MYTH: There is only one type of parallelism.
FACT: There are several different forms of parallel computing: bit-level, instruction-level, data-level, and task-level.
MYTH: Everything can be made parallel.
FACT: There are limits to the amount of parallelism that can be applied to game oriented routines.
MYTH: CPUs have increased IPC by about 5% per gen during the last couple years because Intel rested on their laurels. Now that AMD is competitive again, we will see giant jumps in IPC soon.
FACT: The x86 ISA reached a limit and cannot evolve further.
MYTH: GPUs performance grows faster than CPUs and soon GPUs will replace CPUs.
FACT: GPUs are easier to scale, but cannot replace CPUs.
Parallel computing is a type of computation in which two or more computations are executed at the same time. The concept is well established in computer science and engineering, but there are lots of misunderstandings from the general public in general forums, blogs, social media, and comments sections of news sites.
I see that people often believe that multithreading is the only type of parallel computing.
The advantage of bit-level parallelism is that it is independent of the application, because it is running on the processor level. The programmer writes the operation and the hardware executes the operation in a single step or in several steps.
The ability of executing two or more instructions at the same time. Consider the arithmetic operations.
Since the second operation does not depend on the result of the first operation, both operations can be executed on parallel.
reducing the execution time to one half.
Data parallelism is the execution of multiple data units in the same time by applying the same operation to them. Data parallelism is implemented in SIMD architectures (Single Instruction Multiple Data).
Suppose we want to move a series of objects a fixed distance in the z axis, this is equivalent to adding the distance to the z coordinate of each object.
In a 4-way SIMD architecture, the operation can be applied to four objects at once, reducing the cycle time by a factor of four. First the coordinates are grouped in 4-wide vectors, then vector addition is executed.
A wider architecture can reduce cycle time. An 8-way architecture could do all the operations in a single step.
Task parallelism is a mode of parallelism where the tasks are divided among the processors to be executed simultaneously. Thread-level parallelism is when an application runs multiple threads at once.
Often people in forums accuse game developers of being "lazy" as the reason why games do not scale to 32 cores.
It must be clear that the limitation to the level of parallelism illustrated by this simple example is not a consequence of lazy programming. The problem cannot be parallelized further due to data dependences among different operations.
Some components that define the state of the game can be split from the main algorithm and run on a separate thread as subtasks, two examples are background music and artificial intelligence. However, those subtasks are not fully disconnected from the main algorithm, because they depend on the decisions taken by the player during the gameplay. For instance, going to the left in a shooter, could mean the gamer finds fifty computer-run enemies in the next room, whereas going to the right could mean finding a weapons arsenal. The thread that runs the artificial intelligence subtask must be synchronized at any instant with the thread that runs the main algorithm. Modern games usually consist of a main thread that runs the basis of the game and four or more slave threads that run subtasks synchronized by the main thread. A CPU with twelve cores at 2GHz could run more subtasks than a CPU with six cores at 4GHz, but the CPU with faster cores can execute the main thread much faster. In general, the CPU with six cores at 4GHz is better for gaming.
Therefore, if we want to duplicate the IPC we need to improve the superscalar hardware to analyze four time more instructions! Some hardware structures of the core such as fetching, or decoding will have to be scaled by a factor of four, but other structures need more aggressive scaling. For example to analyze the interdependences between two instruction we need only one comparison, because there is only one possible relationship between two any instructions, but increase the number of instructions to analyze to four and we need six comparisons –we have to compare the first instruction with the second instruction, the first with the third, the first with the fourth, the second with the third, the second with the fourth, and finally the third instruction with the fourth instruction–. For eight instructions the numbers of comparisons needed is twenty-eight. Detecting dependences among 2000 instructions requires almost two million comparisons! This cost obviously limits the number of instructions that can be considered for issue at once. Current cores have of the order of thousands of comparators.
This is not the only scaling problem. Code has branches, non-scientific code has a branch each eight or ten instructions. So, to fetch a sequence of hundreds of instructions the core must know, in advance, what paths will be taken on each bifurcation. This is the task of branch predictors. Imagine a predictor with average accuracy of 90%; this means that the prediction fails only in one branch of each ten. It can seem this very high accuracy solves the problem of branching in code, but accuracy reduces with each consecutive branch, because the probabilities of failed prediction increase with each new bifurcation. Consider a simple code example with only binary branches.
Whereas the probability the core is in the correct path after the first bifurcation point is of 90%, the probability that the core is in the correct path after the second branch is lower; the probability that the hardware correctly predicts the second branch continues being of 90%, but this probability is now bound to the probability of the core already being in the correct path after the first bifurcation. The probability the core is in the correct path after the second bifurcation point is now.
The probability has reduced to 81%. After ten consecutive binary branches the probability reduces to about 35%; this means the core is analyzing an incorrect sequence of instructions in two of three occasions! Current state-of-art predictors are very complex and include correlating tables for predictions; those tables keep a record of the paths taken by the core before and use those tables to improve the branch prediction by predicting a new branch in the context of the former branches leading up to it, instead predicting each branch just in isolation. Current state-of-art predictors are power-hungry and take up valuable space on the core.
One could think that being wrong in 5% of cases has a negligible impact on performance, but that is not correct. Things are not linear. When the core detect has been working in the incorrect path, it must cancel all the speculative work has been doing in advance, flush the pipeline entirely, and start again at the early point in the sequence of instructions where the prediction failed. This affect performance; this is called the mispredict branch penalty. Even with predictors accurate to 95%, the mispredict penalty can reduce the performance of a high-performance core by about one third. In other words, the core is not doing useful work one third of the time.
The superscalar out-of-order microarchitectures behind the x86 ISA have hit a performance wall. No engineering team can break this wall, at best, engineers can spend years working on optimizing current microarchitectures to get a 2% IPC gain here and a 7% here. The only possibility to get a quantum jump on IPC over the current designs is if we change from a serial ISA, as x86, to a new ISA that scales up.
The existence of an IPC wall is not new. Research made decades ago about the limits of instruction level parallelism on code identified a soft wall about 10-wide cores. This wall was the reason why Intel engineers in collaboration with Hewlett Packard engineers developed a new ISA that would be scalable. The new ISA, dubbed EPIC, stands for Explicitly Parallel Instruction Computing. Intel wanted to use the migration from 32 bits to 64 bits to abandon x86 and replace it with EPIC. The plan failed badly because the promises of the new ISA could not be fulfilled. The reason? The ISA was developed around the concept of a smart compiler, which no one could build. Therefore, EPIC-based hardware was penalized by executing non-optimal binaries.
The evolution of GPUs is impressive, but the figure is measuring throughput. A GPU is an TCU (Throughput Compute Unit), whereas a CPU is a LCU (Latency Compute Unit); this is the terminology used by AMD in its HSA specification. GPUs are designed to crunch lots of numbers in a massively parallel way, when branches and the response time (latency) to changes are not relevant; otherwise a CPU is needed, this is the reason why GPUs are not used to execute the operative system.
Why are GPUs easier to scale up? Consider a process node shrink that provides four times higher density; for instance, a 14nm –> 7nm shrink. We could add four times more transistors in the same space. The key here is on how transistors are used in GPUs and CPUs. The relationship between IPC and number of transistors (the complexity of a microarchitecture) is not linear, but approximately given by.
Where a is a parameter and A is the area used by transistors that execute a single thread. This is the so-called Pollack's rule, which was derived empirically by him. For a theoretical derivation of the rule check this work by me.
In the first case, the higher density provided by the new node has been used to increase the performance of each core by 100%; the total throughput has duplicated as well. In the second case, the higher density has been used to quadruplicate the total throughput. The same node shrink can give us two different performance increases; however, keep in mind that the twice higher throughput achieved by quadruplicating the number of cores does not come for free, because the latency is the same, whereas in the first case latency has also reduced by 100%.
I am a scientist, computer designer and builder.


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\begin{document}
	\begin{titlepage}
		\begin{flushright}
			\phantom{arXiv:yymm.nnnn}
		\end{flushright}
		\vspace{1cm}
		\begin{center}
			{\Large\bf Turbulent cascades in a truncation of the\vspace{3mm}\\ cubic Szeg\H{o}{}  equation and related systems}\\
			\vskip 15mm
			{\large Anxo Biasi$^{1}$ and Oleg Evnin$^{2,3}$}
			\vskip 7mm
			{\em $^1$  Institute of Theoretical Physics, Jagiellonian University, Krak\'ow, Poland}
			\vskip 3mm
			{\em $^2$ Department of Physics, Faculty of Science, Chulalongkorn University,
				Bangkok, Thailand}
			\vskip 3mm
			{\em $^3$ Theoretische Natuurkunde, Vrije Universiteit Brussel and\\
				The International Solvay Institutes, Brussels, Belgium}
			\vskip 7mm
			{\small\noindent {\tt anxo.biasi@gmail.com, oleg.evnin@gmail.com}}
			\vskip 10mm
		\end{center}
		\vspace{1cm}
		\begin{center}
			{\bf ABSTRACT}\vspace{3mm}
		\end{center}

		\noindent We introduce a truncated version of the cubic Szeg\H{o}{} equation, an integrable model for deterministic turbulence. In this truncation, a majority of the Fourier mode couplings are eliminated while the signature features of the model are preserved, namely, a Lax pair structure and a hierarchy of finite-dimensional dynamically invariant manifolds. Despite the impoverished structure of the interactions, the turbulent behaviors of our new equation are stronger in an appropriate sense than for the original cubic Szeg\H{o}{} equation. We construct explicit analytic solutions displaying exponential growth of Sobolev norms. We furthermore introduce a family of models that interpolate between our truncated system and the original cubic Szeg\H{o}{} equation, along with other related deformations. These models possess Lax pairs, invariant manifolds, and display a variety of turbulent cascades. We additionally mention numerical evidence, in some related systems, for an even stronger type of turbulence in the form of a finite-time blow-up.

		\vfill

	\end{titlepage}

\begin{flushright}
{\it \hspace{1cm}\\
 ``...the vessel underwent trials on the Paddington Canal in 1837.\\
However, by one of those fortunate accidents, which sometimes\\
occur in the history of science and technology, the propeller\\
was damaged during the trials and about half of it broke off,\\
whereupon the vessel immediately increased its speed.''} \cite{Carlton}\vspace{1.2cm}
\end{flushright}

	\section{Introduction\vspace{3mm}}\label{sec:Intro}

	In studies of Hamiltonian PDEs, the phenomenon of turbulence continues to challenge various communities of mathematicians and physicists. It is defined by transfer of energy from long-wavelength to short-wavelength modes, leading to concentration of energy on arbitrarily small spatial scales. Such {\em turbulent cascades} are usually quantified by the growth of Sobolev norms, and the general question of unbounded increase of Sobolev norms is a long-standing problem \cite{Bourgain}. Historically, much of the effort in this direction has focused on the wave turbulence theory \cite{Nazarenko} which employes averaging over the phases of weakly interacting waves. By contrast, studies of fully deterministic turbulence tend to be more recent, with much of the related efforts focused on the {\em nonlinear Schr\"{o}dinger equation} \cite{Hani1,Hani2,Colliander,Kuksin,Merle, Guardia, Guardia2, Guardia3, Guardia4}. Close parallels exist between this line of research and studies of weak turbulence in Anti-de Sitter spacetimes \cite{BR,rev2}, a topic of interest within general relativity and theoretical high-energy physics.

