{fig1:ion-wake-laser}. Since the ponderomotive force of a laser driver is an outward force for both the electrons and ions, the on-axis density-spike cannot be from this direct force from the laser. Similarly the ion-density-spike at the radial wake-edge in an electron beam driven ion-motion cannot be excited directly by the space-charge force of the beam, and is caused by the electron wake's radial-edge density compression.

The Lagrangian fluid model of the ions in a bubble consists of ion-rings under cylindrical symmetry with $m_{i} d^2r_{i}/dt^2 = \Sigma F_{wk}$ (where $F_{wk}$ is the force of the electron wake on the ions). The bared-ion region inside the bubble is assumed to be a positively charged cylinder under steady-state approximation ($R_{B} > r_{Be}$, back of the bubble electron compression radius). The force on the ions from the non-linear electron compression $\delta n_e = n_{Be}\gg n_0$ in the back of the bubble and radius $r_{Be}$, pulls the ion rings towards the axis; and within the bubble, the mutual space-charge force of the ion-rings leads to the ion-rings being driven outwards, away from the axis. The \enquote{suck-in} force on the ions is $F^{back} = - Z_{i} 2\pi e^2 n_{Be} \frac{r^2_{Be}}{r_{i}}$. The space-charge force on the ions in the cavity is $F^{sc} = Z_{i} 2 \pi e^2 n_0 r_{i}$. The equation of motion is $m_{i}d^2r_{i}/dt^2 - \frac{c\beta_{\phi}}{\lambda_{Np}}(F^{sc}\tau_{cav} - F^{back}\tau_{back}) = 0$ using, $\omega_{pi}^2 = Z_{i} ~ 4\pi e^2 n_0 / m_{i}$, we have,

\begin{align}
\frac{d^2r_{i}}{dt^2} + \beta_{\phi} \frac{\omega^2_{pi}}{2}\left(\frac{n_{Be}}{n_0} \frac{\tau_{back}}{\tau_{cav}} \frac{r^2_{Be}}{r_{i}^2} - 1 \right) r_{i} = 0
\label{ion-ring-motion}
\end{align}

\noindent where we have assumed that $c\tau_{cav}/\lambda_{Np} \simeq 1$. Therefore the ion dynamics is dictated by an equilibrium or a separatrix ion-ring radius, where the inward and the outward impulses balance out, $r_{i}^{eq}=r_{Be}\sqrt{\frac{n_{Be}}{n_0} ~ \mathcal{D} }$. The ion-rings at $r_{i} \le r_{i}^{eq}$ collapse inwards towards the axis resulting in an on-axis density spike. Whereas the ion-rings at $r_{i} \ge r_{i}^{eq}$ move out away from the axis. For $m_{i}/Z_{i}>m_{p}$ the ion-response is slower but similar.

When the radially outward moving ion-rings reach beyond $R_B$, there is excess net negative charge of the wake electrons within the bubble-sheath. As a result the radially propagating ion rings get trapped and start accumulating just inside the bubble-sheath and cannot freely move beyond, forming a density compression at $R_B$. So, the cylindrical ion soliton is formed around $R_B$. This accumulation of the moving ion-rings is shown in Fig.\ref{fig1:ion-wake-laser}, where it is seen that the ion and electron density start forming a peak at $R_B$.

The radial location of the excitation of the ion-soliton in the non-linear electron-wave regime is much greater than a skin depth, $c/\omega_{pe}$; thus the ion-wake starts off with a spatial-scale which is over several $c/\omega_{pe}$. This is due to the balance of opposing radial forces on the plasma electrons from the driver and the ion cavity, resulting in their radial accumulation at $R_B$ \cite{Pukhov-laser-bubble}. In the laser-driven case - the outward ponderomotive force is balanced by the evacuated ion-cavity: $F^{pm}_{las} = -\frac{m_ec^2}{2\gamma_e}{\bf \nabla}_r \lvert{\bf a}_0(r)\rvert^2 \simeq F_{cav} = 2\pi e^2 n_0 R_{B}$ gives $R_{B} \sim (c/\omega_{pe})^2 \frac{1}{\gamma_e} {\bf \nabla}_r \lvert{\bf a}_0(r)\rvert^2$ when simplified using ${\bf \nabla}_r|{\bf a}_0(r)|^2 \simeq a_0^2/R_{B}$ and $\gamma_e \simeq a_0$, $R_{B} \simeq   \sqrt{a_0} ~ \frac{c}{\omega_{pe}}$ (computationally, $\simeq2\sqrt{a_0} ~ c/\omega_{pe}$ \cite{Lu-bubble-regime}). In the electron beam-driven bubble the outward force of the beam on the plasma electrons is balanced by the inward pull of the evacuated ion-cavity: $F_{b}(R_B) = 2\pi e^2 n_b r_b^2 / R_B \simeq F_{cav} = 2 \pi e^2 n_0 R_B$. This gives, $R_B \simeq \sqrt{\Lambda_b/(\pi n_0)}$, where $\Lambda_b = n_b \pi r_b^2$ is the line charge density of the beam, where $r_b$ is the beam-radius computed here as $2.3~\sigma_r$ to account for $95\%$ of beam particles for a radially Gaussian beam profile.

\subsection{Bubble field time-asymmetry driven ion-soliton: \\ simulation results}
\label{excitation-phase-simulations}

The above ion-ring model is verified using $2\frac{1}{2}D$ OSIRIS PIC simulations \cite{osiris-code-2002} of the ion-wake in the bubble regime by simulating various energy-sources - laser-pulses in cartesian coordinates and electron-beams in cylindrical coordinates.  The laser pulse is circularly polarized with radially Gaussian and longitudinally polynomial profile (as in \cite{laser-pulse-profile}) with $a_0=4$ (not shown $a_0=1.0$ to $40.0$), pulse length of $30\frac{1}{\omega_0}$, matched focal spot-size radius of $40\frac{c}{\omega_0}$, and laser frequency $\omega_0=10\omega_{pe}$ (the pulse dimensions are in the FWHM of the field). The electron beam is initialized with $\gamma_b\sim38,000$, $n_b=5n_0$ (not shown $n_b=0.25n_0$ to $50n_0$) and spatial Gaussian-distribution with $\sigma_r=0.5\frac{c}{\omega_{pe}}$ and $\sigma_z=1.5\frac{c}{\omega_{pe}}$ (the beam spatial dimensions are $5\sigma$ in both the dimensions). The smallest spatial scale, $c/\omega_{pe}$ is resolved in the beam case and $c/\omega_0$ in the laser case (laser frequency $\omega_0$), with 20 cells in the longitudinal direction and 50 cells in the transverse direction. Each of the plasma grid cell has 36 particles. The beam is initialized with 64 particles per cell. The plasma is initialized in the Eulerian specification (non-moving window) and pre-ionized with $Z_i=1$. At the longitudinal boundaries we initialize vacuum space of $50 ~ c/\omega_{pe}$ followed by density ramps of 20 $c/\omega_{pe}$ sandwiching the homogeneous plasma. Absorbing boundary conditions are used for fields and particles.

The electron-beam driven ion-wake soliton structure in theory is compared to the simulations in Fig.\ref{fig4:bubble-ion-dynamics}(a) and \ref{fig3:bubble-train}(a). The observed $R_B=2.45 ~ c/\omega_{pe}$ (just behind the beam) whereas the estimated bubble radius is $R_B = \sqrt{n_b/n_0 ~ (2.3\sigma_r)^2} = 2.57 ~ c/\omega_{pe}$ ($r_b= 2.3\sigma_r=1.15c/\omega_{pe}$, the assumption $r_b\ll R_B$ is not strictly satisfied). In Fig.\ref{fig2:ion-wake-beam} which is in the propagation-phase, the observed ion-wake soliton is located at $r \simeq 3.3 ~ c/\omega_{pe}$ at 460 $\omega_{pe}^{-1}$ which is about $1.7 \frac{2\pi}{\omega_{pi}}$. The ion-soliton is excited at an early time around $R_B$ and in the snapshot in Fig.\ref{fig2:ion-wake-beam} it has propagated outwards. The on-axis density spike drops to a minimum at $r_{i}^{eq} \approx 0.45c/\omega_{pe}$ in Fig.\ref{fig2:ion-wake-beam} whereas the estimated $r_{i}^{eq} = 0.5 ~ c/\omega_{pe}$ ($n_{Be}/n_0 \simeq 12$, $\mathcal{D} \simeq 0.1$, $r_{Be}\simeq0.5 ~ c/\omega_{pe}$). The radial ion momentum $p_r-r$ phase-space in Fig.\ref{fig7:radial-phase-spaces}(b) shows the ions accumulate at the axis and the channel edge, at a time corresponding to Fig.\ref{fig2:ion-wake-beam}(b). The ions at the channel edge are seen to have a drift velocity and a thermal spread. The radial electron momentum $p_r-r$ phase-space in Fig.\ref{fig7:radial-phase-spaces}(a) shows that a large density of thermalized electrons are trapped within the ion soliton which is confirmed from the density plots in Fig.\ref{fig2:ion-wake-beam}(a).

