content_image/412255.jpg content_image/412256.jpg content_image/412257.jpg
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about 10% of the second harmonic. 

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. 

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 

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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. 

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 