	Motivated by studies of nonlinear Schr\"{o}dinger equations, G\'erard and Grellier designed a tractable model of non-dispersive evolution, the {\em cubic Szeg\H{o}{} equation} \cite{GG}; see \cite{GG2} for a review. This nonlinear equation on the circle $\mathbb{S}^1$ is Lax-integrable and displays
turbulent behaviors that can be analyzed using integrability. To be more precise, it has been proved that there exist initial conditions with super-polynomial growth of some Sobolev norms; however, there are no explicit examples of such data. Existence of initial conditions with exponential growth of Sobolev norms is another open problem. Explicit solutions can be constructed that show `weak weak turbulence' in the language of \cite{Colliander}, so that the Sobolev norm growth is always bounded but can be arbitrarily enhanced by fine-tuning the initial data. These properties of the cubic Szeg\H{o}{} equation motivated various studies of modifications of this model in order to elucidate whether such features are preserved and/or some additional phenomena emerge due to modifications. Examples that preserve integrability but display initial configurations with unbounded Sobolev norm growth in the past literature are \cite{Pocovnicu,Xu,Thirouin,GG3}. In \cite{Pocovnicu}, the cubic Szeg\H{o}{} equation was placed on the real line $\mathbb{R}$, providing solutions with polynomial growth of Sobolev norms. In \cite{Xu}, the so-called {\em $\alpha$-Szeg\H{o}{} equation} was introduced, with explicit examples of initial configurations with exponentially growing Sobolev norms. A few years later, the quadratic Szeg\H{o}{} equation was introduced in \cite{Thirouin}. This case also displayed solutions with exponentially growing Sobolev norms. Finally, in \cite{GG3}, we can find the damped Szeg\H{o}{} equation, likewise with unbounded Sobolev norm growth.
\newpage
The cubic Szeg\H{o}{} equation and the other systems we discuss here belong to the class of cubic resonant systems. These resonant systems often arise as weakly nonlinear approximations to the dynamics of PDEs whose linearized spectra of normal frequencies are strongly resonant.
Thus, it was shown that the cubic Szeg\H{o}{} equation accurately describes the weakly nonlinear long-time dynamics of some sectors of the half-wave equation \cite{GG,GG2} and the wave guide Schr\"odinger equation \cite{Xu2}. In closer contact with physics applications, resonant systems arise as approximations for the dynamics of Bose-Einstein condensates \cite{GHT,GT,BMP,BBCE,GGT} and in Anti-de Sitter (AdS) spacetimes \cite{FPU,CEV,BMR,CF,BEL,BCE,BEF}, the latter topic extensively studied in relation to AdS instability \cite{BR,rev2}. Resonant systems constructed in this way often show powerful analytic structures and admit special solutions \cite{BBE, breathing}, even in the absence of Lax-integrability that characterizes the cubic Szeg\H{o}{} equation and other systems that we focus on here.

As we have mentioned, considerable attention has been given to modifying the cubic Szeg\H{o}{} equation in a way that strengthens  its turbulent behaviors or makes them more manifest. The main goal of our present exposition is to report our discovery of the {\em truncated Szeg\H{o}{} equation} that possesses such features, together with systems interpolating between this new equation and the original cubic Szeg\H{o}{} equation, and some further related deformations.
Relatively to what has been proposed in the literature before, our systems are close in spirit to those of \cite{Xu} and \cite{GG3}, though the concrete alteration we make in the equations is completely different. One might legitimately ask what is the purpose of introducing yet another modification of the cubic Szeg\H{o}{} equation with such properties. First, for any system as special as the cubic \Sz{} equation, it is important to understand the full range of modifications that preserve its crucial properties (Lax-integrability, invariant manifolds, turbulence), and our results provide an extra four-parameter space of dynamical systems to contribute to this picture. Second, our equation is qualitatively distinct, say, from the modifications introduced in \cite{Xu,GG3}, which are its closest analogs, and may teach us important lessons. The modifications of \cite{Xu, GG3} are designed by adding a single simple term to the cubic \Sz{} equation. By contrast, our truncated Szeg\H{o}{} equation results from {\it removing} most of the terms from (the Fourier representation of) the cubic \Sz{} equation. As turbulent energy transfer relies on mode couplings, the fact that turbulence is strengthened after removing a majority of mode couplings is surprising and counter-intuitive, and makes us think of the curious naval engineering incident from our epigraph, though the route by which we arrived at our equation was more systematic relatively to the XIX century naval precedent. Apart from their direct relevance in the context of the cubic \Sz{} equation{} studies, our findings may invite general re-evaluation of which mode couplings play an essential role in formation of turbulent cascades.

	The article is organized as follows: in section \ref{sec:Szego}, we provide some relevant details on resonant equations in general, as well as the cubic Szeg\H{o}{} and $\alpha$-Szeg\H{o}{} equations, which form the background material for our actual studies. In section \ref{sec:TR_Szego}, we introduce and examine our first model, the truncated Szeg\H{o}{} equation. The Lax pair structure, invariant manifolds and explicit solutions with unbounded Sobolev norms are constructed. In section \ref{sec:beta_Szego}, we do the same with our second model, the $\beta$-Szeg\H{o}{} equation. We provide summary and commentary, and mention some extra tentative results on finite-time turbulent blow-up in section \ref{sec:Discussion}.

	\section{Brief review of resonant equations, the cubic Szeg\H{o}{} equation and the $\alpha$-Szeg\H{o}{} equation}\label{sec:Szego}

	\subsection{Resonant equations}

The cubic \Sz{} equation{} and the new equations we shall introduce in this paper are all of the following algebraic form:
	\begin{equation}
	i\frac{d\alpha_n}{dt}=\hspace{-5mm}\sum_{\begin{array}{c}\vspace{-6mm}\\ \scriptstyle m,k,l=0\vspace{-2mm}\\ \scriptstyle n+m=k+l\end{array}}^{\infty}\hspace{-5mm} C_{nmkl}\bar{\alpha}_m{\alpha}_k\alpha_{l}.
	\label{eq:flow_eq_linear_spectrum}\vspace{-2mm}
	\end{equation}
	Here,  $\alpha_n$ with $n=0,1,2,\ldots$ are complex-valued functions of time, which are our dynamical variables, bars denote complex conjugation, and $C_{nmkl}$ are real numbers that can be called the {\em mode couplings} or the {\em interaction coefficients}. The interaction coefficients are invariant under the following interchanges of the indices: $n\hspace{-0.0cm}\leftrightarrow\hspace{-0.0cm}m$, $k\hspace{-0.0cm}\leftrightarrow\hspace{-0.0cm}l$, $(n,m) \hspace{-0.0cm}\leftrightarrow\hspace{-0.0cm}(k,l)$. Such systems are often called {\em resonant systems} or {\em resonant equations}. Note the {\em resonant condition} $n+m = k+l$ restricting the summations in (\ref{eq:flow_eq_linear_spectrum}). Different representatives of this large class of equations that we shall study are distinguished by different explicit choices of the interaction coefficients $C$.

Equations of the form (\ref{eq:flow_eq_linear_spectrum}) commonly arise via application of
time-averaging or multi-scales analysis \cite{murdock,KM} as weakly nonlinear approximations to PDEs whose linearized spectrum of normal frequencies is highly resonant, as happens to a number of PDEs in harmonic traps or Anti-de Sitter spacetimes. We shall not review such derivations here, as it is quite far apart from our main focus, and will simply refer the reader to \cite{rev2,GHT,BBCE,murdock, KM}. We note that, viewed from this perspective, the dynamical variables $\alpha_n$ originate as the complex amplitudes of the linearized normal modes of the PDE, which acquire slow evolutions under the effect of weak nonlinearities. A system of the form (\ref{eq:flow_eq_linear_spectrum}) then accurately approximates this slow evolution.

The resonant equation (\ref{eq:flow_eq_linear_spectrum}) has a canonical Hamiltonian structure with a Hamiltonian of the form
	\begin{equation}
	\mathcal{H}={\frac12} \hspace{-2mm}\sum_{\begin{array}{c}\vspace{-6.5mm}\\ \scriptstyle n,m,k,l=0\vspace{-2mm}\\ \scriptstyle n+m=k+l\end{array}}^{\infty}\hspace{-3mm} C_{nmkl}\bar{\alpha}_n\bar{\alpha}_m{\alpha}_k\alpha_{l}\vspace{-3mm}
	\label{eq:H_cq}
	\end{equation}
and the symplectic form $i\sum_n d\bar\alpha_n\wedge d\alpha_n$. In addition to the Hamiltonian, there are two generic conserved quantities, irrespectively of the form of the interaction coefficients $C$, which we list together with the associated symmetry transformations:
	\begin{align}
	N = \sum_{n=0}^{\infty}|\alpha_n|^2, & \qquad \alpha_n(t) \to e^{i \phi} \alpha_n(t) \label{eq:N_consered_quantity},\\
	E = \sum_{n=0}^{\infty}n|\alpha_n|^2, & \qquad \alpha_n(t) \to e^{i n \theta} \alpha_n(t).  \label{eq:E_consered_quantity}
	\end{align}
	Additionally, equation (\ref{eq:flow_eq_linear_spectrum}) enjoys the scaling symmetry
	\begin{equation}
	\alpha_{n}(t) \to \epsilon \alpha_n(\epsilon^2 t).
	\end{equation}
For specific choices of the interaction coefficients, the set of symmetries and conserved quantities may, of course, become much bigger. This is the case for the class of systems treated in \cite{BBE}, for the cubic \Sz{} equation, and for the new systems we shall introduce below.

In full generality, equation (\ref{eq:flow_eq_linear_spectrum}) admits an infinite number of dynamically invariant manifolds. Namely, one can choose two mutually prime integers $p$ and $q$, $p<q$, and set to zero all $\alpha_n$ except for those with $n=p\hspace{-1mm}\mod q$. If this restriction is implemented in the initial conditions,  (\ref{eq:flow_eq_linear_spectrum}) guarantees that the modes set to zero will never get excited, which defines an invariant manifold of the evolution. We shall use such $(p\hspace{-1mm}\mod q)$-restrictions in some of our arguments. Again, for special systems within the large class given by  (\ref{eq:flow_eq_linear_spectrum}) , there can be many more invariant manifolds. Thus, for the systems at the focus of our current study, a crucial role is played by an infinite hierarchy of finite-dimensional invariant manifolds that we shall explicitly review below.

	\subsection{The cubic Szeg\H{o}{} equation} \label{subsec:Szego}
	 The cubic Szeg\H{o}{} equation was intensively studied in \cite{GG} and is usually given in the form
	 \begin{equation}
	 i\partial_t u = \Pi\left(|u|^2u\right),
	 \label{eq:Szego}
	 \end{equation}
	 where $\Pi$, the so-called Szeg\H{o}{} projector, acts as a filter of negative Fourier modes
	\begin{equation}
	\Pi\left(\sum_{n\in\mathbb{Z}} \alpha_{n} e^{in\theta}\right) = \sum_{n=0}^{\infty} \alpha_{n} e^{in\theta}.
	\label{eq:Szego_projector}
	\end{equation}
	This equation is placed on the circle $\mathbb{S}^1$ and studied for $u\in L_{+}^2\left(\mathbb{S}^1\right)$; namely, the space of functions $L^{2}\left(\mathbb{S}^1\right)$ where negative Fourier modes are zero, $\alpha_n = 0$ $\forall n < 0$. Then, as $\Pi: L^2\left(\mathbb{S}^1\right)\to L_{+}^2\left(\mathbb{S}^1\right)$, (\ref{eq:Szego}) guarantees that for initial conditions $u_0\in L_{+}^2\left(\mathbb{S}^1\right)$
	\begin{equation}
		u(t, e^{i\theta}) = \sum_{n=0}^{\infty} \alpha_n(t) e^{i n\theta}, \qquad \text{with }\ \sum_{n=0}^{\infty} |\alpha_n|^2 < \infty.
		\label{eq:u_variable}
	\end{equation}
	The space $L^{2}\left(\mathbb{S}^1\right)$ is endowed with the inner product
	\begin{equation}
		\left(u|v\right) := \int_{\mathbb{S}^1}u\bar{v}\frac{d\theta}{2\pi}.
	\end{equation}
	The cubic Szeg\H{o}{} equation admits a Hamiltonian structure with the Hamiltonian
	\begin{equation}
	\mathcal{H}_{Sz}=\frac{1}{4}\int_{\mathbb{S}^1}|u|^4  \frac{d\theta}{2\pi} \qquad \text{with } u\in L_+^2(\mathbb{S}^1).
	\end{equation}

Substituting the Fourier expansion (\ref{eq:u_variable}) into (\ref{eq:Szego}), we rewrite the cubic \Sz{} equation{} in an equivalent form
	\begin{equation}
	i\frac{d\alpha_n}{dt}=\hspace{-5mm}\sum_{\begin{array}{c}\vspace{-6mm}\\ \scriptstyle m,k,l=0\vspace{-2mm}\\ \scriptstyle n+m=k+l\end{array}}^{\infty}\hspace{-5mm} \bar{\alpha}_m{\alpha}_k\alpha_{l}.
	\label{eq:Szego_modes}\vspace{-2mm}
	\end{equation}
This representation manifestly matches the general algebraic structure of the resonant equation (\ref{eq:flow_eq_linear_spectrum}), with the interaction coefficients given by the very simple
expression $C_{nmkl}^{\left(\text{Sz}\right)} = 1$.
We thus note that there are two different representations here to deal with: the position space representation (\ref{eq:Szego}) in terms of the Szeg\H{o}{} projector, and the Fourier space representation (\ref{eq:Szego_modes}). The position space representation is more familiar from the literature on the cubic \Sz{} equation{}, and the Szeg\H{o}{} projector language is effective, for example, for analyzing Lax integrability. The Fourier space representation makes more explicit contact with the physics of resonant systems. We shall rely on both representations in our derivations to highlight various aspects of our considerations.\vspace{5mm}