In the laser-driven bubble simulations the expected and observed $R_B \simeq 4 ~ c/\omega_{pe}$ as shown in Fig.\ref{fig1:ion-wake-laser}(a). In Fig.\ref{fig1:ion-wake-laser}(c) the ion-wake soliton is created at  $r=4.2 ~ c/\omega_{pe}$. The expected and observed on-axis density-spike radius is $r_{i}^{eq} = 0.45 ~ c/\omega_{pe}$ ($n_{Be}/n_0 \simeq 8$, $\mathcal{D} \simeq 0.1$, $r_{Be}\simeq0.5 ~ c/\omega_{pe}$). The model for the excitation of this structure of the non-linear wake has been verified for a range of laser and beam parameters from quasi-linear to strongly non-linear electron wake regime.

\section{Propagation phase: \\ Soliton driven by electron thermal pressure gradient}
\label{ion-soliton-propagation-phase}

As described in section \ref{ion-soliton-excitation-phase} the electron bubble-wake train fields excite a cylindrical ion soliton. Eventually, the electron oscillations phase-mix and thermalize as electron thermal energy on the time-scale of about an ion plasma period. In this section we model the propagation of the cylindrical soliton radially outwards driven by the temperature gradient as shown in eq.\ref{Cylindrical-KdV-equation}. This soliton propagation is modeled using a modified cKdV equation in a non-equilibrium condition such that an electron temperature gradient sustains and drives the cylindrical ion soliton.

\subsection{Thermally-driven ion-acoustic soliton: \\ Analytical model}
\label{driven-soliton-model}

In the linear regime the homogenous ion-acoustic wave equation predicts sinusoidal radial ion oscillations that support the wave. However, the linearized ion-acoustic wave equation is inadequate to describe the propagating solitary density spike at the ion-wake edge, with ion density accumulation many times the background density.

When the density in the ion perturbation begins to rise to the order of the background density, the electrostatic potential due to charge-separation between the ions and the thermal electrons correspondingly rises. This leads to {\it wave-steepening} due to the preferential acceleration of ions in the direction of the ion-acoustic wave velocity. When the potential of the wave is large enough the ions get trapped and co-propagate at the ion-acoustic wave phase velocity, this non-linearity is the basis of the soliton. It should be noted that the linearized kinetic theory does not formally incorporate the trapping of particles at the wave phase-velocity. In this limit the density perturbation shape is therefore not sinusoidal as the co-propagating background ions accumulate and their density perturbation takes the form of an ion-soliton. The co-propagating ion velocity in the soliton can therefore exceed the ion-acoustic phase velocity, $v_i > c_s$ and $\mathcal{M}-1 > 0$ where $\mathcal{M} = v_i/c_s$ is the Mach number. Therefore, non-linear acoustic waves are in the form of a soliton and propagate faster than the ion-acoustic velocity.

To second-order, the non-linear ion-density spike $n_{i}(r,t) > n_0$ propagation is governed by the Korteweg-de Vries (KdV) equation \cite{KdV-non-linear-ion-waves} which has propagating solutions of the form $\mathcal{U}(r-\mathcal{M}c_st)$ \cite{Berezin-Karpman-1964} where $\mathcal{U}$ is the ion-acoustic waveform, a soliton solution and $\mathcal{M}$ is the Mach number ($= v_i/c_s$) of the propagating solution. Higher-order contributions to the KdV equation have also been considered by earlier works. However, the more important and relevant here is that the standard form of the cKdV equation assumes an isothermal plasma whereas the bubble-wake phase-mixes into a plasma with a radial electron temperature gradient, whereas the ions are initially cold. In a non-isothermal plasma the effect of trapped electrons in the ion-soliton have been considered using the Bernstein-Greene-Kruskal (BGK) model at the ion-acoustic velocity \cite{Schamel-trapped-particle-modes}.

It is also known that a single ion-soliton under the appropriate conditions can break-up into multiple solitons leading to a N-soliton solution. This is also a phenomenon we observe in the simulations shown in the ion density of the beam-driven case at $z = 60\frac{c}{ \omega_{pe} }$ in Fig.\ref{fig2:ion-wake-beam}.

We consider a description of the non-linear cylindrical ion-acoustic waves with a radial temperature gradient. We assume that the background electron trapping does not significantly modify the distribution function. We assume that the temperature changes slowly in vicinity of the ion soliton. This assumption is validated by the PIC simulations in Fig.\ref{fig5:elec-temperature-profile}.

\begin{figure}[ht!]
	\begin{center}
   	\includegraphics[width=5in]{Fig-5-electron-temperature-profile.jpg}

	\end{center}
\caption{ \small {\bf Radial profile of the root-mean-square radial electron momentum (proportional to the square-root of the electron temperature, $\sqrt{T_e}$) at 460 $\omega_{pe}^{-1}$ for the beam-driven ion-wake in Fig.\ref{fig2:ion-wake-beam}}. The blue curve shows the root-mean-squared radial electron momentum, $p^e_{th}(r) = \sqrt{ \left[ \Sigma_k ~ p^2_r(k,r) ~ 2\pi r ~ n_e(k,r) \right] / \Sigma_{k} ~ 2\pi r ~ n_e(k,r) }$, profile of the wake electrons corresponding to the time in Fig.\ref{fig2:ion-wake-beam} at 460 $\omega_{pe}^{-1}$. This represents the square-root of the electron temperature, $p^e_{th} ~ \propto ~ \sqrt{T_e}$. The radial gradient of the temperature, $\frac{ \partial }{ \partial r } T_e$ is thus computed at the peak of the soliton (red) and in its vicinity (green). It is interesting to note that $\frac{\partial }{\partial r}T_e^{(1)}\biggr\rvert_{peak} = 0$. }
\label{fig5:elec-temperature-profile}
\end{figure}

To obtain the KdV equation \cite{Sahai-dissertation} in cylindrical coordinates with radial temperature gradient we normalize with respect to the local electron temperature, the radius: $\hat{r} = \frac{r}{\lambda_D}$, time: $\hat{t} = \omega_{pi}t = t\sqrt{\frac{4\pi e^2 n_0}{m_i}}$, electric field: $\hat{E} = \frac{e \lambda_{De} }{k_B T_e} E$, potential: $\phi = \frac{e}{k_B T_e} \Phi$, ion-density perturbation: $\hat{n}_i = n_i/n_0$, electron-density perturbation: $\hat{n}_e = n_e/n_0$, ion-fluid velocity: $\hat{v}=\frac{v_i}{c_s}$. Under this normalization the cylindrical coordinate equations transform as: electron Boltzmann distribution equation $\frac{\partial \hat{n}_e}{\partial \hat{r} } = -\hat{n_e} \hat{E} - \hat{n}_e \phi \frac{ \partial }{ \partial \hat{r} } ~ \mathrm{ln} T_e$, ion-fluid continuity equation $\frac{\partial }{\partial \hat{t} } \hat{n}_i + 2 \frac{ \hat{n}_i \hat{v}}{\hat{r}} + 2\frac{\partial}{\partial \hat{r}} \hat{n}_i \hat{v} = 0$, ion-fluid equation of motion $\frac{\partial }{\partial \hat{t} } \hat{v} + \hat{v} \frac{\partial}{\partial \hat{r}} \hat{v} = \hat{E}$ and the Poisson equation $\nabla^2\Phi = \frac{1}{\hat{r}} \frac{\partial}{\partial \hat{r}} (\hat{r}\hat{E})  = \hat{n}_i - \hat{n}_e$. The electric field $\hat{E}$ is both due to the thermal pressure and the radial fields of the wake, $\hat{E}_{wk} + \hat{E}_{th}$. But, in the following analysis the propagation of a non-linear ion-acoustic wave is considered, so we assume that the electron oscillations are thermalized and thus the effect of the fields of the wake is negligible, $\hat{E}_{wk} \rightarrow 0$.

We look for a propagating disturbance of $\hat{n}_e$, $\hat{n}_i$, $\hat{v}$ and $\hat{E}$ in a stationary background plasma with uniform background density $n_0$. We consider weakly non-linear ion-acoustic wave and expand all the wave quantities in the powers of $\delta = \mathcal{M} - 1$. We perturbatively expand $\hat{n}_i$, $\hat{n}_e$, $\hat{E}$, $\phi$, $T_e$ and $\hat{v}_i$ and retain all terms up to the order of $\delta^2$. Note that we have assumed that before the electron wake excitation the plasma is cold, $T_e^{(0)} \simeq 0$.

