content_image/434023.jpg content_image/434031.jpg content_image/434022.jpg content_image/434024.jpg content_image/434032.jpg content_image/434033.jpg content_image/434025.jpg content_image/434027.jpg content_image/434028.jpg content_image/434030.jpg content_image/434026.jpg content_image/434029.jpg content_image/434034.jpg
## II. FORMULATION OF THE PROBLEM 

Fluxon dynamics in a long Josephson junction is usu- ally described by the following modified sG equation $^{8–12}$: 

where φ denote the quantum phase difference across the barrier, α is the loss factor, and γ is the bias current density. The spatial coordinate x has been normalized to the Josephson penetration depth λ$_{J}$ and the time co- ordinate t to the inverse plasma frequency ω − 1 0 , where λ$_{J}$ = (¯ h/ 2 µ$_{0}$edj$_{c}$ ) 1 / $^{2}$, ω$_{0}$ = (2 ej$_{c}$/ ¯ hC ) 1 / $^{2}$, j$_{c}$ is the critical current density and C denotes the junction capacitance per unit area. 

However, to describe more adequately a real physical situation, one can consider a more general form $^{16,17}$: 

<img src='content_image/434023.jpg'>

where β denotes the surface loss parameter and we as- sume both α and γ to be x -dependent, taking into ac- count both a spatial modulation of the loss factor due to the self-pumping and a nonuniform distribution of the current density along the junction. 

For the overlap geometry one can neglect the self-fields and assume simple open-circuit boundary conditions $^{9}$: 

where h = H$_{ext}$/j$_{c}$λ$_{J}$ , H$_{ext}$ denotes the external mag- netic field, and L ≫ 1 is the normalized junction length. Following Refs. 11–14,18 we look for an approximate 

solution in a form of a dense fluxon train traveling on a rotating background. Thus, a linearized solution of Eq. (2) can be written as 

where φ$_{0}$ = θ ( x ) + Ω t is the background term (linear in time) and ψ denotes a quasi-linear term (usually small) representing the motion of fluxons and plasma waves within the junction. 

Substituting Eq. (4) into Eq. (2) we find 

<img src='content_image/434031.jpg'>

We are interested in a steady-state, time periodic so- lution, thus the background frequency Ω should be equal to the fundamental frequency of the oscillating term ψ . Both experimental data and numerical simulations show that the output signal of a real FF oscillator is nearly sinusoidal $^{1,15}$. Thus, it is reasonable to assume the time- dependence of the term ψ to be harmonic in Ω t , and neglect any higher harmonics generated by the nonlin- ear term sin( φ$_{0}$ + ψ ). Consequently, Eq. (5) can be split into a time-independent part and a part oscillating with frequency Ω, while the boundary conditions (3) can be written separately for θ ( x ) and ψ ( x, t ): 

<img src='content_image/434022.jpg'>

<img src='content_image/434024.jpg'>

<img src='content_image/434032.jpg'>

<img src='content_image/434033.jpg'>

<img src='content_image/434025.jpg'>

## III. APPROXIMATE ANALYTICAL SOLUTION 

The method for solving Eq. (5) with the boundary con- ditions (6) and (7) is similar to that reported recently (see Ref. 18 for details). Thus, below we discuss briefly only the main points of the analysis. 

It can be easily shown that for ψ oscillating with fre- quency Ω, the nonlinear term sin( φ$_{0}$ + ψ ) contributes also to the time-independent part of Eq. (5), and such a con- tribution yields, in fact, the first-order approximation of the I - V characteristic of the junction. Thus, the time- independent equation can be written as: 

where 

<img src='content_image/434027.jpg'>

and T = 2 π/ Ω. 

Solving Eq. (8) for θ ( x ) we find 

<img src='content_image/434028.jpg'>

Without loss of generality C$_{2}$ can be set to zero while from the boundary condition (6) we find C$_{1}$ = h and 

where γ$_{0}$ , α$_{0}$ denote the averaged current density and loss factor, respectively: 

<img src='content_image/434030.jpg'>

It follows from the relation (4) that the time-averaged value of φ$_{t}$ , (being proportional to the constant voltage) is equal to the frequency Ω. Thus, the expression (11) can be regarded as the current-voltage ( I - V ) characteristic of the Josephson junction. One can see that the total current density γ$_{0}$ consists of a linear (Ohmic) part and a nonlinear contribution related to the Josephson current. 

For ψ sufficiently small we can write 

## A. Time-independent equation 

<img src='content_image/434026.jpg'>

<img src='content_image/434029.jpg'>

## B. Time-dependent equation 

<img src='content_image/434034.jpg'>