	Some relevant properties of the cubic Szeg\H{o}{} equation are \cite{GG}:
	\begin{itemize}
		\item This system possesses two Lax pairs; see (\ref{LaxSz}) below.
		\item Sobolev norms\footnote{In terms of the Fourier modes, Sobolev norms can be expressed in the form $$\|u\|_{H^{s}}^2 = \sum_{n=0}^{\infty}\left(1+n\right)^{2s}|\alpha_n|^2.$$} of solutions of (\ref{eq:Szego}) subject to initial conditions $u_0\hspace{-0.12cm} \in\hspace{-0.12cm} H_{+}^{s}\left(\mathbb{S}^1\right)$  with $s\hspace{-0.1cm}>\hspace{-0.1cm}1$ (where $H_{+}^{s}\left(\mathbb{S}^1\right)\hspace{-0.12cm}:=\hspace{-0.12cm} H^s\left(\mathbb{S}^1\right)\hspace{-0.1cm}\cap\hspace{-0.1cm} L_{+}^2\left(\mathbb{S}^1\right)$), cannot grow faster than exponentially,
		\begin{equation}
			\|u(t)\|_{H^{s}} \leq C_{s}\|u_0\|_{H^{s}} e^{C_s \|u_0\|^2_{H^{s}}|t|} \qquad \text{with } C_{s}>0.
		\end{equation}
		\item There exist complex invariant manifolds $\mathcal{V}(d)$ consisting of functions of the form
		\begin{equation}
			u(z) = \frac{A(z)}{B(z)},
			\label{eq:V_manifold}
		\end{equation}
		where, $A(z)$ and $B(z)$ are polynomials in $z$ with no common factors, $B(0) = 1$, $B(z)$ has no zeros in the closed unit disk and
		\begin{itemize}
			\item If $d = 2D$ is even, the degree of $A(z)$ is at most $D\hspace{-0.1cm}-\hspace{-0.1cm}1$ and the degree of $B(z)$ is exactly $D$.
			\item If $d = 2D+1$ is odd, the degree of $A(z)$ is exactly $D$ and the degree of $B(z)$ is at most $D$.
		\end{itemize}
	These manifolds are linked to the Lax-integrability structure, as we shall explain below.
	\item Sobolev norms for initial conditions $u_0 \in \mathcal{V}(d)$ remain bounded for all times,
		\begin{equation}
			\forall s > \frac{1}{2} \qquad \underset{t\in\mathbb{R}}{\sup} \|u(t)\|_{H^{s}} < \infty.
			\label{eq:Szego_bond_infty}
		\end{equation}
		\item There are families of initial conditions $u_0^{\epsilon} \in \mathcal{V}(d)$ for a given $d$, with $\sup_{\epsilon} \|u_0^{\epsilon}\|_{H^s}<\infty$ for all $s$, such that
		\begin{equation}
			\forall s > \frac{1}{2} \qquad \underset{\epsilon}{\sup}\  \underset{t\in\mathbb{R}}{\sup} \|u^{\epsilon}(t)\|_{H^{s}} = \infty;
			\label{eq:Szego_epsilon_singular}
		\end{equation}
		namely, despite (\ref{eq:Szego_bond_infty}), we can fine-tune the initial conditions inside the same $\mathcal{V}(d)$ to get Sobolev norms with $s\hspace{-0.06cm}>\hspace{-0.06cm}1/2$ that grow as much as we please. (This is sometimes known as `weak weak turbulence' \cite{Colliander}.)
		In particular, the three-dimensional invariant manifold $\mathcal{V}(3)$ contains such solutions and will be of special interest in this paper.
		\item There exist initial conditions $u_0$ in $C^{\infty}\left(\mathbb{S}^1\right)\cap L_{+}^{2}\left(\mathbb{S}^1\right)$ such that for all $s> 1/2$
		\begin{align}
			\underset{t\to\infty}{\lim\text{sup}}\frac{\|u(t)\|_{H^s}}{t^{k}}= \infty& \qquad \forall k\geq 1,\\
			\underset{t\to\infty}{\lim\text{inf}}\|u(t)\|_{H^{s}} < \infty. & \
		\end{align}
		Namely, there is an infinite sequence of exchanges of energy back and forth between low and high modes, and the flow of energy to high modes provides for a super-polynomial growth of Sobolev norms with $s> 1/2$. We remark that currently there are no explicit examples of these initial data.
		\item The existence of initial conditions $u_0$, such that Sobolev norms with $s>1/2$ display an exponential growth is an open problem.
	\end{itemize}

	Now, to define the Lax pairs for (\ref{eq:Szego}), consider the following operators:
	\begin{equation}
	H_uh=\Pi(u\bar h),\qquad T_bh=\Pi(bh),\qquad Sh=e^{i\theta}h.
	\label{eq:H_T_S_operators}
	\end{equation}
	In components, their action is
	\begin{equation}
	(H_u h)_n=\sum_{m=0}^\infty \alpha_{n+m}\bar h_m,\qquad (T_b h)_n=\sum_{m=0}^\infty b_{n-m} h_m, \qquad (Sh)_n=h_{n-1}.
	\label{eq:Hu_def}
	\end{equation}
	Note that $S$ is simply a shift changing the sequence $\{\alpha_0,\alpha_1,\ldots\}$ into $\{0,\alpha_0,\alpha_1,\ldots\}$, while the corresponding conjugate $S^\dagger$ does the opposite shift from $\{\alpha_0,\alpha_1,\ldots\}$ into $\{\alpha_1,\alpha_2,\ldots\}$. In particular, $S^\dagger S=1$.

	The two Lax pairs given in Theorem 3 in the third reference of \cite{GG} are
	\begin{equation}
	\frac{dH_u}{dt}=[B_u,H_u],\qquad \frac{dK_u}{dt}=[C_u,K_u],
	\label{LaxSz}
	\end{equation}
	with
	\begin{equation}
		K_u = S^{\dagger}H_u=H_uS=H_{S^\dagger u}, \qquad B_u= \frac{i}2 H_u^2-i T_{|u|^2}, \qquad C_u=\frac{i}2 K_u^2-i T_{|u|^2},
		\label{eq:C_B_operators}
	\end{equation}
	whenever the equations of motion for $u$ are satisfied. The existence of invariant manifolds (\ref{eq:V_manifold}) can be understood in terms of these Lax operators. Actually, they correspond to the choices of $u$ for which the sum of the ranks of the operators $H_u$ and $K_u$ is $d$.

	\subsection{The $\alpha$-Szeg\H{o}{} equation} \label{subsec:alpha_Szego}
	The $\alpha$-Szeg\H{o}{} equation \cite{Xu} was constructed as a deformation of the cubic \Sz{} equation{} by a term proportional to the lowest mode $\alpha_0 = (u|1)$:
	\begin{equation}
	i\partial_t u = \Pi\left(|u|^2u\right) + \alpha\left(u|1\right),
	\label{eq:alpha_Szego_v1}
	\end{equation}
	 where $\alpha\in\mathbb{R}$ and $u,\ \Pi$ and the operators $K_u$ and $C_u$ (that will appear later in this section), have the same definition as for the cubic Szeg\H{o}{} equation. For any $\alpha~\neq~0$, the continuous dependence on this parameter can be absorbed by the rescaling $\tilde{u}(t) = \sqrt{|\alpha|}u(|\alpha|t)$, leaving the $\alpha$-Szeg\H{o}{} equation as
	\begin{equation}
	i\partial_t \tilde{u} = \Pi\left(|\tilde{u}|^2\tilde{u}\right) + \text{sgn}(\alpha)\left(\tilde{u}|1\right).
	\label{eq:alpha_Szego_v2}
	\end{equation}

Note that this model cannot be literally expressed in terms of the resonant equation (\ref{eq:flow_eq_linear_spectrum}), but the necessary deviation from the algebraic structure of (\ref{eq:flow_eq_linear_spectrum}) is small and only appears in the equation for the lowest mode:
		\begin{equation}
		i\frac{d\alpha_0}{dt}\mp \text{sgn}(\alpha)\alpha_0 =\hspace{-5mm}\sum_{\begin{array}{c}\vspace{-6mm}\\ \scriptstyle m,k,l=0\vspace{-2mm}\\ \scriptstyle m=k+l\end{array}}^{\infty}\hspace{-5mm} \bar{\alpha}_m{\alpha}_k\alpha_{l},
		  \qquad \text{and} \qquad  i\frac{d\alpha_n}{dt}=\hspace{-5mm}\sum_{\begin{array}{c}\vspace{-6mm}\\ \scriptstyle m,k,l=0\vspace{-2mm}\\ \scriptstyle n+m=k+l\end{array}}^{\infty}\hspace{-5mm} \bar{\alpha}_m{\alpha}_k\alpha_{l} \qquad \text{for }n \geq 1.
		  \label{eq:alpha_Szego_modes}
		\end{equation}
		The properties of this system are slightly different from the case of the cubic Szeg\H{o}{} model \cite{Xu}:
		\begin{itemize}
			\item The $\alpha$-Szeg\H{o}{} model possesses one Lax pair
			\begin{equation}
				\frac{dK_{u}}{dt} = \left[C_u, K_u\right].
				\label{eq:Lax_pair_truncate_Szego}
			\end{equation}
			\item Sobolev norms of solutions of (\ref{eq:alpha_Szego_v2}) subject to initial conditions $u_0 \in H_{+}^{s}\left(\mathbb{S}^1\right)$ with $s>1$, cannot grow faster than exponentially,
			\begin{equation}
			\|u(t)\|_{H^{s}} \leq \|u_0\|_{H^{s}} e^{C |t|} \qquad \text{with } C>0.
			\end{equation}
			\item There exist complex invariant manifolds $\mathcal{L}(D)$ defined by the choices of $u$ such that the rank of $K_u$ is $D$. They consist of rational functions of the form
			\begin{equation}
				u(z) = \frac{A(z)}{B(z)},
				\label{eq:L_odd}
			\end{equation}
			where $A(z)$ and $B(z)$ are polynomials in $z$ of degree at most $D$, with no common factors, $\text{deg}(A) = D$ or $\text{deg}(B) = D$ and $B(z)$ has no zeros in the closed unit disk.