We transform to a moving frame of the steepened ion density perturbation using the coordinate transform $\xi=\delta^{1/2}(\hat{r} - \hat{t})$ and $\tau = \delta^{3/2}\hat{t}$. Using this, $\hat{r} = \delta^{-1/2}(\xi + \delta^{-1}\tau)$ and $\frac{\partial \xi}{\partial \tau} = \frac{\partial \xi}{\partial \hat{t} } \frac{\partial \hat{t} }{\partial \tau} = -\frac{1}{\delta}$. We renormalize the electric field as, $\tilde{E} = \delta^{-1/2}\hat{E}$. Note that in the moving frame the potential gradient is, $E = -\frac{\partial }{\partial \hat{r}} \Phi = -\delta^{1/2} \frac{\partial }{\partial \xi} \Phi $, so $\tilde{E}$ is a more appropriate quantity.

Under the assumption that in the moving-frame the quantities of the disturbance change with small $\delta = \mathcal{M} - 1$, the terms in equations are perturbatively expanded and the terms with same powers of $\delta$ are collected. From the $\delta^{1}$ order terms of all the equations above, we infer $\Phi^{(1)} = n^{(1)}_{e} = v^{(1)} = n^{(1)}_{i}\equiv \mathcal{U}$ and $\frac{\partial }{\partial \xi}\mathcal{U} = -\tilde{E}^{(1)}$.

By collecting the $\delta^{2}$ terms from the Boltzmann's equation we obtain $\tilde{E}^{(2)} = -\frac{\partial }{\partial \xi} n^{(2)}_{e} + \mathcal{U}\frac{\partial }{\partial \xi}\mathcal{U} - \mathcal{U}\frac{\partial }{\partial \xi}T_e^{(1)}$. Similarly, collecting the $\delta^{2}$ terms from the ion-fluid equation of motion we obtain $\frac{\partial }{\partial \xi} \hat{v}^{(2)} - \frac{\partial }{\partial \xi} n^{(2)}_{e} = \frac{\partial }{\partial \tau} \mathcal{U} + \mathcal{U} \frac{\partial }{\partial \xi} T_e^{(1)}$ and from the Poisson equation we obtain $\frac{\partial^3 }{\partial \xi^3} \mathcal{U} = - \frac{\partial }{\partial \xi}(n^{(2)}_{i} - n^{(2)}_{e})$. Taking the $\delta$-order terms of the continuity equation and substituting $\mathcal{U}$ we obtain, $\mathcal{U} + \tau \left( \frac{\partial }{\partial \tau} \mathcal{U} + 2\mathcal{U}\frac{\partial }{\partial \xi}\mathcal{U} + \left[ \frac{\partial }{\partial \xi} v^{(2)} - \frac{\partial }{\partial \xi} n^{(2)}_{e} \right] \right) - \delta \left(U^2 + v^{(2)} \right) = 0$.  Neglecting quantities with $\delta$ times the second-order terms and using the $\frac{\partial }{\partial \tau} \mathcal{U}$ result above, $\frac{\mathcal{U}}{\tau} + 2\frac{\partial }{\partial \tau} \mathcal{U} + 2\mathcal{U}\frac{\partial }{\partial \xi}\mathcal{U} + \left[\frac{\partial }{\partial \xi} n^{(2)}_{e} - \frac{\partial }{\partial \xi} n^{(2)}_{i}\right] = - \mathcal{U}\frac{\partial }{\partial \xi}T_e^{(1)}$.

Using the $\delta^2$ terms of the Poisson equation in the above result and using the self-similarity property of the ion-soliton, we obtain the driven Korteweg-de Vries equation in cylindrical coordinates \cite{Maxon-cyl-soliton} (a more detailed derivation of this modified cKdV model can be found in \cite{Sahai-dissertation}),

\begin{align}
\nonumber & \Phi^{(1)} = n^{(1)}_{e} = v^{(1)} = n^{(1)}_{i}\equiv \mathcal{U} \\
& \frac{\mathcal{U}}{\tau} + 2\frac{\partial }{\partial \tau} \mathcal{U} + 2\mathcal{U}\frac{\partial }{\partial \xi}\mathcal{U} + \frac{\partial^3 }{\partial \xi^3} \mathcal{U} = - \mathcal{U}\frac{\partial }{\partial \xi}T_e^{(1)}
\label{Cylindrical-KdV-equation}
\end{align}

It differs from the cartesian-KdV equation by the term $\frac{\mathcal{U}}{\tau}$ and the temperature-gradient based driver term $- \mathcal{U}\frac{\partial }{\partial \xi}T_e^{(1)}$. The cartesian KdV equation can be analytically solved to obtain two classes of solutions: (a) self-similar solutions which are shown in \cite{Berezin-Karpman-1964} to be Airy functions and (b) soliton solutions. A \enquote{soliton} is a single isolated pulse which retains its shape as it propagates at some velocity, $v_{soliton}$. This means that for a soliton-like solution the $\mathcal{U}$ only depends upon the soliton-frame variable, $\zeta = \xi - \mathcal{M}c_s \tau$ and not on space-like $\xi$ and time-like $\tau$ variables separately. The solution of the cartesian KdV equation in this co-moving frame is $\mathcal{U}(\zeta)=3 v_s ~ \mathrm{sech}^2(\sqrt{ \frac{v_s}{2} } \zeta)$ \cite{Berezin-Karpman-1964}.

The cKdV equation and the driven cKdV equation obtained here cannot be solved analytically. However, the numerical analysis and experimental verification \cite{cyl-soliton-observation} of the cylindrical-KdV (cKdV) equation show that it supports functions of the form $\mathcal{U} \propto \mathrm{sech}^2(r-\mathcal{M}c_st)$ in the form of a cylindrical ion-soliton. But, the amplitude of the cylindrical soliton changes as it propagates. Here, we show that the wake electron temperature drives the ion soliton for much longer distances than possible in an isothermal plasma. The velocity of the soliton in the cylindrical case is higher than in cartesian case \cite{Maxon-cyl-soliton}. Since the ion-wake is excited in a non isothermal plasma its velocity changes as it is driven. The mean electron temperature reduces as the soliton propagates radially outwards because the electron thermal energy is distributed over a larger volume. The cKdV equation is also known to support an N-soliton solution, and simulations show N-soliton forming during the propagation phase. We computationally seek the dependence of the non-linear ion-density spike on $(r-\mathcal{M}c_st)$-coordinate.

It should be noted that such soliton solutions are supported under certain limiting condition on the Mach number, $\mathcal{M}$. The strict condition on the existence and stability of ion-soliton arises from a threshold limit on the magnitude of soliton potential to continue trapping the background ions.

Here we find that the speed of ion soliton is nearly equal to and only slightly higher that the ion-acoustic speed calculated using the mean temperature. As this is not an isothermal plasma, there is no well-defined ion-acoustic speed. So, the ion-acoustic wave is phase-mixed and its velocity also changes as it propagates.

The local electron temperature of the ion soliton as shown in Fig.\ref{fig5:elec-temperature-profile}, is used to calculate the Mach number, $\mathcal{M}$ and thus a stability criterion can be derived. This problem is represented using the condition on the Sagdeev psuedo-potential, $\mathcal{V}(\phi) = - \left( \mathrm{exp}(\phi) -1 + \mathcal{M}(\mathcal{M}^2 - 2\phi)^{1/2} - \mathcal{M}^2 \right)$ that it has to be a real number. This condition is satisfied when $\mathcal{M}^2 - 2\phi \geq 0$ therefore, $\phi < \mathcal{M}^2/2$ and $\phi_{max} = \mathcal{M}^2/2$. Using this we find the well-known condition,

\begin{align}
\nonumber & 1 < \mathcal{M} < 1.6, \quad v_i < 1.6 ~ c_s \\
& \Phi < \frac{\mathcal{M}_{max}^2}{2} = 1.28
\label{KdV-solution-existence}
\end{align}

As will be shown later, we find from simulations that the Mach number calculated using the mean temperature is well within these bounds, and thus the soliton is stable.

