5 FIG. 1: (Color online) Current-voltage characteristic calcu- lated analytically (solid line) and numerically (open circles) for γ ( x ) = const, L = 5, h = 3, α 0 = 0 . 2: (a) β = 0, (b) β = 0 . 01. The dashed line shows the first-order approxima- tion (24). FIG. 2: (Color online) the same as in Fig. 1 but for h = 5. tion (dashed line) is fairly accurate. On the other hand, the self-consistent solution (solid line) reproduces very accurately all the details of the numerical solution. As the next example, let us consider a more realistic case of a very long junction with small damping. Follow- ing Ref. 17 we choose L = 40, α$_{0}$ = 0 . 033, β = 0 . 035, and a slightly asymmetric current profile γ ( x ) depicted in Fig. 3. The bias electrode ( x$_{1}$ ≤ x ≤ x$_{2}$ ) is shorter than the total junction length. Consequently, the current distribution is assumed parabolic for x$_{1}$ ≤ x ≤ x$_{2}$ , and exponential in the unbiased tails ( x ≤ x$_{1}$ , x ≥ x$_{2}$ ). Contrary to the previous example, now we cannot use the analytical approximation (24). First, the current dis- tribution is x -dependent, what means that θ ( x ) departs from a simple linear dependence θ ( x ) = hx , and rele- vant integrals cannot be calculated analytically. Second, it appears (even for a uniform current profile) that the first-order approximation is not accurate enough for very long junctions, and consecutive iterations are necessary to obtain a self-consistent solution. Fig. 4 shows the I - V characteristics calculated for junc- tion parameters specified above and for two values of the external magnetic field h = 2 . 5 and h = 3 . 5. As before, open circles correspond to numerical simulations while the solid line represents a self-consistent solution. One can see that the agreement between numerical and analytical results is rather poor for h = 2 . 5. To ex- plain this discrepancy we should recall the main assump- tion (see Sec. II) that Eq. (5) can be separated into a time-independent part and a part sinusoidal in Ω t while neglecting higher-order harmonics. The Fourier analy- sis shows, however, that the anharmonic contribution to ψ ( t ) is rather large for h = 2 . 5. E.g. for γ$_{0}$ = 0 . 3 we find the content of the second harmonic to be about 35%. Physically, it means that the fluxon train is not sufficiently dense for h = 2 . 5 and its time-dependence although periodic is not strictly sinusoidal. Contrary to the case h = 2 . 5, for h = 3 . 5 we observe ex- cellent agreement between numerical and analytical data. Now the I - V characteristic is smooth and the Fiske steps disappeared completely as a combined result of surface losses and self-pumping effects. The Fourier analysis of ψ ( t ) shows now that the anharmonicity diminishes rather quickly with increasing h and for h = 3 . 5 we observe only GLYPH<s50>GLYPH<s46>GLYPH<s53> GLYPH<s51>GLYPH<s46>GLYPH<s48> GLYPH<s51>GLYPH<s46>GLYPH<s53> GLYPH<s52>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s55> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s57> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s32> GLYPH<s48> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s50>GLYPH<s46>GLYPH<s53> GLYPH<s51>GLYPH<s46>GLYPH<s48> GLYPH<s51>GLYPH<s46>GLYPH<s53> GLYPH<s52>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s55> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s57> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s52>GLYPH<s46>GLYPH<s48> GLYPH<s52>GLYPH<s46>GLYPH<s53> GLYPH<s53>GLYPH<s46>GLYPH<s48> GLYPH<s53>GLYPH<s46>GLYPH<s53> GLYPH<s54>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s57> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s49> GLYPH<s49>GLYPH<s46>GLYPH<s50> GLYPH<s49>GLYPH<s46>GLYPH<s51> GLYPH<s32> GLYPH<s48> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s52>GLYPH<s46>GLYPH<s48> GLYPH<s52>GLYPH<s46>GLYPH<s53> GLYPH<s53>GLYPH<s46>GLYPH<s48> GLYPH<s53>GLYPH<s46>GLYPH<s53> GLYPH<s54>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s57> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s49> GLYPH<s49>GLYPH<s46>GLYPH<s50> GLYPH<s49>GLYPH<s46>GLYPH<s51> GLYPH<s48> GLYPH<s40>GLYPH<s97>GLYPH<s41>