			\item For $\alpha < 0$ and initial conditions $u_0 \in \mathcal{L}(D)$, Sobolev norms are bounded
			\begin{equation}
			\forall s \geq 0 \qquad  \|u(t)\|_{H^{s}} < C.
			\end{equation}
			\item For $\alpha > 0$ and some $u_0 \in \mathcal{L}(1)$, Sobolev norms with $s\hspace{-0.08cm} >\hspace{-0.08cm} \frac{1}{2}$ grow exponentially for large enough~time,
			\begin{equation}
			\forall s > \frac{1}{2}, \qquad \|u(t)\|_{H^{s}} \underset{t\to\infty}{\simeq} e^{c(2s-1)|t|}.
			\end{equation}
			Hence, the $\alpha$-Szeg\H{o}{} equation has solutions with unbounded Sobolev norms.
			\item The $\alpha$-Szeg\H{o}{} equation contains the cubic Szeg\H{o}{} equation. Even for $\alpha\neq 0$, by restricting the initial conditions to odd modes (setting even modes to 0), (\ref{eq:alpha_Szego_modes}) is reduced to (\ref{eq:Szego_modes}). Hence, the $\alpha$-Szeg\H{o}{} equation has subsectors of initial conditions with the properties displayed in subsection~\ref{subsec:Szego}.
		\end{itemize}

	\section{The truncated Szeg\H{o}{} equation}\label{sec:TR_Szego}

With the above preliminaries, we proceed with the key point of our presentation, which is the introduction of the {\em truncated Szeg\H{o}{} equation}. Starting from the cubic \Sz{} equation, one simply sets to zero a specific large set of the interaction coefficients (a majority of them, in fact), while leaving the remaining ones intact, according to the pattern
	\begin{equation}
	C_{nmkl}^{\left(tr\right)} = \begin{cases}
		1 & \text{if } n m k l = 0,\\
		0 & \text{if } n m k l \neq 0.
	\end{cases}
	\label{eq:C_BE}
	\end{equation}
The condition that the product $nmkl$ must vanish evidently implies that at least one of the mode numbers $n$, $m$, $k$ or $l$ must vanish in order for the corresponding coupling coefficient $C$ to be nonzero.
	We have labelled the interaction coefficients of this truncated Szeg\H{o}{} system  by $C^{\left(tr\right)}$ for future reference. By truncation, we simply mean eliminating interactions between modes (it should not be confused with restricting the dynamics of a given system to one of its invariant manifolds). Given the expression for $C^{\left(tr\right)}$, the equations of motion (\ref{eq:flow_eq_linear_spectrum}) take the form
	\begin{equation}
	i\frac{d\alpha_0}{dt} = \sum_{m=0}^{\infty} \sum_{k=0}^{m}\bar{\alpha}_m \alpha_k \alpha_{m-k}, \qquad i\frac{d\alpha_{n}}{dt} = \bar{\alpha}_0 \sum_{k=0}^{n}\alpha_k \alpha_{n-k} + 2\alpha_0 \sum_{m=1}^{\infty} \bar{\alpha}_m \alpha_{n+m} \quad \text{for } n \geq 1.
	\label{eq:resonant_eq_BE}
	\end{equation}
	 One can rewrite (\ref{eq:resonant_eq_BE}) in position space, i.e., in terms of $u$ given by (\ref{eq:u_variable}):
	\begin{equation}
	i\partial_t u = \Pi(|u|^2u)-S\Pi(|S^\dagger u|^2 S^\dagger u),
	\label{eq:truncated_Szego_eq}
	\end{equation}
	where $\Pi$ is the Szeg\H{o}{} projector (\ref{eq:Szego_projector}) and $S$ the shift operator defined in (\ref{eq:H_T_S_operators}). The corresponding Hamiltonian is
	\begin{equation}
	\mathcal{H}_{tr}=\frac{1}{4}\int_{\mathbb{S}^1}\left(|u|^4-|S^\dagger u|^4\right) \frac{d\theta}{2\pi} = \frac{1}{4}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{k=0}^{n+m}C_{nmk(n+m-k)}^{\left(tr\right)}\bar\alpha_n\bar\alpha_m\alpha_k\alpha_{n+m-k}.
	\label{eq:Hamiltonian_truncated_Szego}
	\end{equation}
This Hamiltonian can be understood as the cubic Szeg\H{o}{} system minus a ``shifted" cubic Szeg\H{o}{} system, with $\{\alpha_0,\alpha_1,...\}$ replaced with
$\{\alpha_1,\alpha_2,...\}$. The same pattern can be noticed in (\ref{eq:truncated_Szego_eq}). This structure of the model will be important for deriving properties of the truncated Szeg\H{o}{} equation from the results previously known for the cubic \Sz{} equation.

	We now list the main properties of the truncated Szeg\H{o}{} equation, which are the central technical results of our paper, and which will be proved in the remainder of this section:
	\begin{itemize}
		\item The truncated Szeg\H{o}{} equation possesses one Lax pair, defined through the operators
(\ref{eq:H_T_S_operators}-\ref{eq:C_B_operators}):
		\begin{equation}
		\frac{dK_{u}}{dt} = \left[C_u-B_{S^\dagger u}, K_u\right].
\label{Laxtr}
		\end{equation}
		(See appendix A for an explicit action of these operators in  the mode representation.) Note that the presence of two Lax pairs in the original cubic \Sz{} equation{} is essentially used in the construction of its general solution in the third reference of \cite{GG}. Such derivations do not immediately generalize to the $\alpha$-Szeg\H{o}{} equation or our system.
		\item Sobolev norms of solutions of (\ref{eq:truncated_Szego_eq}) subject to initial conditions $u_0 \in H_{+}^{s}\left(\mathbb{S}^1\right)$ with $s>1$, cannot grow faster than exponentially, which excludes a finite-time blow-up:
		\begin{equation}
		\|u(t)\|_{H^{s}} \leq \|u_0\|_{H^{s}} e^{C |t|} \qquad \text{with } C>0.
		\label{eq:truncated_Szego_global_bound}
		\end{equation}
		\item There exist complex invariant manifolds $\mathcal{L}(D)$ defined in (\ref{eq:L_odd}).
		\item For some $u_0\hspace{-0.1cm}\in\mathcal{L}(1)$, Sobolev norms with $s\hspace{-0.08cm} >\hspace{-0.08cm} \frac{1}{2}$ grow exponentially at late times:
		\begin{equation}
		\forall s > \frac{1}{2}, \qquad \|u(t)\|_{H^{s}} \underset{t\to\infty}{\simeq} e^{(2s-1)c|t|}.
		\label{eq:truncated_Szego_exponential_growth_Sobolev}
		\end{equation}
	\end{itemize}
	Before proceeding with our proofs, we add some extra comments:
	\begin{itemize}
		\item Integrability is a very fragile property, and a priori, one expects any modifications in the
mode coupling pattern to upset it. The specific modification of the mode couplings used to define the truncated Szeg\H{o}{} equation is very special in this regard, as integrability is preserved (although we lose one of the two Lax pairs of the cubic \Sz{} equation{} equation). Note that the first Lax operator $K_u$ is common to the original and the truncated Szeg\H{o}{} equations. As a consequence, one can construct a Lax pair for an arbitrary linear combination of $C^{\left(tr\right)}$ and $C^{\left(\text{Sz}\right)}$. This is the key idea behind our $\beta$-Szeg\H{o}{} system to be introduced in section \ref{sec:beta_Szego}.
		\item The exponential growth of Sobolev norms (\ref{eq:truncated_Szego_exponential_growth_Sobolev}) is very surprising given the apparently impoverished structure of (\ref{eq:C_BE}). Interactions only through high modes have been completely eliminated, so that all the interactions must involve the lowest mode. Paradoxically, this enhances
the turbulent phenomena in $\mathcal{L}(1)$.
		\item In view of the exponential growth of Sobolev norms (\ref{eq:truncated_Szego_exponential_growth_Sobolev}) the bound (\ref{eq:truncated_Szego_global_bound}) is optimal.
		\item In this work, we will focus our attention on the properties of solutions in $\mathcal{L}(1)$. Dynamics in general $\mathcal{L}(D)$ is an open problem.
		\item The properties displayed here for the truncated Szeg\H{o}{} equation and the ones for the\\ $\alpha$-Szeg\H{o}{} equation showed in section \ref{subsec:alpha_Szego} are similar; however, we remark that the mode coupling structures of systems (\ref{eq:alpha_Szego_modes}) and (\ref{eq:resonant_eq_BE}) are very different. We shall return to further comparisons between these systems in our concluding section.
	\end{itemize}

	 \subsection{Lax pair, bounded Hamiltonian and $\mathcal{L}(D)$ invariant manifolds}
	 The local and global well-posedness of the truncated Szeg\H{o}{} equation for initial conditions $u_0 \in H_{+}^{s}\left(\mathbb{S}^1\right)$ with $s> 1/2$, as well as the exponential upper bound (\ref{eq:truncated_Szego_global_bound}), come from the results obtained in \cite{GG} for the Szeg\H{o}{} equation and in \cite{Xu} for the $\alpha$-Szeg\H{o}{} equation, essentially because the right-hand sides of (\ref{eq:resonant_eq_BE}) consist of subsets of terms that would have appeared in the case of the cubic \Sz{} equation. Specifically, after integrating (\ref{eq:truncated_Szego_eq}) in time
	 \begin{equation}
	 	u(t) = u_0 - i \int_{0}^{t}\left(\Pi(|u(s)|^2u(s))-S\Pi(|S^\dagger u(s)|^2 S^\dagger u(s))\right)ds,
	 \end{equation}
	 we have to use the estimates (where $\|u\|_{W} := \sum |\alpha_n|$ is the Wiener norm)
	 \begin{align*}
	 	& \|\Pi\left(|u|^2u\right)\hspace{-0.05cm}-\hspace{-0.05cm}S\Pi\left(|S^\dagger u|^2 S^\dagger u\right)\hspace{-0.15cm}\|_{H^{s}}  \hspace{-0.05cm} \leq \hspace{-0.05cm} \|\Pi\left(|u|^2u\right)\hspace{-0.15cm}\|_{H^{s}}\hspace{-0.05cm}+\hspace{-0.05cm}\|S\Pi\left(|S^\dagger u|^2 S^\dagger u\right)\hspace{-0.15cm}\|_{H^{s}}  \hspace{-0.05cm},\\
		& \|\Pi\left(|u|^2u\right)\hspace{-0.15cm}\|_{H^{s}} \leq c \|u\|_{L^\infty}^2 \|u\|_{H^s} \leq c \|u\|_{W}^2\|u\|_{H^s},\\
		& \|SS^\dagger u\|_{L^{\infty}} = \| u - (u|1)\|_{L^{\infty}} \leq \|u\|_{L^{\infty}}+\|(u|1)\|_{L^{\infty}} \leq 2\|u\|_{L^{\infty}},\\
		& \|S\Pi\left(|S^\dagger u|^2S^\dagger u\right)\hspace{-0.15cm}\|_{H^{s}} \hspace{-0.05cm} \leq \hspace{-0.05cm} \|\Pi\left(|SS^\dagger u|^2 SS^\dagger u\right)\hspace{-0.15cm}\|_{H^{s}} \hspace{-0.05cm} \leq \hspace{-0.05cm} \hat{c} \| u\|_{L^\infty}^2 \|u\|_{H^s} \hspace{-0.05cm} \leq \hspace{-0.02cm} \hat{c} \|u\|_{W}^2 \|u\|_{H^s},
		\end{align*}
		the Brezis-Gallou\"et type estimate\footnote{A proof of this estimate can be found in the appendix of the first reference of \cite{GG}.}
		$$ \|u\|_{L^{\infty}} \leq C \|u_0\|_{H^{\frac{1}{2}}} \left(\log \left(2 + \frac{\|u\|_{H^{s}}}{\|u_0\|_{H^{\frac{1}{2}}}}\right)\right)^{1/2},
		$$
		the upper bound for the Wiener norm for $s>1$ \cite{Xu},
		\begin{equation} \underset{t\in\mathbb{R}}{\text{sup}} \|u\|_{W} \leq \tilde{c}_s \|u_{0}\|_{H^s},
		\end{equation}
	 and the Gronwall lemma. Note that the bound on the Wiener norm is proved in \cite{Xu} only relying on the fact that $K_u$ is a Lax operator. Since that holds true for our current model, as we shall immediately demonstrate, the proof of \cite{Xu} directly translates to our case.