\subsection{Thermally-driven ion-acoustic soliton: \\ simulation results }
\label{driven-soliton-simulations}

\begin{figure}[ht!]
	\begin{center}
   	\includegraphics[width=5in]{Fig-6-soliton-time-evolution.jpg}

	\end{center}
\caption{ \small {\bf Time evolution of the cylindrical ion soliton}. (a) electron (black) and ion (red) spike radial positions (in terms of $c\omega_{pe}^{-1}$) with time and a third-order fit (green) for the position of the ion density-spike of the soliton. (b) radial wakefields of the electron bubble oscillations (in terms of $m_e c \omega_{pe}e^{-1}$) at the electron density spike (magenta) and at the ion density spike (blue). (c) radial velocity of the ion density spike of the soliton calculated from the third-order fit curve (red). An estimate of the sound speed (green) using the mean temperature, between the axis \& the soliton location (green) and in the vicinity of the soliton peak (blue), in the expression $c_s = \sqrt{k_B \langle T_e \rangle / m_i}$. Since the plasma is not isothermal the mean temperature is calculated by averaging the temperature of electrons over the indicated spatial region. (d) gradient of the electron temperature at the soliton ion density peak (blue) and in the vicinity of the peak (red). The vicinity of the ion density peak of the soliton is defined as shown in Fig.\ref{fig5:elec-temperature-profile}. }
\label{fig6:ion-motion-lineouts}
\end{figure}

The channel-edge density spike, with a form similar to the cKdV-solution in the $r-\mathcal{M}c_st$ frame as shown in Fig.\ref{fig1:ion-wake-laser} and Fig.\ref{fig2:ion-wake-beam} is seen to be propagating radially outwards. The propagation phase starts around $t = 200 \omega_{pe}^{-1}$ as the radial electric fields $\mathbf{E}_{wk} \rightarrow 0$ as shown in Fig.\ref{fig6:ion-motion-lineouts}(b). The propagation phase is evident in Fig.\ref{fig6:ion-motion-lineouts}(a) where the red curve is the position of the peak of the ion-soliton in time. The radial position of the peak of the ion-soliton from each of the PIC electrons and ion density snapshot is obtained in the post-processing scripts and this is shown in Fig.\ref{fig6:ion-motion-lineouts}(a). The cylindrical ion-soliton has propagated from  $r_{soliton}(460 \omega_{pe}^{-1}) = 3.3 c/\omega_{pe}$  (also seen in Fig.\ref{fig2:ion-wake-beam}) to $r_{soliton}(1100 \omega_{pe}^{-1}) = 4.1 c/\omega_{pe}$ which corresponds to an average speed of $\langle v_{soliton} \rangle = 0.0013c$.

We compare the time-averaged soliton speed $\langle v_{soliton} \rangle$ to the average speed of sound, $c_s/c = p^{e}_{th} \sqrt{\frac{\Upsilon}{2}\frac{m_e}{m_i}}$ where the average $p^{e}_{th} \simeq 0.06$ from the electron phase-space (not shown). This gives $c_s \simeq 0.001c$ ($\Upsilon = 2$ for 2D) in agreement with the average soliton velocity. Using this time-averaged analysis we see that $\mathcal{M} \simeq 1.3$ and so the stability criteria in eq.\ref{KdV-solution-existence} is satisfied.

However, as the soliton moves out the volume between the axis and the soliton edge increases. Thus the electron thermal energy re-distributes and spreads over a larger volume. This leads to the reduction in the temperature with time. The soliton is not freely propagating but is driven by the radial gradient of the electron temperature as shown in eq.\ref{Cylindrical-KdV-equation}. The soliton speed thus changes in time as shown in the red curve of Fig.\ref{fig6:ion-motion-lineouts}(c). The sound speed also varies with time and it is estimated using the temperature at that instant using $c_s(t) = \sqrt{k_B \langle T_e(t) \rangle / m_i}$. The plasma is not in thermal equilibrium and its temperature varies radially as shown in Fig.\ref{fig5:elec-temperature-profile}. The root-mean-square radial momentum is used to estimate the temperature at an instant of time, and is calculated over radial dimension from the $p_r-r$ phase-space. The mean temperature is calculated by taking the average of the rms radial momentum - over the entire channel: channel-$\langle p^e_{th} \rangle = \left[ \Sigma^{r_{sol}}_{r=0} ~ p_{th}(r) ~ 2\pi r ~ n_e(r) \right] / \Sigma^{r_{sol}}_{r=0} ~ 2\pi r ~ n_e(r) ~ \propto ~ \mathrm{channel} - \sqrt{\langle T_e \rangle }$ or in the vicinity of the soliton: soliton-$\langle p^e_{th} \rangle = \left[ \Sigma^{r_{sol}+\epsilon}_{r_{sol}-\epsilon} ~ p_{th}(r) ~ 2\pi r ~ n_e(r) \right] / \Sigma^{r_{sol}+\epsilon}_{r_{sol}-\epsilon} ~ 2\pi r ~ n_e(r) ~ \propto ~ \mathrm{soliton} - \sqrt{\langle T_e \rangle }$. The instantaneous sound speed, $c_s(t)$ computed with channel-$\langle p^e_{th} \rangle$ is in the green curve in Fig.\ref{fig6:ion-motion-lineouts}(c) and $c_s(t)$ computed with soliton-$\langle p^e_{th} \rangle$ is in the blue curve in Fig.\ref{fig6:ion-motion-lineouts}(c). The extent of the vicinity ($\epsilon$) around the soliton peak is shown in Fig.\ref{fig5:elec-temperature-profile}.

We compare the curves in (i) red: $v_{soliton}(t)$ (from the 3rd-order polynomial curve-fit of the radial position of the ion-density peak as a function of time), (ii) green: $c_s(t)$ from channel-$\langle p^e_{th}(t) \rangle$ and (iii) blue: $c_s(t)$ from soliton-$\langle p^e_{th}(t) \rangle$ in Fig.\ref{fig6:ion-motion-lineouts}(c). From the comparison it is observed that they are in good agreement. It can be seen that the velocity of the soliton estimated using the location of the ion-density peak (red) lies between $c_s(t)$ calculated using the average temperature over the channel (green) which is the upper limit and $c_s(t)$ calculated using the average temperature over the soliton (blue) which is the lower limit.

We also present the radial gradient of the electron temperature, $\frac{ \partial }{ \partial r } T_e(r,t)$ in Fig.\ref{fig6:ion-motion-lineouts}(d). It is interesting to note from the blue curve in Fig.\ref{fig6:ion-motion-lineouts}(d) that the temperature gradient at the peak of the ion-soliton is zero, $\frac{ \partial }{ \partial r } T_e(r,t) \big\rvert_{peak} = 0$. In the vicinity of the soliton peak we see that the gradient of the temperature follows the variation in the ion soliton velocity, this follows from eq.\ref{Cylindrical-KdV-equation}. The vicinity of the soliton peak is shown as the green curve overlaid on the thermal momentum curve in Fig.\ref{fig5:elec-temperature-profile}.

In Fig.\ref{fig2:ion-wake-beam}(b) N-soliton formation is observed in the ion-density at around $z \simeq 60 \frac{c}{\omega_{pe}}$. The single-ion soliton is seen splitting into several solitons. The N-soliton solution can explained by the seeding of different initial momentum of the ion-rings because ion-rings driven in the ``push-out" phase have a radial position dependent defocussing force acting on them, $F^{sc}(r_{i}) = Z_{i} 2 \pi e^2 n_0 r_{i}$. This is shown in Fig.\ref{fig4:bubble-ion-dynamics}(d). Thus the ion-rings originating at a larger radii from the axis are pushed outwards with a force of a higher magnitude and the rings originating at a smaller radii just outside $r_i^{eq}$ are pushed outwards by a smaller force. So, over a longer time the set of ion-rings with a higher initial momentum propagate radially outwards at a larger radial velocity. This break-up of a single ion-soliton into N-solitons occurs over a longer time-scale because the difference in momentum is small compared to the average momentum.

The thermal momentum, $p_{e}^{th}$ at this time is less than one-tenth of the peak wake quiver momentum. There are several reasons for the cooling, such as, transfer of the wake energy to the ions and the trapped electrons \cite{phase-mixing-trapping}, escape of the highest energy electrons and un-trapped ions from the channel edge, energy loss to the bow-shock and the re-distribution of the energy over an expanding volume. The peak radial ion-momentum is $\simeq0.005$ which shows that not all the radially propagating ions are trapped. The un-trapped free-streaming ions at $\simeq 7c/\omega_{pe}$ can be distinguished from the ions at the channel-edge in $p_r-r$ phase-space.

\begin{figure}[ht!]
	\begin{center}
   	\includegraphics[width=5in]{Fig-7-radial-phase-spaces.jpg}

	\end{center}
\caption{ \small {\bf Radial phase-space snapshots of the electron and ion density in Fig.\ref{fig2:ion-wake-beam}}. (a) electron $p_r-r$ radial momentum phase-space showing the accumulation of thermalized electrons within the ion-soliton. (b) ion $p_r-r$ radial momentum phase-space showing the on-axis and ion-wake edge ion accumulations. }
\label{fig7:radial-phase-spaces}
\end{figure}

It should be noted that the long-term stability of the on-axis ion-density spike of the non-linear ion-acoustic wave is not fully modeled here. The on-axis ion-density spike will disintegrate due to mutual Coulomb repulsion of the ions over the sub skin-depth spike radius. This effect of the collapse of the on-axis density-spike will be addressed in future work. We expect that the disintegration of the central structure to be further by azimuthal asymmetries not included in the cylindrically symmetric simulations. Earlier disintegration is seen in cartesian simulations not shown.

