II. FORMULATION OF THE PROBLEM III. APPROXIMATE ANALYTICAL SOLUTION 2 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 $^{16,17}$: 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 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 ): 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 where γ$_{0}$ , α$_{0}$ denote the averaged current density and loss factor, respectively: Without loss of generality C$_{2}$ can be set to zero while from the boundary condition (6) we find C$_{1}$ = h and Solving Eq. (8) for θ ( x ) we find and T = 2 π/ Ω. where 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: 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. A. Time-independent equation B. Time-dependent equation φ$_{xx}$ − φ$_{tt}$ − αφ$_{t}$ = sin φ − γ, (1) φ$_{xx}$ − φ$_{tt}$ − α ( x ) φ$_{t}$ + βφ$_{xxt}$ = sin φ − γ ( x ) , (2) φ$_{x}$ ( ± L/ 2) + βφ$_{xt}$ ( ± L/ 2) = h, (3) φ = φ$_{0}$ + ψ, (4) θ$_{xx}$ = α ( x )Ω + S ( x ) − γ ( x ) , (8) S ( x ) = 〈 sin( φ$_{0}$ + ψ ) 〉 T = (1 /T ) ∫ T 0 sin( φ$_{0}$ + ψ ) dt, (9) θ = ∫ x − L/ 2 [ ∫ η − L/ 2 [ α ( ξ )Ω + S ( ξ ) − γ ( ξ )] dξ ] dη + C$_{1}$x + C$_{2}$. (10) γ$_{0}$ = α$_{0}$ Ω + 1 L ∫ L/ 2 − L/ 2 S ( x ) dx, (11) γ$_{0}$ = 1 L ∫ L/ 2 − L/ 2 γ ( x ) dx, α$_{0}$ = 1 L ∫ L/ 2 − L/ 2 α ( x ) dx. (12) θ$_{xx}$ + ψ$_{xx}$ − ψ$_{tt}$ − α Ω − αψ$_{t}$ + βψ$_{xxt}$ = sin( φ$_{0}$ + ψ ) − γ. (5) θ$_{x}$ ( ± L/ 2) = h, (6) ψ$_{x}$ ( ± L/ 2) + βψ$_{xt}$ ( ± L/ 2) = 0 . (7) sin( φ$_{0}$ + ψ ) ≃ sin φ$_{0}$ + ψ cos φ$_{0}$. (13)