	 To establish the Lax pair (\ref{Laxtr}), we recall the ``Szeg\H{o}{}-minus-shifted-Szeg\H{o}{}" structure of the truncated Szeg\H{o}{} system. Then, one can effectively reuse the Lax pairs (\ref{LaxSz}).
	 \begin{proposition}
		Let $u\in C\left(\mathbb{R},H^{s}\left(\mathbb{S}^1\right)\right)$ for $s> 1/2$, if $u$ solves the truncated Szeg\H{o}{} equation (\ref{eq:truncated_Szego_eq}), then $\left(K_u, D_u\right)$ with $D_u = C_u-B_{S^\dagger u}$ satisfy
		\begin{equation}
			\frac{dK_{u}}{dt} = \left[D_u, K_u\right],
\label{DK}
		\end{equation}
i.e., they provide a Lax pair.
	 \end{proposition}
	 \begin{proof}
	 	For this proof, we will use the Lax pairs (\ref{LaxSz}) for the cubic Szeg\H{o}{} equation. We consider $dK_u/dt$ evaluated with the equation of motion (\ref{eq:truncated_Szego_eq}). Since $K_u$ is linear in $u$, we have
	 	\begin{equation}
	 	\frac{dK_u}{dt}=-i K_{\Pi(|u|^2u)}+iK_{S\Pi(|S^\dagger u|^2 S^\dagger u)}.
	 	\end{equation}
	 	The first term is simply what one would have gotten for the cubic \Sz{} equation{} itself, and hence it equals $[C_u,K_u]$ by (\ref{LaxSz}). The second term can be written as follows, taking into account (\ref{LaxSz}),
	 	\begin{equation}
	 	-iK_{S\Pi(|S^\dagger u|^2 S^\dagger u)}=-iH_{S^\dagger S\Pi(|S^\dagger u|^2 S^\dagger u)}= -iH_{\Pi(|S^\dagger u|^2 S^\dagger u)}=[B_{S^\dagger u}, H_{S^\dagger u}]=[B_{S^\dagger u}, K_{u}].
	 	\end{equation}
	 	Hence,
	 	\begin{equation}
	 	\frac{dK_u}{dt}=[C_u-B_{S^\dagger u},K_u],
	 	\end{equation}
	 	and a Lax pair for the truncated Szeg\H{o}{} system is given by $K_u$ and
	 	\begin{equation}
		 	 D_u=C_u-B_{S^\dagger u}=iT_{|S^\dagger u|^2}-iT_{|u|^2}=-iT_{|u|^2-|S^\dagger u|^2}.
	 	 \end{equation}
(We give an alternative verification of the Lax pair in appendix A using the language of the mode space. This derivation is lengthier but more straightforward and self-contained.)
	 \end{proof}

	 An important fact that we must clarify is whether the Hamiltonian (\ref{eq:Hamiltonian_truncated_Szego}) is bounded from below. While the Hamiltonian for the cubic \Sz{} equation{} is positive, removing interaction coefficients to obtain $C^{\left(tr\right)}$ could undermine the existence of a lower bound. This does not in
	 fact occur:
	 \begin{proposition}
	 	Given initial conditions $\{\alpha_n(0)\}$ such that the conserved quantities N and E defined in (\ref{eq:N_consered_quantity}-\ref{eq:E_consered_quantity}) are finite, the Hamiltonian (\ref{eq:Hamiltonian_truncated_Szego}) is bounded from below by
	 	\begin{equation}
		 	\mathcal{H}_{tr} \geq - N E.
		 	\label{eq:H_bounde_truncated_Szego}
	 	\end{equation}
	 \end{proposition}
	\begin{proof}
		The Hamiltonian (\ref{eq:Hamiltonian_truncated_Szego}) is rewritten in the following form
		\begin{equation}
		\mathcal{H}_{tr}=\frac{|\alpha_0|^4}{4}+|\alpha_0|^2\sum_{n=1}^\infty |\alpha_n|^2+ \frac{1}{2}\sum_{n=2}^\infty \sum_{k=1}^{n-1} \left(\bar{\alpha}_0 \bar{\alpha}_{n} \alpha_k \alpha_{n-k}+ \bar{\alpha}_k \bar{\alpha}_{n-k}\alpha_0 \alpha_{n}\right).
		\label{Htr}
		\end{equation}
		Note that
		$$
		\sum_{n=2}^\infty \sum_{k=1}^{n-1} \left(\bar{\alpha}_0 \bar{\alpha}_{n} \alpha_k \alpha_{n-k}+ \bar{\alpha}_k \bar{\alpha}_{n-k}\alpha_0 \alpha_{n}\right)=\sum_{n=2}^\infty\left[\Big|\alpha_0 \alpha_{n}+\sum_{k=1}^{n-1} \alpha_k \alpha_{n-k}\Big|^2-|\alpha_0\alpha_n|^2-\Big|\sum_{k=1}^{n-1} \alpha_k \alpha_{n-k}\Big|^2\right],
		$$
		and also that
		$$
		\sum_{n=2}^\infty |\alpha_0\alpha_n|^2 \le |\alpha_0|^2\sum_{n=1}^\infty |\alpha_n|^2.
		$$
		At the same time, by the Cauchy-Schwartz inequality,
		$$
		\Big|\sum_{k=1}^{n-1} \alpha_k \alpha_{n-k}\Big|^2\le \sum_{k=1}^{n-1} |\alpha_k|^2 |\alpha_{n-k}|^2 \sum_{l=1}^{n-1} 1=(n-1) \sum_{k=1}^{n-1} |\alpha_k|^2 |\alpha_{n-k}|^2= \sum_{k=1}^{n-1} (2k-1)|\alpha_k|^2 |\alpha_{n-k}|^2.
		$$
		Hence,
		\begin{align*}
		&\sum_{n=2}^\infty \Big|\sum_{k=1}^{n-1} \alpha_k \alpha_{n-k}\Big|^2\le \sum_{k=1}^{\infty} (2k-1)|\alpha_k|^2  \sum_{l=1}^{\infty}|\alpha_{l}|^2\\
&\hspace{3.5cm}=(2E-N+|\alpha_0|^2)(N-|\alpha_0|^2)=2NE-2E|\alpha_0|^2-(N-|\alpha_0|^2)^2.
		\end{align*}
		Combining everything together, we obtain the bound (\ref{eq:H_bounde_truncated_Szego}).
	\end{proof}

	As the existence of invariant manifolds $\mathcal{L}\left(D\right)$ is linked to the Lax operator $K_u$, they are predictably respected by the truncated Szeg\H{o}{} evolution:
\begin{proposition}
	For every positive integer $D$ there exists a complex manifold $\mathcal{L}(D)$, given in (\ref{eq:L_odd}), that is invariant under the evolution of the truncated Szeg\H{o}{} equation (\ref{eq:resonant_eq_BE}).
\end{proposition}
\begin{proof}
	Since our $K_u$ is exactly the same as for the cubic Szeg\H{o}{} and $\alpha$-Szeg\H{o}{} equations, we could simply verbatim recapitulate the analysis of \cite{Xu}.
Instead of reproducing this proof, we shall sketch here an elementary proof that remains valid for as long as no two simple poles of $u$ collide. We are going to show that the truncated Szeg\H{o}{} equation reduces to $2D+1$ ODEs for $2D+1$ variables, making the restriction to such manifolds consistent with the evolution, when the initial conditions are restricted to $\mathcal{L}(D)$ and assuming that $u(t=0,z)$ only has simple poles. The Fourier coefficients of $u(z)$ are then of the form
	\begin{equation}
		\alpha_{n \geq 1}(t) = \sum_{k=1}^{D}c_k(t) p_k(t)^n, \qquad \hspace{-0.5cm} \alpha_0(t) = b(t) +  \sum_{k=1}^{D}c_k(t), \quad \hspace{-0.3cm} |p_k|<1 , \quad \hspace{-0.3cm} p_i\neq p_j.
		\label{eq:L_D_single_poles}
	\end{equation}

	We start our analysis with the equations for $\alpha_{n \geq 1}$, returning to the equation for $\alpha_0$ at the end. The left-hand side of (\ref{eq:resonant_eq_BE})
		reduces to $D$ terms of the form $i \left(\dot{c}_i + n c_i \frac{\dot{p}_i}{p_i}\right) p_i^n$. Now we are going to show that the right-hand side is decomposed into $D$ terms with the same structure; namely, $p_i^n$ times a linear function of $n$. After substituting (\ref{eq:L_D_single_poles}) on the right-hand side of (\ref{eq:resonant_eq_BE}), we can show that the second sum is decomposed into terms of the form $A_i(b,p,c)p_i^n$ where $A_i(b,p,c)$ do not depend on $n$. The first sum has terms of the form
		\begin{equation}
			B_{ij}(b,c,p)\sum_{k=0}^{n}p_i^{k}p_j^{n-k}.
		\end{equation}
		When $i = j$ it becomes $B_{jj}(b,p,c) (n+1) p_j^n$ and when $i\neq j$ (remember that $p_i\neq p_j$ and $|p_k| < 1$)
		\begin{equation}
		B_{ij}(b,c,p)\sum_{k=0}^{n}p_i^{k}p_j^{n-k} = B_{ij}(b,p,c) \frac{p_i^{n+1}-p_j^{n+1}}{p_i-p_j}.
		\end{equation}
		As a consequence, gathering all the terms with $p_i^{n}$ on the right-hand side, we get $D$ terms whose $n$-dependence is of the form $p_i^{n}$ times a linear function of $n$. Matching the left-hand side and the right-hand side, we obtain $2D$ equations for $2D+1$ variables. The remaining equation, the one for $b(t)$, comes from expressing $\dot b$ using the equation of motion for ${\alpha}_0$ and the $2D$ equations for $\dot p_i$ derived above.

A more complete (but less elementary) proof that remains valid even if poles collide can be given based on the properties of $K_u$ as in \cite{Xu}.
	\end{proof}

	\subsection{Explicit blow-up in $\mathcal{L}(1)$}\label{subsec:TR_Szego_3d-manifold}

Having described the structure of invariant manifolds, we shall now focus on the dynamics in $\mathcal{L}(1)$, and demonstrate strong explicit turbulent behaviors there, which is a proof of (\ref{eq:truncated_Szego_exponential_growth_Sobolev}). To this end, solutions in $\mathcal{L}(1)$ are parametrized as
	\begin{equation}
	\alpha_0(t) = b(t), \qquad \alpha_{n\geq 1}(t) = \left(b(t)p(t) + a(t)\right)p(t)^{n-1},
	\label{eq:ansatz_3d}
	\end{equation}
	where $b,\ a$ and $p$ are complex functions of time. With this ansatz, the equations of motion in (\ref{eq:resonant_eq_BE}) are reduced to a coupled system of ODEs given by
	\begin{align}
	i \dot{p} = & \left(N - \left(1-|p|^2\right)E\right) p + a\bar{b}, \label{eq:truncated_szego_pdot}\\
	i \dot{b} = & (N+E)b + E a \bar{p}, \\
	i \dot{a} = & (N-E) a - E  |p|^2 b p, \label{eq:truncated_szego_adot}
	\end{align}
	where we have made use of the following expressions for the conserved quantities (\ref{eq:H_cq}-\ref{eq:E_consered_quantity})
	\begin{equation}
	E = \frac{|bp+a|^2}{\left(1-|p|^2\right)^2}, \qquad N = |b|^2 + \left(1 - |p|^2\right)E,
	\label{eq:E_J_3d_ansatz}
	\end{equation}
	\begin{equation}
	\mathcal{H}_{tr} = \frac{1}{4}\left(N^2 + 2 E N - 3 E^2 + 2E S\right), \qquad \text{with } S = (N+E)|p|^2 + \frac{E}{2} |p|^4 +  (a \bar b \bar p + \bar a b p).
	\label{eq:H_S}
	\end{equation}
	Note that the conservation of $N,\ E$ and $\mathcal{H}_{tr}$ implies the conservation of $S$. Through these expressions for the conserved quantities, the evolution of $|\alpha_n(t)|^2$ can be written in terms of $|p(t)|^2$ as
	\begin{equation}
	|\alpha_0(t)|^2 = N - \left(1 - |p(t)|^2\right)E, \qquad |\alpha_{n\geq 1}(t)|^2 = \left(1 - |p(t)|^2\right)^2 E |p(t)|^{2n-2}.
	\label{eq:alpha_n_as_N_E_x}
	\end{equation}
	 Therefore, our study will be focused on the equation for $|p(t)|^2$.
	After elementary algebra utilizing (\ref{eq:E_J_3d_ansatz}-\ref{eq:H_S}), and using the notation $x(t) := |p(t)|^2$, this equation can be expressed in the form
	\begin{equation}
	\dot{x}^2 + c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4 x^4 = 0,
	\label{eq:xdot_eq_1}
	\end{equation}
	with the coefficients
	\begin{equation*}
	c_4 =  -\frac{7}{4}E^2, \qquad c_3 =  3E^2 - N E, \qquad c_2 =  N^2 + 2 N E - 3 E^2 + 3E S,
	\end{equation*}
	\begin{equation}
	c_1 =  4E^2 + 2 N S - 4NE - 6 E S, \qquad c_0 =  S^2.
        \label{Veffc1}
	\end{equation}
	This equation can be understood as the energy conservation for zero energy trajectories of a quartic nonlinear oscillator, which can be manifested by rewriting it as
	\begin{equation}
	\dot{x}^2 + V_{\text{eff}}(x) = 0,
	\label{eq:xdot_eq_2}
	\end{equation}
with $V_{\text{eff}}$ read off  (\ref{eq:xdot_eq_1}-\ref{Veffc1}).
	Consequently, given an initial condition, through a standard analysis of $V_{\text{eff}}(x)$ we can determine whether $|\alpha_n(t)|^2$ is static, periodic or whether Sobolev norms with $s > 1/2$ are unbounded. See fig.~\ref{fig:shapes_Veff} for illustrative pictures of $V_{\text{eff}}(x)$ associated with each of these behaviors.