In summary, the ion-wake is a near-void channel with sub-skin-depth density-spikes on-axis and at the bubble-edge located at the bubble-radius, $R_{B}$ \cite{Pukhov-laser-bubble} of several $c/\omega_{pe}$. The ion accumulation in both the density-spikes is many times the background density, and the outre spike propagates outwards as a solitary structure at slightly above the speed of sound.

The time-scale of dissipation of ion-wake and relaxation of the plasma distribution to $v_{th}/c\sim 0$ sets an upper limit on the repetition-rate \cite{hot-plasma-wake} of the future plasma colliders. It is well known that the ion acoustic wave is damped by collisions and ion-wave Landau damping. The ion dynamics opens  questions upon the plasma container walls and the distance needed to avoid damaging them by the significant radially outward ion-flux.

It was earlier suggested that the wakefield energy in the plasma wave could be replenished and sustained \cite{beam-wakefield-SLAC} by a train of energy source in order to achieve high repetition rate. However, as shown in this paper due to ion motion this is not possible.

\section{Positron acceleration: \\ ``Crunch-in" regime in the Ion-wake channel}
\label{ion-soliton-positron-wake}

We explore the use of the ion-wake channel for positron-beam driven positron wakefield acceleration in a novel and relevant \enquote{crunch-in} regime where the channel radius is of a few $c/\omega_{pe}$ as is the case for the ion-wake channel. Such channels are also promising \cite{hollow-channel-chiou}\cite{hollow-acclerator-1998}\cite{hollow-accelerator-2013} for exciting the well-studied purely electromagnetic electron-wakefields. These pure electromagnetic fields driven in a hollow-channel have proven to have zero focusing forces when driven by relativistic particles \cite{hollow-acclerator-1998}. Here we show that in the Crunch-in regime driven even in an ion-wake channel, strong accelerating and focusing fields of electrostatic nature are excited by the electron rings crunching in from the channel wall.

The ion-wake enables the \enquote{Crunch-in} regime because as it slowly propagates radially outwards the channel radii scans over a variety of $c/\omega_{pe}$, while its length is the energy-source plasma interaction length. Meter-scale propagation of electron beams and few centimeter-scale propagation of laser beams in plasmas while exciting nonlinear electron-waves has been well characterized in experiments. The theoretical model presented above thus provides a mechanism to generate long channels of several skin-depth radii. As we show below, the non-linear \enquote{crunch-in} regime requires such channels to optimally match with the driving energy-source.

It is well known \cite{positron-accln-2001} that in a homogeneous plasma positron beam driven wakes have two major problems \cite{positron-IPAC-2015} - (i) The plasma electrons collapsing to the axis from different radii arrive at different times, preventing optimal compression. This is because the radial force of the positron beam driving the ``crunch-in" decreases with the radii. (ii) The plasma ions located in the path of the positron beam result in a de-focussing force on it. The transport of the positron in a positron-beam driven wake is thus not ideal in a homogenous plasma and has to rely on external focusing optics ahead of the plasma. The use of hollow plasma channels with a few $c/\omega_{pe}$ is shown here to provide possible pathways to overcome these fundamental problems.

The formation of much shorter plasma channels excited by significantly different processes have been shown previously. These processes including using a collimated laser with annular profile \cite{milchberg-axicon-channel}\cite{bessel-beam-2011}, using a hollow capillary discharge \cite{hollow-channel-discharge}, among others \cite{ting-smlwfa-channel}\cite{ponderomotive-pdpwfa-wake-channel} \cite{ponderomotive-linear-wake-channel}.

As the ion-wake channel is a practical realization of the hypothesized ideal hollow-channel plasma \cite{positron-accln-2001} of a few skin-depth channel radius, we examine its excitation by a positron-beam and possible use for positron acceleration \cite{positron-IPAC-2015}. In this section we analyze whether the positron-beam driven wake-fields excited in the ion-wake can be used for the acceleration and transport of a positron beam.

\subsection{Non-linearly driven Ion-wake channel: \\ analytical model }
\label{driven-ion-wake-analytical}

Positron acceleration using the ion-wake channel is explored in the non-linear ``crunch-in" regime of perturbed electron oscillation radii, $\delta r_e \geq r_{ch}$ under the condition that the peak beam density $n_{pb} > n_0$.

We use the analytical model of the radial electron ``crunch-in" based excitation of a positron beam wake in the plasma . The equation of motion of the plasma electron rings at $r$ from the axis, under the positron beam crunch-in force but neglecting the space-charge force of the collapsing electron rings is: $\frac{\operatorname d^2}{\operatorname d\xi^2}r \propto - \frac{1}{r} n_{bp}(\xi) r_{bp}^2(\xi)$, where $\xi = c\beta_{pb}t - z$ is the space just behind the positron beam with velocity $c\beta_{pb}$ driving the collapse. This is a non-linear second-order differential equation of the form, $r'' = f(r,r',\xi)$ where $f$ is not linear in $r$. Under the assumption about the positron-beam properties, $n_{bp}(\xi)$ and $r_{bp}(\xi)$ being constant during the entire interaction of the positron-beam with the hollow-channel over its full length. So, upon dropping the dependence on $\xi$ the equation simplifies to its {\it special case} which has analytical solutions, $r'' = f(r,r')$. The solution to this equation is \cite{positron-IPAC-2015}, $r_{ch}\sqrt{\pi} ~ \mathrm{erf} \left( \sqrt{\mathrm{ln}( r_{ch} /r)} \right) = -\sqrt{2\mathcal{C}} ~ \xi$,
where $\mathcal{C}=\frac{1}{2\pi\beta_b^2} \frac{n_{bp}}{n_0} \pi \left(\frac{r_{bp}}{c/\omega_{pe}}\right)^2$. Therefore, the collapse time-duration is $\xi_{coll} =  -r_{ch}\sqrt{\frac{\pi}{2\mathcal{C}}} $. We note that there is an anomaly that exists in our problem formulation and the solution because we have not taken into account the space-charge force of the compressing electrons as they collapse to the axis and this force balances the crunch-in force of the positron beam. Under these approximations the collapse time in a homogeneous plasma is \cite{positron-accln-2001}:

\begin{align}
\tau_{c}=\sqrt{\pi} \frac{ r_{ch}}{\omega_{pe} \sqrt{ n_{bp}/n_0 } r_{pb} }
\label{positron-beam-electron-collapse}
\end{align}

\noindent This expression shows that the collapse-time even in a homogeneous plasma depends strongly on the properties of the beam and the radius from which the rings are collapsing in.

Also, note that we have neglected the initial expansion velocity of the channel, $dr_{ch}/dt$). For optimal compression avoiding phase-mixing, the electron rings should collapse over, $\tau_c \simeq \mathcal{D}\lambda_{Np}/c$ where $\lambda_{Np}$ is the non-linear wavelength of the positron-driven wake and $\mathcal{D}$ is the duty-cycle of compression phase. So, the optimal channel radius is $r_{ch}^{opt} \simeq  2\sqrt{\pi}\mathcal{D}\frac{\lambda_{Np}}{\lambda_{pe}}\frac{\omega_{pb}}{\omega_{pe}} r_{pb}$. The scaling of the $r_{ch}^{opt}$ with positron beam parameters is shown in \cite{positron-IPAC-2015}.

\begin{figure}
	\begin{center}
   	\includegraphics[width=5in]{Fig-8-positron-acc-ion-wake-channel.jpg}
	\end{center}
\caption{ \small {\bf 2D-PIC simulations (a) electron density in real-space showing electron density excitation in the ``crunch-in'' regime (b) corresponding radial profile of wakefields excited by a positron-beam in an ideal-channel versus an ion-wake channel}. The radial profile of the normalized electron density (black) in an ion-wake channel (normalized to the maximum electron compression) at longitudinal location of the peak accelerating wakefield ($r_{pb} = 2.3c/\omega_{pe}$, $\gamma_{pb}=38000$, $n_{pb}=1.3 n_0$). The radial profile of the accelerating-wakefield and normalized focussing-wakefield potential (radial field integrated from the edge of the box to a radius).}
\label{fig8:positron-acc-field-radial-profiles}
\end{figure}

\subsection{Non-linearly driven Ion-wake channel: \\ simulation result }
\label{driven-ion-wake-simulation}

Using 2-$\frac{1}{2}$D PIC simulations in a moving window we study the positron beam driven wakefields in cylindrical geometry. We compare positron acceleration in an ideal (Heaviside density function, $n_0\mathsf{H}(r-r_{ch})$) and an ion-wake channels (on-axis and channel-edge density-spike, channel minimum density of $0.1n_0$) with $r_{ch}= 2.5 ~ c/\omega_{pe}$. For non-linear wake parameters $r_{pb}=2.3 c/\omega_{pe}$, $n_{pb} = 1.3 n_0$ and $r_{ch}^{opt}\simeq 2.3 c/\omega_{pe}$ ($\mathcal{D}\frac{\lambda_{Np}}{\lambda_{pe}}=0.25$). Fig.\ref{fig8:positron-acc-field-radial-profiles} shows that the peak on-axis accelerating field is $0.4~m_ec\omega_{pe}e^{-1}$ for an ideal channel and $0.2~m_ec\omega_{pe}e^{-1}$ for the ion-wake channel. Fig.\ref{fig8:positron-acc-field-radial-profiles} also shows that the focussing potential (normalized to 27.6 $m_ec^2e^{-1}$) is similar and overall focussing in both cases. However, in the ion-channel the radial field is defocussing around the on-axis ion-spike.

