GLYPH<s45>GLYPH<s50>GLYPH<s48> GLYPH<s45>GLYPH<s49>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s50>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s50> GLYPH<s49>GLYPH<s46>GLYPH<s52> GLYPH<s49>GLYPH<s46>GLYPH<s54> GLYPH<s49>GLYPH<s46>GLYPH<s56> GLYPH<s120> GLYPH<s50> GLYPH<s120> GLYPH<s120> GLYPH<s120> GLYPH<s49> GLYPH<s49>GLYPH<s46>GLYPH<s50> GLYPH<s49>GLYPH<s46>GLYPH<s52> GLYPH<s49>GLYPH<s46>GLYPH<s54> GLYPH<s49>GLYPH<s46>GLYPH<s56> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s50>GLYPH<s46>GLYPH<s50> GLYPH<s50>GLYPH<s46>GLYPH<s52> GLYPH<s50>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s48> GLYPH<s104> GLYPH<s61>GLYPH<s50>GLYPH<s46>GLYPH<s53> GLYPH<s104> GLYPH<s61>GLYPH<s51>GLYPH<s46>GLYPH<s53> GLYPH<s98> GLYPH<s45>GLYPH<s50>GLYPH<s48> GLYPH<s45>GLYPH<s49>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s50>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s48>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s48>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s48>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s48>GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s49>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s49>GLYPH<s50> GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s101> GLYPH<s102> GLYPH<s102> GLYPH<s40> GLYPH<s120> GLYPH<s41> GLYPH<s120> GLYPH<s48> GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s49> 6 FIG. 5: (Color online) Spatial distribution of α eff ( x ) calcu- lated analytically (solid line) and numerically (dotted line) for h = 3 . 5 and two values of the averaged current density γ 0 = 0 . 1 and γ 0 = 0 . 3. The remaining junction parameters are the same as in Fig. 4. FIG. 3: Normalized bias current distribution similar to that assumed in Ref. 17. L = 40, x 1 = − 9, x 2 = 5 . 5. FIG. 4: (Color online) Current-voltage characteristic calcu- lated analytically (solid line) and numerically (open circles) for two values of the external magnetic field: h = 2 . 5 and h = 3 . 5. The current profile is shown in Fig. 3 and the remain- ing junction parameters are: L = 40, α 0 = 0 . 033, β = 0 . 035. γ$_{dc}$ (3Ω) = 3 γ$_{dc}$ (Ω), thus a nonzero contribution to γ$_{eff}$ appears at 3Ω = Ω$_{g}$, giving rise to a step at Ω$_{b}$ = Ω$_{g}$ / 3. As shown in Subsection III.E, an effective loss factor α$_{eff}$ ( x ) is spatially modulated due to the self-pumping, and its average value becomes significantly larger than α$_{0}$ . Fig. 5 shows the spatial distribution of α$_{eff}$ plotted for γ$_{0}$ = 0 . 1 and γ$_{0}$ = 0 . 3. The junction parameters are assumed as before. i.e. L = 40, h = 3 . 5, α$_{0}$ = 0 . 033, β = 0 . 035. Now the dotted lines correspond to numeri- cal simulations and the solid lines denote self-consistent (analytical) solutions. One can see that the agreement is excellent, the self-consistent solution following closely all the details of the numerical simulations, taken here as a reference. It is clear that α$_{eff}$ generally grows with increasing value of γ$_{0}$ . E.g. for γ$_{0}$ = 0 . 3 we obtain an averaged value of α$_{eff}$ ( x ) about three times larger than the “un- pumped” value α$_{0}$ . Such an effect together with surface losses makes the I - V curve smooth, damping effectively the Fiske steps. about 10% of the second harmonic. It is interesting that the value of Ω$_{b}$ can also be deduced directly form Eq. (30). Indeed, noting that γ$_{dc}$ ( − Ω) = − γ$_{dc}$ (Ω) one can see the quadratic term in Eq. (30) to vanish, provided γ$_{dc}$ (Ω) is a linear function. However, for a nonlinearity starting at Ω = Ω$_{g}$ we find Another interesting feature which is visible for h = 3 . 5 is an additional small step at Ω$_{b}$ = 1 . 9. Such a step ap- pears in experimental results 17,26 and can be attributed to the self-pumping effect described in Subsection III.E. As shown in Ref. 26, the position of the current step fol- lows from a simple relation Ω$_{b}$ = Ω$_{g}$ / 3, where Ω$_{g}$ denotes the gap voltage. Assuming typical junction parameters 26 we find a normalized dimensionless gap voltage Ω$_{g}$ = 5 . 7, hence Ω$_{b}$ = 1 . 9. In this paper a semi-analytical approach has been sug- gested, making possible to solve a modified sG equa- tion (2) with both surface losses and self-pumping effects taken into account. The solution, as given by Eqs. (4), (10), and (18), consists of a rotating background and a unidirectional fluxon train accompanied by two plasma waves traveling in opposite directions with the velocity close to the critical value Ω /δ$_{0}$ ≃ ± 1. Having obtained an analytical solution to Eq. (2) one can determine some macroscopic, directly measurable quantities, such as the constant bias current and constant voltage, making pos- V. CONCLUSIONS