	\begin{figure}[t!]
		\centering
		\begin{subfigure}[b]{0.5\textwidth}
			\centering
			\includegraphics[width=8cm]{Potential_1.pdf}
			\caption{\small Stationary solution.}
			\label{fig:shapes_Veff_a}
		\end{subfigure}%
		\begin{subfigure}[b]{0.5\textwidth}
			\centering
			\includegraphics[width=8cm]{Potential_2.pdf}
			\caption{\small Periodic solution.}
			\label{fig:shapes_Veff_b}
		\end{subfigure}%

		\begin{subfigure}[b]{0.5\textwidth}
			\centering
			\includegraphics[width=8cm]{Potential_3.pdf}
			\caption{\small Infinite-time blow-up solution.}
			\label{fig:shapes_Veff_c}
		\end{subfigure}%
		\begin{subfigure}[b]{0.5\textwidth}
			\centering
			\includegraphics[width=8cm]{Potential_4_v2.pdf}
			\caption{\small Finite-time blow-up solution.}
			\label{fig:shapes_Veff_d}
		\end{subfigure}%
		\caption{\small The shapes of the effective potential for different sets of initial conditions, corresponding to different dynamical regimes of our system. The shaded areas do not correspond to valid configurations of the dynamical variables (they were included to provide a more complete picture of $V_{\text{eff}}(x)$).  The center of the black circle denotes the origin $(0,0)$ and $x_{\text{min}}$ ($x_{\text{max}}$) represents the minimum (maximum) value reached by $x(t)$. Dynamics of the type $(a),\ (b)$ and $(c)$ actually occurs in the evolution of the truncated Szeg\H{o}{} equation, while $(d)$ is impossible due to the upper bound (\ref{eq:truncated_Szego_global_bound}), and the plot is crossed out in red to highlight this fact. (One should view $(d)$ as an illustration of an imaginary shape of $V_{\text{eff}}(x)$ for which a finite-time blow-up would emerge.)}
		\label{fig:shapes_Veff}
	\end{figure}

	We shall now construct explicit initial configurations within the ansatz (\ref{eq:ansatz_3d}) whose evolution displays unbounded Sobolev norms. The exponential bound (\ref{eq:truncated_Szego_global_bound}) prevents any finite-time blow-up, a feature that is reflected in the potential as $V_{\text{eff}}(1)\geq 0$ and $V_{\text{eff}}(1)=0\Leftrightarrow V_{\text{eff}}'(1)~=~0$. The only kind of turbulence present is in the  form of unbounded Sobolev norms growth over an infinite range of time. The corresponding type of the effective potential can be seen in fig.~\ref{fig:shapes_Veff_c}. This set of configurations must satisfy $V_{\text{eff}}(1)=0$, a condition that, in terms of the conserved quantities, takes the form
	\begin{equation}
		3E-2(N+S) = 0
	\end{equation}
	and in terms of the dynamical variables in $\mathcal{L}(1)$,
	\begin{equation}
		2|b+a\bar{p}|^2 = |bp+a|^2(1+|p|^2).
	\end{equation}
	The solution is
	\begin{equation}
	a = b p \left(1 - e^{i\lambda} \sqrt{2 \left(1+ \frac{1}{|p|^2} \right)}\right) \qquad \text{with }  \lambda \in \mathbb{R},
	\label{eq:a0_general_ID}
	\end{equation}
	indicating that only initial conditions within a lower-dimensional submanifold in $\mathcal{L}(1)$ exhibit unbounded Sobolev norm growth. This situation is directly parallel to the $\alpha$-Szeg\H{o}{} equation \cite{Xu}.
	Assuming (\ref{eq:a0_general_ID}), the effective potential of (\ref{eq:xdot_eq_2}) takes the form
	\begin{equation}
	V_{\text{eff}}(x) = - |b(0)|^4 F \left(1-x\right)^2 (x_{\text{min}} - x) (c - x),
	\end{equation}
	where $c,\ x_{\text{min}}$ and $F$ are functions of $|p(0)|^2$ and $\lambda$ satisfying the bounds $F>0$, $0\leq x_{\text{min}} < 1$ and $c < x_{min}$ for $0\leq|p(0)|^2 < 1$. Furthermore, $x_{\text{min}}$ denotes the minimum value of $x(t)$, and $c$, another real zero of $V_{\text{eff}}(x)$. The solution of equation (\ref{eq:xdot_eq_2}) subject to this structure of $V_{\text{eff}}(x)$ is
	\begin{equation}
	x(t) = \frac{\left(c - x_{\text{min}}\right)\cosh{\left(\omega t + \phi\right)} - c - x_{\text{min}} + 2 c x_{\text{min}}}{\left(c - x_{\text{min}}\right)\cosh{\left(\omega t + \phi\right)} +c + x_{\text{min}} - 2}, \quad \text{with} \quad \omega = |b(0)|^2\sqrt{F(1-c)(1-x_{\text{min}})}
	\end{equation}
	and $\phi$ such that $x(0) = |p(0)|^2$. The inequalities $c< x_{\text{min}}$, $0\leq x_{\text{min}} < 1$ and $F > 0$ guarantee that if $|b(0)|\neq 0$, then $\omega>0$ and also that $(c - x_{\text{min}})\neq 0$. Hence, $x(t)\equiv |p(t)|^2$ exponentially approaches $1$ at late times:
	\begin{equation}
	|p(t)|^2 \sim 1 - \mathcal{O}\left(e^{-\omega t}\right).
	\end{equation}
	Then, using the expressions for $|\alpha_n(t)|^2$ in terms of $N,\ E$ and $|p(t)|^2$ given in (\ref{eq:alpha_n_as_N_E_x}), Sobolev norms for $s > 1/2$ have the following exponential growth at late times
	\begin{equation}
	\|u(t)\|_{H^{s}} \underset{t\to\infty}{\simeq} e^{(2s-1)\frac{\omega}{2} |t|}.
	\label{eq:u_H_s_exp_growth_truncated_szego}
	\end{equation}
	In fig.~\ref{fig:collapse_wave_function}, we show the evolution of
	\begin{equation}
	v(t,\theta) = \sum_{n=1}^{\infty} \alpha_n(t) e^{in\theta}
	\label{eq:v_function}
	\end{equation}
for one of the initial conditions displaying such exponential growth of Sobolev norms. This function shows a concentration phenomenon, meaning that $v(t,\theta)$ tends to 0 at large $t$ everywhere, except for a single value of $\theta$, where it tends to $4E$. Evidently, such behavior must incur unbounded growth of the derivatives of $v$, which is in turn reflected in the growth of Sobolev norms. Note that $v(t,\theta)$ itself could not possibly blow up by conservation of $N$ and $E$. The blow-up necessarily enters through the derivatives of $v$.

	\begin{figure}[t!]
	\centering
	\begin{subfigure}[b]{0.5\textwidth}
		\centering
		\includegraphics[width=8cm]{Collapse_Wave_Function_1_labels_v2.pdf}
		\caption{\small $\left(\theta,|v(t,\theta)|^2\right)$}
		\label{fig:collapse_wave_function_1}
	\end{subfigure}%
	\begin{subfigure}[b]{0.5\textwidth}
		\centering		\hspace{2cm}\includegraphics[width=8cm]{Collapse_Wave_Function_derivative_v2.pdf}
		\caption{\small $\left(\theta,|\partial_\theta v(t,\theta)|^2\right)$}
		\label{fig:collapse_wave_function_2}
	\end{subfigure}%
		\caption{\small Concentration of the function $v(t,\theta)$ given by (\ref{eq:v_function}) and its derivative at a point for initial data within the family (\ref{eq:a0_general_ID}) characterized by exponential growth of Sobolev norms.  Fig.~\ref{fig:collapse_wave_function_1} shows that $|v(t,\theta)|^2$, despite becoming concentrated at a point, remains finite, converging to $4E$ there (black dashed line). Fig.~\ref{fig:collapse_wave_function_2} shows that the first derivative of $v(t,\theta)$ also becomes concentrated at a point, but in contrast with $v(t,\theta)$, its value at this point is not bounded.}
		\label{fig:collapse_wave_function}
	\end{figure}

	\section{The $\beta$-Szeg\H{o}{} equation}\label{sec:beta_Szego}

	Since the truncated Szeg\H{o}{} equation inherits many properties of the cubic \Sz{} equation, including a common Lax operator $K_u$ and a hierarchy of finite-dimensional invariant manifolds, it is natural to ask whether an interpolating family can be constructed connecting these two equations, retaining such special properties. To explore this question, we define
	\begin{equation}
	C_{nmkl}^{\left(\beta\right)} = \begin{cases}
		1 & \text{if } n m k l = 0\\
		1-\beta & \text{if } n m k l \neq 0
	\end{cases}
	\label{eq:C_beta}
	\end{equation}
	with $\beta \in \mathbb{R}$. Note that these coefficients are simply the linear combination
	\begin{equation}
	C^{(\beta)}=\beta C^{(tr)}+(1-\beta)C^{(\text{Sz})}.
	\label{eq:C_beta_linear_combination}
	\end{equation}
	Therefore, for $\beta = 0$ and $1$ we recover the original systems $C^{(0)} = C^{\left(\text{Sz}\right)}$ and $C^{(1)}~=~C^{\left(tr\right)}$. Additionally, in the limits $\beta \to \pm \infty$, one can rescale  $C^{\left(\beta\right)}$ by $\pm 1/\beta$ to obtain a ``shifted" Szeg\H{o}{} system (the cubic \Sz{} equation{} with  $\{\alpha_0,\alpha_1,...\}$ replaced by $\{\alpha_1,\alpha_2,...\}$). Using the standard Szeg\H{o}{} projector and  the shift operators, we can represent the  resonant system (\ref{eq:flow_eq_linear_spectrum}) with the interaction coefficients (\ref{eq:C_beta}) in position space as
	\begin{equation}
	i\partial_t u = \Pi(|u|^2u)-\beta S\Pi(|S^\dagger u|^2 S^\dagger u).
	\label{eq:beta_Szego_eq}
	\end{equation}
	We shall call this system the $\beta$-Szeg\H{o}{} equation in analogy to the $\alpha$-Szeg\H{o}{} equation, although there are important differences between these two deformations; see section \ref{sec:Discussion} for further discussion. The main properties of the $\beta$-Szeg\H{o}{} equation, which we will analyze below, are:
	\begin{itemize}
		\item These systems possess one Lax pair
		\begin{equation}
		\frac{dK_{u}}{dt} = \left[C_u-\beta B_{S^\dagger u}, K_u\right].
		\label{eq:Lax_pair_beta_Szego}
		\end{equation}
		\item Sobolev norms of solutions of (\ref{eq:beta_Szego_eq}) subject to initial conditions $u_0 \in H_{+}^{s}\left(\mathbb{S}^1\right)$ with $s>1$, cannot grow faster than exponentially,
		\begin{equation}
			\|u(t)\|_{H^{s}} \leq \|u_0\|_{H^{s}} e^{C |t|}\qquad \text{with } C>0.
			\label{eq:beta_Szego_gobal_bound}
			\end{equation}
		\item There exist complex invariant manifolds $\mathcal{L}(D)$ given in (\ref{eq:L_odd}).
		\item For $\beta < 0$, the Sobolev norms for $u_0 \in \mathcal{L}(1)$ remain bounded,
		\begin{equation}
			\forall s \geq 0, \qquad \|u(t)\|_{H^{s}} \leq C, \qquad \text{with }\ C > 0.
			\label{eq:beta_Szego_bound_beta_less_zero}
		\end{equation}
		\item For $\beta > 0$, there exist $u_0 \in \mathcal{L}(1)$ such that the Sobolev norms with $s\hspace{-0.08cm} >\hspace{-0.08cm} \frac{1}{2}$ grow exponentially at late times,
		\begin{equation}
		\forall s > \frac{1}{2}, \qquad \|u(t)\|_{H^{s}} \underset{t\to\infty}{\simeq} e^{(2s-1)c|t|}.
		\label{eq:beta_Szego_exponential_bound}
		\end{equation}
		\item For $\beta \in (9,\infty)$, there exist $u_0 \in \mathcal{L}(1)$ such that Sobolev norms with $s\hspace{-0.08cm} >\hspace{-0.08cm} \frac{1}{2}$ have a polynomial growth at late times,
		\begin{equation}
		\forall s > \frac{1}{2}, \qquad \|u(t)\|_{H^{s}} \underset{t\to\infty}{\simeq} t^{(2s-1)}.
		\label{eq:beta_Szego_polynomial_bound}
		\end{equation}
		\item For $\beta \neq 1$, the $\beta$-Szeg\H{o}{} equation contains the cubic Szeg\H{o}{} equation as one of its invariant manifolds. It can be observed by restricting the initial conditions to odd modes (setting even modes to 0). Hence, if $\beta\neq 1$, the $\beta$-Szeg\H{o}{} equation has subsectors with all the properties displayed in section \ref{subsec:Szego}.
	\end{itemize}
	A majority of these properties straightforwardly arise from the fact that the $\beta$-Szeg\H{o}{} equation is a combination of the cubic Szeg\H{o}{} and the truncated Szeg\H{o}{} equations given by (\ref{eq:C_beta_linear_combination}). For the rest of our treatment, we shall focus on proving (\ref{eq:beta_Szego_bound_beta_less_zero}-\ref{eq:beta_Szego_polynomial_bound}), which requires explicit computations.\footnote{We shall not explore the dynamics in $u\in\mathcal{L}(D)$ for $D>1$. One may expect that these manifolds are integrable in the Liouville sense by analogy with \cite{GG,Xu,Thirouin}; namely, that they admit $2D+1$ conserved quantities in involution, the same as the number of dimensions. This is not a priori guaranteed by the Lax pair, and requires further analysis. The Lax pair provides $D$ conserved quantities, which in addition to the Hamiltonian and the $L^2$ norm give $D+2$ conservation laws. When $D>1$, this number by itself is insufficient for Liouville integrability. For the cubic Szeg\H{o}{} equation, the remaining conserved quantities come from the second Lax pair (\ref{LaxSz}), while for the $\alpha$-Szeg\H{o}{} and the quadratic Szeg\H{o}{} equations they were found using different methods.}