Thus the non-linearly driven ion-wake channel is useful for accelerating and transporting positrons despite the lower accelerating and focusing fields in comparison with the ideal channel. We also note that the on-axis density spike has detrimental effect on the focussing fields near the axis. However, the on-axis density spike is unstable over longer time-scales and collapses \cite{Sahai-dissertation}. The cylindrical simulations used in the current work to model the ion-wake ignore any azimuthal asymmetries in the distribution of electrons and ions in the on-axis density spike. Exploring the collapse of the on-axis density spike will be addressed in future-work. Ideal channels of a few $c/\omega_{pe}$ are technologically challenging whereas the ion-wake channel of radius $r_{ch}\gtrsim R_B$ is formed behind every bubble-wake.

\section{Conclusion}
In conclusion, using theory and PIC simulations we have shown the dynamics of the formation and evolution of a non-linear ion-wake excited by the well-characterized time-asymmetric electron bubble-wakefields independent of the type of energy-source. We have shown that the non-linear ion-wake has a characteristic cylindrical ion-soliton solution and evolves to an N-soliton solution over longer time as described by a driven cKdV equation. Thus over the period of persistence of the ion-soliton, a second electron bunch cannot be accelerated in the plasma. This establishes an upper limit on the repetition rate of a plasma collider. We have also shown the feasibility of using the ion-wake channel for positron acceleration in the positron-beam driven ``crunch-in" regime within an experimentally relevant parameter regime.

\begin{acknowledgments}
Work supported by the US Department of Energy under DE-SC0010012 and the National Science Foundation under NSF-PHY-0936278. I acknowledge the hospitality of the Dept. of Physics at the Imperial College London and the John Adams Institute, while making corrections to the manuscript. I acknowledge the OSIRIS code \cite{osiris-code-2002} for PIC simulations presented here. I acknowledge support for experiments by FACET group at Stanford Linear Accelerator Laboratory and Prof. M. Downer's group at University of Texas at Austin. I acknowledge the 256-node {\it Chanakya} server at Duke university.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


The survey of 447 businesses of all sizes was conducted over the summer by systems integrator Champion Solutions Group. It found that 53% of those businesses haven't implemented a formal BYOD policy, while more than one-fourth confessed they have no systematic security approach, much less a formal policy.
The survey findings are "ridiculous … surprising," said Champion CEO Chris Pyle, in an interview. Mobile security best practices have been promulgated by analysts and security firms for more than a decade to protect sensitive corporate data, but there is apparently widespread variation about how companies implement security for BYOD workers.
"The evidence is indisputable that a growing need exists for more stringent application of security policies and procedures in modern businesses," Champion wrote in an 18-page white paper describing the survey's findings.
The value of allowing workers to use their personal smartphones and tablets while at work is now well understood by companies, and has been tied to greater productivity because workers find and use applications and services for their phones that they personalize to become more efficient.
But Pyle said there can be a downside. "You need to have workplace freedom, but you need to have a framework as well," he said.
In addition to the lack of formal BYOD policies, the survey found that only 21% of businesses are using multifactor authentication (MFA) to verify a user's identity when granting access to critical enterprise applications and data. MFA covers a wide category of techniques to require two or more methods of authentication from independent categories of credentials when a person logs in from a device.
MFA techniques deployed by U.S. businesses rarely include biometric authentication, where a fingerprint or iris scan is used to support a user name and password to authenticate a user's access to corporate data, Pyle said. Sometimes, instead, a unique code, or token, is sent to a smartphone for each entry into a company's applications or other data.
More often, companies rely on enterprise rights management software to grant a group of users — such as sales managers — access to a certain set of data from their phones — such as sales in a certain district. But that approach doesn't guarantee that every instance of access is from the authorized user. It could be from someone else who may have stolen a phone or used it temporarily, unbeknownst to the owner.
While newer iPhones and many Android phones, like Samsung Galaxy smartphones, use fingerprint scanners for a user to access the phone itself, companies are only just beginning to consider using the fingerprint scanner to access enterprise apps or data, Milgram said. Some companies are relying on partitioning of personal from work data within the operating system of some newer phones, but even that approach may not be secure enough, depending on the level of risk that a company can tolerate.
Champion has 2,300 business customers, primarily in health care, distribution and finance. It operates a business unit called MessageOps that helps companies deploy Microsoft Cloud Services, including Office 365, Enterprise Mobility and Azure services. Champion also works as a system integrator to help companies deploy other enterprise mobility management products such as VMware's AirWatch and IBM's Maas360.
The low adoption of MFA for mobile security noted in the survey doesn't surprise Pyle. "There's a lot of good talk by companies about what's coming and what's going to be available, but people aren't implementing what's available today, let alone what's coming tomorrow," he said.
The survey also found that 30% of companies don't even require alphanumeric passwords (those using both the alphabet and numbers). "That's a pretty basic precaution," Pyle added.
Champion plans to repeat the survey next year and expects to see a greater focus on security, he said.