	\subsection{Explicit blow-up in $\mathcal{L}(1)$}\label{subsec:Deformation_Szego_3d-manifold}

In order to  prove (\ref{eq:beta_Szego_bound_beta_less_zero}-\ref{eq:beta_Szego_polynomial_bound}) and provide explicit examples of such solutions,
we essentially repeat our previous analysis of the truncated Szeg\H{o}{} equation within $\mathcal{L}(1)$, but now at generic values of $\beta$. To this end, we write $\mathcal{L}(1)$ in the form (\ref{eq:ansatz_3d}), so that the equations of motion are reduced to
	\begin{align}
	i\dot{p} =& \left(N -\beta (1-|p|^2)E\right) p + a\bar{b},\\
	i\dot{b} =& (N+E)b + E a\bar{p},\\
	i\dot{a} = & \left(N - \beta E\right) a - \beta E |p|^2  b p.
	\end{align}
	While $\beta$ appears on the right-hand side of these equations, the equation for $\dot{x}$ does not explicitly depend on this parameter (remember the definition $x(t) := |p(t)|^2$)
	\begin{equation}
	i\dot{x} = \left(a\bar b\bar p - \bar a b p\right).
	\end{equation}
	Expressions for $N$ and $E$ in (\ref{eq:E_J_3d_ansatz}) are also $\beta$-independent, and the only dependence comes from the Hamiltonian, which can be reduced to the conserved quantity
	\begin{equation}
	S = (N+E) x + \frac{\beta}{2} E x^2 + (a\bar b \bar p + \bar a b p),
	\end{equation}
	as was done in (\ref{eq:H_S}). In analogy with (\ref{eq:xdot_eq_2}), we make use of an effective potential $V_{\text{eff}}(x)$ to analyze the evolution of $x(t)$. It is a quartic polynomial for a generic $\beta$, but for specific values of this parameter it can be reduced to a cubic polynomial, or for $\beta = 0$ $\left(C^{\left(\text{Sz}\right)}\right)$, to a quadratic polynomial. Therefore, the family of models $C^{(\beta)}$ displays different phenomena for different values of $\beta$ and $p(0)$. Focusing our attention on initial conditions potentially displaying unbounded Sobolev norms, we observe that $V_{\text{eff}}(1)=0\Rightarrow V_{\text{eff}}'(1)=0$ and also that $V_{\text{eff}}(1) \geq 0$, preventing any blow-up in finite time, in agreement with the bound (\ref{eq:beta_Szego_gobal_bound}). As in the case of the truncated Szeg\H{o}{} equation, turbulent solutions must satisfy $V_{\text{eff}}(1)=0$, a condition that, in terms of the conserved quantities, takes the form
	\begin{equation}
		(2+\beta)E - 2(N+S) = 0.
		\label{eq:S_func_N_E_beta}
	\end{equation}
	In terms of the dynamical variables,
	\begin{equation}
		2|b+a\bar p|^2 = \beta |bp+a|^2(1+|p|^2).
		\label{eq:constraint_V1_0_beta}
	\end{equation}
	It shows that, for $\beta < 0$, this conditions is not satisfied for nontrivial configurations and therefore, Sobolev norms of $u_0\in\mathcal{L}(1)$ remain bounded. For $\beta = 0$, the cubic Szeg\H{o}{} system, this constraint is only satisfied for $b = -a\bar{p}$, which corresponds to stationary solutions as we will see later. The case $\beta > 0$ is different:  (\ref{eq:constraint_V1_0_beta}) is solved by
	\begin{equation}
	a = b p \frac{\left(\left(1+|p|^2\right)\beta-2\right) +\sqrt{2\beta}e^{i\lambda} (1-|p|^2)\sqrt{1+\frac{1}{|p|^2}}}{(2-\beta)|p|^2 - \beta} \qquad \text{with } \lambda \in \mathbb{R},
	\label{eq:a_blow-up_beta}
	\end{equation}
	where certain combinations of $\beta$, $p$ and $\lambda$ lead to $\|u(t)\|_{H^{s}}\to\infty$ as $t\to \infty$ for $s>1/2$. Fig.~\ref{fig:phase_diagram_beta_x0} shows a sketch of the different regions in the $(\beta,p(0))$-plane with $\lambda = 0$ and $\pi$.
	\begin{figure}[t!]
		\centering
		\begin{subfigure}[b]{0.5\textwidth}
			\centering
			\includegraphics[width=8cm]{Phase_Diagram_beta_x0_lambda_0_v2.pdf}
			\caption{\small $\lambda = 0$}
			\label{fig:lambda_0}
		\end{subfigure}%
		\begin{subfigure}[b]{0.5\textwidth}
			\centering
			\includegraphics[width=8cm]{Phase_Diagram_beta_x0_lambda_pi_v2.pdf}
			\caption{\small $\lambda = \pi$}
			\label{fig:lambda_pi}
		\end{subfigure}%
		\caption{\small Regions in the $(\beta,\ p(0))$-plane with different dynamics of initial conditions subject to (\ref{eq:a_blow-up_beta}) with $|b|\neq 0$ and $\lambda = 0,\pi$. Blue areas consist of initial configurations for which the Sobolev norms are bounded. Green areas consist of initial configurations with exponential growth of $\|u(t)\|_{H^{s}}$ for $s>1/2$. Black solid lines represent the boundary region with a polynomial growth. Black dashed lines are placed at $\beta= 0$, namely $C^{\left(\text{Sz}\right)}$, and demarcate a transition between systems with bounded and unbounded Sobolev norms. The pictures are naturally extended to infinity to the left and to the right.}
		\label{fig:phase_diagram_beta_x0}
	\end{figure}

	We shall now focus the discussion on the case $\lambda=0$, where the equations are simple enough to extract explicit expressions and this is a good representative of the behavior of the initial conditions (\ref{eq:a_blow-up_beta}) for generic $\lambda$.  In this case, under condition (\ref{eq:a_blow-up_beta}), the potential $V_{\text{eff}}(x)$ becomes
	\begin{equation}
	V_{\text{eff}}(x) = - |b(0)|^4 F \left(1-x\right)^2\left(x_{0} - x\right)\left(c - x\right),
	\end{equation}
	where $x_0 = x(0)$ and $F$ and $c$ are functions of $p(0)$ and $\beta$. For $0<\beta < 16$ we find that $V_{\text{eff}}(x)< 0$ for $x_0 < x < 1$ and Sobolev norms $\|u(t)\|_{H^{s}}$ with $s>1/2$ have exponential growth. For $\beta \geq 16$ there are three possibilities depending on $\beta$ and $p(0)$:
	\begin{itemize}
		\item  The additional zero $c \notin \left[x_0,1\right]$. In this case, the Sobolev norms with $s>1/2$\\ $\|u(t)\|_{H^{s}} \simeq e^{\left(2s - 1\right)\omega|t|/2}$ at late times.
		\item The additional zero $c \in [x_0,1)$. In this case, all Sobolev norms are bounded.
		\item The additional zero $c = 1$ (at the threshold between the two previous behaviors). In this case, the Sobolev norms with $s>1/2$ grow as $\|u(t)\|_{H^{s}} \simeq t^{\left(2s-1\right)}$ at late times.
	\end{itemize}
	 It can be shown that the condition $c=1$ is solved by
	\begin{equation}
	x_0 = \frac{\left(\sqrt{\beta} - 4\right)^2}{\beta - 8}, \qquad \text{for }\ \beta \geq 16.
	\label{eq:x0_beta_polynomial}
	\end{equation}
	The explicit expressions for $x(t)$ as a function of time are then
	\begin{equation}
	x(t) = \begin{dcases}
		\frac{(c-x_0) \cosh \left(\omega t\right) - c - x_0 + 2 c x_0}{(c - x_0) \cosh\left(\omega t\right) + c + x_0 - 2} & \text{for } x_0 \in [0,1) \quad \text{if } 0 < \beta < 16 \\[15pt]
		\frac{(c-x_0) \cos \left(\omega t\right) - c - x_0 + 2 c x_0}{(c - x_0) \cos\left(\omega t\right) + c + x_0 - 2} & \text{for } x_0 \in \big{[}0,\frac{\left(\sqrt{\beta} - 4\right)^2}{\beta - 8}\bigg{)} \quad \text{if } \beta > 16 \\[15pt]
		\hspace{2.5cm}\frac{ \tilde{c} t^2 + x_0}{\tilde{c} t^2 + 1} & \text{for } x_0 = \frac{\left(\sqrt{\beta} - 4\right)^2}{\beta - 8} \quad \text{if } \beta \geq 16 \\[15pt]
		\frac{(c-x_0) \cosh \left(\omega t\right) - c - x_0 + 2 c x_0}{(c - x_0) \cosh\left(\omega t\right) + c + x_0 - 2} & \text{for } x_0 \in \left(\frac{\left(\sqrt{\beta} - 4\right)^2}{\beta - 8},1\right) \quad \text{if } \beta \geq 16
	\end{dcases}
	\end{equation}
	where $\omega = |b(0)|^2|\sqrt{F(1-c)(1-x_0)}|$ and $\tilde{c} = - |b(0)|^4 F (1-x_0)^2/4$ (with $\tilde{c} > 0$ for (\ref{eq:x0_beta_polynomial})).

	 As can be observed in fig.~\ref{fig:phase_diagram_beta_x0}, for $\lambda = \pi$, we reach to similar conclusions, but in this case the transition ($c=1$) between bounded and unbounded Sobolev norms is placed at
	\begin{equation}
	x_0 = \frac{\left(\sqrt{\beta} - 4\right)^2}{\beta - 8}, \qquad \text{for }\ \beta \in (9,16],
	\label{eq:x0_beta_polynomial_v2}
	\end{equation}
	and also displays polynomial growth of Sobolev norms. For generic $\lambda$, the scenario is the same: we find combinations of the parameters $\beta$ and $p(0)$ that lead to exponential growth of Sobolev norms, other combinations for which these norms are bounded, and a curve (depending on $\lambda$), separating the two previous behaviors, that displays polynomial growth of Sobolev norms. Note that the cases $\lambda= 0$ and $\pi$ show that $\beta\in(9,\infty)$ is sufficient for the existence of initial conditions $u_0\in\mathcal{L}(1)$ with a polynomial growth of Sobolev norms. This range of values of $\beta$ is also necessary. It can be seen after imposing the appropriate conditions over $V_{\text{eff}}(x)$; namely, $V_{\text{eff}}(1) = V_{\text{eff}}''(1) = V_{\text{eff}}(x_0) = 0$ and also $V_{\text{eff}}(x) < 0$ for $x\in(x_0,1)$ with $x_0 \in[0,1)$. As we explained above, the first condition is satisfied by (\ref{eq:S_func_N_E_beta}). For the second condition, after expressing $V_{\text{eff}}(x)$ in terms of $N$, $E$ and $x$ with (\ref{eq:S_func_N_E_beta}), we obtain the relation
		\begin{equation}
			N = (1\pm \sqrt{\beta})^2 E.
			\label{eq:N_func_E_beta}
		\end{equation}
		Due to (\ref{eq:S_func_N_E_beta}) and (\ref{eq:N_func_E_beta}), the above conditions involving $x_0$ are only satisfied for $\beta > 9$.

	\begin{figure}[t!]
		\centering
		\begin{subfigure}[b]{0.5\textwidth}
			\centering
			\includegraphics[width=8cm]{Veff_Limit_beta_0_plus_v2_p0_1over5.pdf}
			\caption{\small $\beta \geq 0$}
			\label{fig:Veff_Limit_beta_0_b}
		\end{subfigure}%
		\begin{subfigure}[b]{0.5\textwidth}
			\centering
			\includegraphics[width=8cm]{Veff_Limit_beta_0_minus_v2_p0_1over5.pdf}
			\caption{\small $\beta \leq 0$}
			\label{fig:Veff_Limit_beta_0_a}
		\end{subfigure}%
		\caption{\small Effective potential for the initial configuration (\ref{eq:a_blow-up_beta}) with $b = (2-\beta)p\bar{p} - \beta$, $p=1/5$. For positive $\beta$, the potential has zeros at $0$ and $1$; unlike negative $\beta$, where it has zeros at $0$ and at a value less than 1. These plots show that $V_{\text{eff}}(x)$ converges to the x-axis for $\beta \to 0$. This behavior has been observed for generic $|p|\in [0,1)$, indicating the existence of a family of stationary solutions at the threshold $\left(C^{\left(\text{Sz}\right)}\right)$ of models with bounded and unbounded Sobolev norms. The center of the black ring is placed at $(0,0)$ and the shaded areas do not correspond to valid configurations of the dynamical variables.}
		\label{fig:Veff_Limit_beta_0}
	\end{figure}

	As we can see in fig.~\ref{fig:phase_diagram_beta_x0}, the cubic \Sz{} equation{} forms a transition point between systems with bounded and unbounded Sobolev norms. It is natural to wonder which solutions of cubic \Sz{} equation{} separate these behaviors. Fig.~\ref{fig:Veff_Limit_beta_0} provides a visual illustration of the transition. Setting $\beta = 0$ and $b = -\bar p$, the initial condition (\ref{eq:a_blow-up_beta}) reduces to a family of stationary solutions for the cubic \Sz{} equation{} \cite{GG}. Its expression is more recognizable from the literature in terms of the generating function
	\begin{equation}
	u(t,z) = e^{-i t}\frac{\bar p - z}{1 - p z}, \qquad \text{with }\ u(t,z) = \sum_{n=0}^{\infty} \alpha_n(t) z^n.
	\label{eq:stationary_solution_Szego}
	\end{equation}
	This result makes us wonder whether one can find other modifications of the cubic \Sz{} equation{} such that (\ref{eq:stationary_solution_Szego}) changes into a turbulent solution.

	In addition to our analysis of the $\beta$-Szeg\H{o}{} equation, we present, in appendix B, an even bigger family of deformations of the cubic \Sz{} equation{} that retain some of its Lax integrability and invariant manifolds and exhibit unbounded Sobolev norm growth. The mode couplings are of the following form (and they can be freely combined with an arbitrary $\alpha$-deformation):
\begin{equation}
\begin{cases}
	C_{0000} = \gamma, &\\
	C_{n0n0} = \delta_1 + \delta_2 n, & \text{for } n\neq 0,\\
	C_{nmkl} = 1, & \text{other cases with } nmkl = 0,\\
	C_{nmkl} = 1-\beta, & nmkl \neq 0.
\end{cases}
\label{eq:delta_gamma_deformations_v2}
\end{equation}

It would be interesting to study more systematically which deformations of the cubic \Sz{} equation{} respect the invariant manifolds $\mathcal{L}\left(D\right)$ or admit a Lax pair based on the operator $K_u$, but we shall not pursue it here. Similar type of analysis was performed in \cite{BBE} for a related question, namely, which resonant systems of the form (\ref{eq:flow_eq_linear_spectrum}) respect another explicitly defined three-dimensional invariant manifold, different from $\mathcal{L}\left(1\right)$.

	\section{Discussion and outlook}\label{sec:Discussion}

We have presented a large family of modifications of the cubic \Sz{} equation{} beyond the $\alpha$-Szeg\H{o}{} equation of \cite{Xu} that retain its Lax pair structure
and a hierarchy of finite-dimensional dynamically invariant manifolds. A central role in this family is played by the truncated Szeg\H{o}{} system (\ref{eq:resonant_eq_BE}-\ref{eq:truncated_Szego_eq}), where a majority of the Fourier mode couplings present in the original cubic \Sz{} equation{} have been eliminated. The systems we have introduced can be explicitly analyzed within the simplest three-dimensional invariant manifold given by (\ref{eq:ansatz_3d}), and display a variety of turbulent cascades, including unbounded exponential or polynomial growth of Sobolev norms.
These cascades are stronger than what is seen in the original cubic \Sz{} equation, which is particularly striking for the truncated Szeg\H{o}{} equation, since naively, one would imagine that eliminating couplings
between different sets of Fourier modes should weaken rather than strengthen turbulent cascades. One is thus encouraged to rethink the role played
by mode couplings in turbulent phenomena.

	Our systems display parallels to other deformations of the cubic \Sz{} equation{} exhibiting unbounded Sobolev norm growth, such as the $\alpha$-Szeg\H{o}{} \cite{Xu} and the damped Szeg\H{o}{} \cite{GG3} equations; nevertheless, there are significant differences. We highlight the main differences between the $\alpha$-Szeg\H{o}{} and the $\beta$-Szeg\H{o}{} equations (similar remarks could be made about the damped Szeg\H{o}{} equation, which is further apart from our models):
	\begin{itemize}
		\item The $\alpha$-Szeg\H{o}{} model (\ref{eq:alpha_Szego_modes}) introduces a deformation in the linear part of the equation for the lowest mode, while the $\beta$-Szeg\H{o}{} model implements a modification of the cubic part, keeping the system within the resonant class  (\ref{eq:flow_eq_linear_spectrum}).
		\item The $\alpha$-deformation, or its extension in (\ref{eq:alpha_deformation_extension}), only explicitly affect the lowest mode. The $\beta$-deformation nontrivially modifies the equations of motion for all modes.
		\item After an appropriate rescaling, the $\alpha$-Szeg\H{o}{} equation is reduced to three relevant systems (\ref{eq:alpha_Szego_v2}) only depending on $\text{sgn}(\alpha)$. In the case of the $\beta$-Szeg\H{o}{} equation, changes in $\beta$ cannot be absorbed into rescaling, leaving an essentially continuous family of systems.
		\item No explicit solutions with polynomial growth of Sobolev norms are known for the $\alpha$-Szeg\H{o}{}, specifically for initial conditions in $\mathcal{L}\left(1\right)$. Such solutions are seen for the $\beta$-Szeg\H{o}{} system for some values of the parameters, see fig.~\ref{fig:phase_diagram_beta_x0}.
		\item The $\alpha$ and $\beta$ deformations can be implemented simultaneously, together with a few further deformations described in appendix B.
	\end{itemize}

We would like to make a further  brief digression that highlights, from a perspective rather different from our main treatment, the distinction between the $\alpha$-Szeg\H{o}{} equation and the $\beta$-deformations, in particular,
the truncated Szeg\H{o}{} equation. All of the systems we have considered here are Hamiltonian, and the standard procedure of {\it quantization} may be applied to such systems, according to the basic principles of quantum mechanics. The generalities of quantization of resonant systems of the form
(\ref{eq:flow_eq_linear_spectrum}) have been considered in \cite{EP}, with connections to the extensive lore of the quantum chaos theory \cite{QCh}. One then studies the corresponding quantum energy spectra, which are in turn
expected to display different distributions of distances between neighboring levels, depending on the integrability properties of the system. By doing so, one discovers that the cubic Szeg\H{o}{} system is extremely special, displaying a purely integer energy spectrum \cite{EP}, while a generic integrable system is expected
to display a Poissonian distribution of energy level distances \cite{QCh}. One is thus led to believe that the cubic \Sz{} equation{} possesses structure beyond ordinary integrability
(an explicit example of that is two inequivalent Lax pairs, as opposed to just one). If one turns on the $\alpha$-deformation (or the related $\delta$-deformations from appendix B), the quantum energy spectrum is no longer integer, but the distribution of energy level distances is nowhere close to Poissonian, with too many small energy level gaps. On the other hand, the truncated Szeg\H{o}{} system, in its quantum version, displays a perfectly
Poissonian distribution of energy level spacings, making it an excellent candidate for a generic Lax-integrable system within the resonant class  (\ref{eq:flow_eq_linear_spectrum}). This may make the truncated Szeg\H{o}{} system an attractive playground for quantum chaos and integrability studies of the type undertaken in \cite{EP}, quite far from the topics that initially stimulated our search for this system.

	In all deformations considered in our treatment, a special role is played by mode 0. One could ask what happens if this role is swapped with another mode.
For example, instead of (\ref{eq:C_beta}), we could consider the following modification of the interactions that do not involve mode 1:
	\begin{equation}
	C_{nmkl} = \begin{cases}
		1 & \text{if } (n-1) (m-1) (k-1) (l-1) = 0,\\
		1-\beta & \text{if } (n-1) (m-1) (k-1) (l-1) \neq 0.
	\end{cases}
	\label{eq:C_mode_1}
	\end{equation}
While we did not analyze the general properties of this system, we know that, after restricting the initial conditions to odd modes, the dynamics of (\ref{eq:C_mode_1}) is governed by $C^{\left(\beta\right)}$. Hence, for $\beta> 0$ it has solutions with unbounded Sobolev norms. This trivial argument can be extended to other similar deformations of the Szeg\H{o}{} equation anchored on other modes.

We conclude with a `teaser' regarding finite-time turbulent blow-up, a question that has indirectly led us to the main discoveries presented in this article.
Finite-time turbulent blow-up is known (from numerical simulations) to take place in extremely complicated (and physically interesting) resonant systems
emerging in Anti-de Sitter spacetimes \cite{FPU,CEV,BMR}. It would be very desirable to have a simple explicit resonant system in which this
phenomenon can be analyzed. Finite-time blow-up cannot happen in the cubic \Sz{} equation, or any of the other systems considered in this article, on account
of the exponential upper bounds on Sobolev norm growth. Our numerical experiments indicate, however, that finite-time turbulent blow-up does
happen in a few simple closely related systems within the resonant class (\ref{eq:flow_eq_linear_spectrum}). More specifically, we have considered
interaction coefficients of the form
	\begin{equation}
	C_{nmkl} = (n+1)^G(m+1)^G(k+1)^G(l+1)^G \qquad \text{with }G>0,
	\end{equation}
	(note that $G=0$ is the cubic \Sz{} equation), as well as a truncated version of these systems analogous to the truncated Szeg\H{o}{} equation (if all of the indices are non-zero, the corresponding $C$ is replaced by zero, otherwise it remains intact). We  have focused on numerical simulations of two-mode initial data
	\begin{equation}
	|\alpha_0(0)| \neq 0, \quad |\alpha_1(0)| \neq 0, \quad |\alpha_{n\geq 2}(0)| = 0
	\end{equation}
for the cases $G = 1/2$ and $1$.
	The blow-up manifests itself as the following asymptotic behavior at large $n$:
	\begin{equation}
	\alpha_{n \gg 1}(t) \sim c(t) n^{-\gamma} e^{-\rho(t)n},
	\label{eq:asymptotic_ansatz}
	\end{equation}
	with $\rho(t) \to 0$ as $t\to t^* < \infty$ and $\gamma=2$ ($\gamma = 5/2$) for $G=1/2$ (for $G=1$).
	Other systems that we have considered and observed similar phenomena are
	\begin{equation}
	C_{nmkl} = (n+m+1)^{G} \qquad \text{with } G> 0,
	\end{equation}
	as well as their truncated versions. These strong and simple blow-up behaviors beg for an analytic explanation.

	\section*{Acknowledgments}

We are indebted to Piotr Bizo\'n for collaboration on related subjects, numerious discussions, and comments on the manuscript.
This work has been supported by the Polish National Science Centre grant number 2017/26/A/ST2/00530, by CUniverse research promotion project at Chulalongkorn University (grant CUAASC), and by FWO-Vlaanderen through project G006918N.

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