text:
LexTalk World show with Swarnendu Chatterjee, Founder, Law Chambers of Swarnendu Chatterjee.
Lextalk World Interviews Swarnendu Chatterjee. Swarnendu Chatterjee is the Founder of Law Chambers of Swarnendu Chatterjee. He is an Advocate-on-Record, Supreme Court of India since March 2020. He has previously worked with Saraf and Partners, L&L Partners and MAP Corporate Legal as Principal Associate in the Dispute Resolution Teams of the aforesaid firms. He has extensive experience spanning over almost a decade, including being chamber junior to Late Shri P.P. Rao from 2013-2017 and erstwhile ASG Hon'ble Justice P S Narasimha (Judge, Supreme Court of India)from October 2017-April 2018.
Swarnendu specializes in Civil Law, Constitutional Law, Insolvency Laws, Taxation Laws, Labour Law, Criminal Law and Arbitration and is a qualified Advocate-On-Record designated by Supreme Court of India in March 2020.
He has worked in many fields of law and his experience of close to a decade extends to virtually all aspects of litigation and arbitration. Swarnendu has extensive experience in both domestic and international jurisdictions. Swarnendu has extensively advised on debt recovery, insolvency, banking and financial issues, service matters, matters of constitutional importance and commercial and corporate disputes and real estate in various Courts all over India.
Host: Tell us about a complex legal issue you worked on. Describe the complexity and tell us how you approached it?
Swarnendu:Law is a dynamic subject and is ever evolving. In our social strata, lawyers and doctors are always contacted whenever there is a deep-rooted problem. There is no case is which cannot be termed as "not complex," it depends on the lawyers as to how they would handle the situation and the case. The degree of complexity increases with the nature of the issues involved and the forum in question; meaning thereby, when we are before the Hon'ble High Courts/Supreme Court, then the degree of complexity and the legal issues involved reach the highest pedestal.
I appear before different forums across the Country including the Hon'ble Supreme Court of India. If we speaking about appearance before District Courts, I have appeared before District Courts of New Delhi, Uttar Pradesh (Nodia), Gurugram, Mumbai (Esplanade Court) and Bhopal. Then, I have appeared before different High Courts apart from Delhi High Court (Calcutta, Allahabad, Madhya Pradesh, Andhra Pradesh, P&H, Orissa), National Company Law Tribunals (Mumbai, Delhi, Allahabad, Hyderabad, Chandigarh) and National Company Law Appellate Tribunal (Delhi). The degree of complexity and the approach of the Courts/Benches differ from place to place and forum to forum.
lead member of the Committee of Creditors, State Bank of India who had briefed one of the best senior advocates in Delhi, Mr. Ramji Srinivasan. It was a very tough task to face a senior of that stature. However, my clients had reposed faith in my abilities and I had faith in God and the training of my late senior, Shri PP Rao (Sr. Adv). I still remember the enormous reading I had to undertake to take the case to its logical conclusion. The victory was obviously very sweet and August 19, 2019 (date of the verdict) shall always be etched in my memory. The second matter, which I had dealt with was the insolvency resolution process case of the Jet Airways. The matter is sub-judice before the Hon'ble NCLAT and the judgment has been reserved by Hon'ble NCLAT. We are awaiting the outcome of the verdict. The matter was hard fought before NCLAT and it had required endless preparations. I was up against the stalwarts in the profession, Mr. Krishnendu Datta (Sr. Adv), Mr. Rohan Rajadhyakshya (Adv) and Mr. Krishnan Venugopal (Sr. Adv). The preparation required a lot of reading of numerous precedents of different forums, long conference hours, etc. However, the learning which I had in this process was beyond comparison. Irrespective of the outcome, I shall cherish this journey forever. The third matter, which I had dealt recently was the matter of one Mr. Kamaluddin (Senior Partner, Paragon Enterprises). The journey started with me being engaged to get the FIR registered against Mr. Kamaluddin in Ambala quashed. We had moved a petition under Section 482 of the Code of Criminal Procedure, before the Punjab and Haryana High Court. The outcome was not such favourable and being aggrieved we moved the Hon'ble Supreme Court. The interesting part was, the Haryana Police was at the doorstep of Mr. Kamaluddin to arrest him and he was detained at the Chitpur GRPS Police Station due to a look-out circular being issued against him. The remand application filed by Haryana Police was listed on the same day as that of the Special Leave Petition before the Hon'ble Supreme Court. The matter before the Hon'ble Supreme Court was heard at length from 4.10pm on 20th July 2022. Being satisfied with the arguments, the Hon'ble Court was pleased to issue notice in the matter and granted the interim relief of protection from arrest and thereafter has issued notice in the matter of issuance of Look-Out Circular. The matter is sub-judice before the Hon'ble Supreme Court. The aforesaid instance was the second one in my career, where I had managed to save a person from incarceration. The first one being Mr. Jaideep Singh (MD, GMV Pvt Limited, Bhopal) wherein, with the team work and support I had from my partner, Mr. Wasim Beg (Partner – Luthra and Luthra, Law Offices, New Delhi) during my tenure there, was commendable, where we had filed a petition before Hon'ble Delhi High Court and had successfully argued to get a relief so that the Hon'ble Sessions Court had in a way had to grant relief to my client. This was again a memorable case.
In my career, whether single-handedly or as a junior to Late Shri PP Rao and being a Principal Associate to two reputed firms in India (Luthra and Luthra) and (Saraf and Partners) there were many such occasions and cases, where with our strategy and background work, we were successful in getting relief for our clients.
When you get relief for your client and they thank for your work, it is the biggest treasure in a professional's life.
Host: The pandemic saw some courts begin moving towards more remote proceedings and availability. Is this sustainable, and a possible way to increase access to justice, in your opinion?
Swarnendu: The pandemic has taught us many things professionally and personally. It was very hard initially as our systems were never equipped with the virtual set up. We always had a belief and stronghold on the physical set up atleast in Courts and in the legal fraternity. The softwares which were there (zoom, Webex etc) came to our rescue. The hearings when it had started initially on the online platforms, were very cumbersome, as we being the stakeholders were not so tech savvy. The notifications of the Courts that regarding urgent hearings made things go crazy for the first few months, as we know it for a fact that for every litigant, their matters are urgent and with the interim orders being extended owing to covid (pandemic) as well as the limitation period being extended, the lawyers from every strata felt the toughness of the pandemic and the economic slowdown.
Even top tier law firms felt that hardness of the situation (economic slowdown owing to the pandemic).
Slowly towards the end of 2020, the softwares were fully functional and in place. The Courts were also equipped with the efiling procedures. However, this was only for the courts in urban or semi-urban areas. The Courts in rural or semi-rural areas suffered a lot due to lack of infrastructure and this is precisely when the High Courts (on their administrative side) and the Supreme Court took cognizance of the same and passed orders for improvement of the infrastructure.
That is when we saw the Governments stepping up to the task and the Courts were given full digital access and situation somehow improved. The mantra was, "Access to Justice cannot be stopped".We overcame the situation and now most of the Courts including the Trial Courts are equipped to conduct hearings and even trials online without any adjournments in some cases. However, still I feel, according to me the hybrid system is fine, so as to reduce the chances of undue adjournments, as the counsels can appear even during their travel across the globe for any work or otherwise.
There is always a scope for improvement and the Courts must ensure that the Virtual Platform must be used to cut down on unnecessary adjournments, especially during trials, so that the witnesses can be examined and trials can be conducted seamlessly without any delay. The goal of the Judiciary and the Government is to cut down / reduction of the pendency of the case. Therefore, unless we as lawyers and being the officers of the Court guide our clients to the right path (avoiding adjournments unless it is absolutely unavoidable) then only the call and the wish of Hon'ble Dr. Justice DY Chandrachud (Judge, Supreme Court of India) of the Courts in India, not being "tareek pe tareek courts" can be fulfilled.
Technology is a boon for the Indian Judicial System which can be used to reduce the unnecessary adjournments and conduct the proceedings without any delay.
The High Courts being in a supervisory jurisdiction over the Trial Courts, must in their administrative capacities ensure that the e-courts systems are in place, the soft copies of the daily orders and judgments are uploaded and access to justice along with speedy justice, which is a facet of Article 21 of the Constitution is available to all the citizens in the Country.
Host: How would you rate the current legal system's drive towards encouraging access to justice? Is there tangible movement in closing the justice gap?
Swarnendu: Continuing from my aforesaid answer, Indian Judiciary has certainly gained a lot of momentum in its drive towards in making justice accessible to all. However, a lot of work is yet to be done. As our Hon'ble CJI has mentioned, Legal Aid must not only be "naam ke vaaste", we lawyers need to step up to provide some pro bono service to the needy clients who can really be our bhagyavidhaata.
My Late Senior used to say, "paying clients are our anyadaata and non-paying clients are our bhagyavidhaata." I follow the same principle. The amount of labour I put in for the cases are exactly the same for each and every matter. It does not depend on the quantum of fees I get. Quoting Martin Luther King, " Injustice anywhere is a threat to justice everywhere," is apt in this situation. We as officers of the Court owe a lot of duty to this nation and being stakeholders of this fraternity we must ensure that justice is served and that too within a time bound manner.
The quality of legal education has improved considerably, yet the students should be inspired to take up the cause of the people and strive towards litigation more as they can serve the society in a better manner. In my opinion, after having worked as a Principal Associate in two leading firms in this country, I dare to say that, Corporate Jobs can feed your pocket and keep you happy economically, but litigation and service to the fraternity can help you get complete job satisfaction, professional satisfaction. I represent a lot of Workers' Union, they may not be rewarding you financially, when you compare it to a corporate client, however

redpajama_set_name: RedPajamaCommonCrawl


Leftist tech companies are trying to divide and conquer.
Wow getting shots from both sides. My point was one is a short description, the other is a longer in depth article meant to give examples and some explanation. So of course there would be differences. Even if the Webster had a second page as long as the wikipedia page it would likely have some pretty big differences than the few words of a dictionary definition.
A dictionary definition alone often inst that useful, and what your point is meant to be just doesn't come across. As for building up fascists, im an anarchist but nice assumption.
I attend the Catholic university of Leuven, 48th in world rankings and 31st in my subject. Why would they provide a dictionary in this day and age, its not a useful reference tool.


text:
College Football Awards, Week 2 (2017) September 11, 2017
Posted by intellectualgridiron in Sports.
Tags: Arkansas, Art Briles, Auburn, Baylor, Big 12, Big XII, Bobby Petrino, Brian Kelly, Bronco Mendenhall, Buckeyes, BYU, Cal, Charlotte, Clemson, Dabo Swinney, Dino Babers, Duke, Eastern Kentucky, Eastern Michigan, Florida, Florida State, Georgia, Georgia Southern, Georgia State, Gus Malzahn, Horned Frogs, Idaho, Jeff Brohm, Kansas State, Kentucky, Kirby Smart, Lincoln Riley, Louisville, LSU, Matt Rhule, Miami Hurricanes, Middle Tennessee, Mississippi State, Morgan State, New Hampshire, New Mexico, Nicholls, North Carolina A&T, Northwestern, Notre Dame, Ohio State, Oklahoma, Oklahoma State, Ole Miss, Oregon, Penn State, Purdue, Randy Edsall, Razorbacks, Rutgers, San Jose State, Sooners, Syracuse, TCU, Tennessee, Texas, Texas A&M, Toledo, Tulsa, UConn, Urban Meyer, USC, Utah, Utah State, West Virginia, Wisconsin, Wyoming
(Note: All rankings are current AP [week 2] unless otherwise noted.)
Wish I were him: Lincoln Riley, Oklahoma
Glad I'm not him: Urban Meyer, Ohio State
Lucky guy: Kirby Smart, Georgia
Poor guy: Brian Kelly, Notre Dame
Desperately seeking a wake-up call: Urban Meyer, Ohio State
Desperately seeking a P.R. man: Jeff Brohm, Purdue
Desperately seeking sunglasses and a fake beard: Dino Babers, Syracuse
Desperately seeking … anything: Matt Rhule, Baylor
Thought you'd kick butt, you did: No. 19 Kansas State (defeated Charlotte 55-7)
Thought you'd kick butt, you didn't: Kentucky (defeated Eastern Kentucky 27-16)
Thought you'd get your butt kicked, you did: San Jose State (lost to Texas 56-0)
Thought you'd get your butt kicked, you didn't: Nicholls (lost to Texas A&M 24-14)
Thought you wouldn't kick butt, you did: Duke (defeated Northwestern 41-17)
Dang, they're good: USC
Dang, they're bad: Baylor
Can't Stand Prosperity: Ohio State
Did the season start? Texas A&M
Can the season end? New Mexico
Can the season never end? Oklahoma
Play this again: No. 15 Georgia 20, No. 24 Notre Dame 19
Play this again, too: Utah 19, BYU 13
Never play this again: Utah State 51, Idaho State 13
Close call: No. 3 Clemson 14, No. 13 Auburn 6
What? Middle Tennessee 30, Syracuse 23
Huh? New Hampshire 22, Georgia Southern 12
Are you kidding me?? Eastern Michigan 16, Rutgers 13
Oh – my – God: No. 5 Oklahoma 31, No. 2 Ohio State 16
(rankings are current AP (post-week 2, pre-week 3)
Ticket to die for: No. 3 Clemson @ No. 14 Louisville
Also: Texas @ No. 4 USC
Best non-Power Five vs. Power Five matchup: Oregon @ Wyoming
Best non-Power Five matchup: Utah State @ Wake Forest
Upset alert: No. 10 Wisconsin @ BYU
Must win: No. 23 Tennessee @ No. 24 Florida
Offensive explosion: Tulsa @ Toledo
Defensive struggle: No. 12 LSU @ Mississippi State
Great game no one is talking about: Purdue @ Missouri
Intriguing coaching matchup: Bobby Petrino of Louisville vs. Dabo Swinney of Clemson
Also: Randy Edsall of UConn vs. Bronco Mendenhall of Virginia
Who's bringing the body bags? Georgia State @ No. 4 Penn State
Why are they playing? Mercer @ No. 15 Auburn
Plenty of good seats remaining: North Carolina A&T @ Charlotte
They Shoot Horses, Don't They? Morgan State @ Rutgers
Week 1 Take-aways:
This week leaves us with more questions than answers. For one, Louisville had fewer penalties against North Carolina than they did against Purdue. All well and good, but is that enough improvement at this rate to be ready for Clemson at home next week? Regarding the TCU-Arkansas game, are the No. 23 Horned Frogs that good, or are the Razorbacks that mediocre? The Auburn-Clemson game was a surprising defensive struggle. What was the bigger surprise: that Auburn's defense held the Tigers to only two touchdowns, or that Auburn's offense – supposedly a specialty under head coach Gus Malzahn – could only muster a measly six points? Moreover, what does this portend for Auburn's offense during the rest of the season?
The shocker of the week was Oklahoma's upset over Ohio State in Columbus. The question becomes, are the Sooners that good, or are the Buckeyes overrated? Ohio State has plenty of NFL-potential bodies on both sides of the ball. What accounts for their lackluster offense this game, and their defensive collapse in the 4th quarter? Actually, there is an answer. The Buckeyes are currently experiencing an identity crisis on offense. Until they get that cleared up, they'll continue to fail to play up to their potential this season, and that will be a genuine shame.
Questions aside, let us take a glance at the Big XII Conference. Simply put, they're looking good right now. The Sooners are rolling after their huge win over the Buckeyes. Oklahoma State has two wins with impressive margins. TCU embarrassed Arkansas on the road today. Kansas State won convincingly, even though it was a body bag game. West Virginia is playing quite strongly right now, though a body bag game against Delaware State next week will obviously be meaningless. It all adds up to a conference that is playing well and giving the rest of college football cause for notice. The ironic weak links are Baylor and Texas. Concerning the Bears, it would only stand to reason that Matt Rhule has not forgotten how to coach. The turmoil surrounding the player sexual assault scandals, the sudden firing of Art Briles, and the havoc wrought by Hurricane Harvey have all combined to take a serious toll on the program. Baylor looks shell-shocked right now, and it will be interesting to see if Rhule, who brought Temple to respectability, can keep things afloat at a program with greater potential but higher expectations, too.
Speaking of Hurricane Harvey, that might also account for Texas A&M has not been playing up to their potential, as well as for Texas' gigantic miscue against Maryland last week. After all, many players for these two programs, as well as for Baylor, have come out of the Houston area, which is still reeling in the wake of the hurricane damage and the residual flooding damage. The latter of which alone has for longer-lasting implications than the former. Let us all pray for those who have been afflicted by that terrible storm, as well as for those who are being afflicted by Hurricane Irma in Florida. As the floodwaters recede and the area rebuilds and moves forward in general, perhaps the morale of the aforementioned Texas teams shall improve, along with their performances.
Speaking of Hurricane Irma, that storm shall leave implications long into the season, given all the games that have already been postponed. One notable example is No. 16 Miami vs. No. 10 Florida State. That game would have been one of the best of the upcoming week. Little doubt lingers that they'll find a time to reschedule such a matchup that is A) a heated, in-state rivalry, and B) a game with conference standing implications. If both teams keep playing to their potential, perhaps both will be ranked even MORE highly by the time they finally butt heads. Let us stay tuned the rescheduling on Oct. 7!
One final note about an overlooked game for the upcoming week: Ole Miss at Cal, which kicks off at 10:30 PM Eastern Daylight Time. While both teams are currently unranked, it does not matter, for it's always a treat to watch SEC vs. Pac-12 matchups!


It's a thrill familiar to anyone who's gone thrift or vintage shopping: your hand skims along a rack packed with garments on hangers, fingers sliding over silk, leather, polyester, wool, lace. And then, without warning or fanfare, there it is: the perfect jean jacket that you've always longed for. In the dressing room, you nervously try it on, and yes! It's like it was expertly stitched just for you, like it's always been yours. Vintage victory!
Right now, Philadelphia is home to dozens of vintage and thrift shops catering to every taste and price point. Of course, there are still the old-school shops, crammed with polyester dresses, wild ties, faded military fatigues and every wale of corduroy. There are also shops that resemble fancy boutiques, with tightly edited collections and a minimalist vibe. Is motorcycle boho witch chic your thing? There's a Philly vintage shop for that. Is a flawless 1930s trench coat on your wish list? A 60s letterman jacket? Nearly every neighborhood flaunts at least one great shop where you can excavate something one-of-a-kind, something that has survived for generations, something to add history to your closet.
Every vintage shop owner has his or her own outlook and approach to vintage, and their own story to tell, but common threads tie them all together: love of fashion, respect for and fascination with the past, and a strong belief that buying previously worn clothes—whatever the motivation—beats buying new any day.
Wilbur is influenced by his studies in art history, the TV shows on the Style Network he watched as a kid, as well as old-school vintage shops in Greenwich Village in New York. "I always wanted to have pretty, colorful things around me," he says. Wilbur's reputation as a first-rate shop is owed as much to the store's charming displays and hidden gems, as toas the owner himself. You're likely to find him helping a customer discover the perfect frock, de-pilling a sweater while gabbing with a friend or tidying up while '70s disco pulses from the shop's speakers. "I just love creating an atmosphere," Wilbur says.
MON–FRI: 1:00–8:00 p.m; SAT–SUN: 12:00 – 6:00 p.m.
"Menswear doesn't change—a suit's a suit," says David Lochner, managing partner at Briar Vintage in Old City. "What changes is where the buttons stand, the width of the lapels and neckties, and the fullness of the cut." If not for the copious tattoos and the iPhone in his hand, Lochner himself looks like he might have time-traveled from the 1930s: he's got on a chambray shirt tucked into high-waisted chinos, wire-framed spectacles and a striped engineer's cap. He's been captivated with old clothing since the age of 13, informed by golden era Hollywood, and especially by the 1993 World War II drama Swing Kids. "I wanted to dress like that, listen to that music and punch Nazis in the face," he says.