Extend soliton-gas statistical framework to nonzero backgrounds and small solitons (b=0)

Develop a statistical description—such as density of states, effective velocity, and collision rate—for left step-type KdV solutions generated via the continuous binary Darboux transformation in the cases of a nonzero background potential q(x,t) ≠ 0 and spectral measures supported on intervals [−a², −b²] with b=0, thereby extending the current understanding established for zero background q(x,t)=0 and b>0.

Background

The authors note that, in current soliton gas studies, statistical quantities are typically analyzed for left step-type KdV solutions on zero backgrounds with absolutely continuous nonnegative spectral measures supported on intervals [−a², −b²] where b>0. This setup fits naturally within their continuous binary Darboux framework and has been the focus of recent rigorous work.

They explicitly state that incorporating nonzero backgrounds q(x,t) ≠ 0 and allowing small solitons (b=0) into this statistical picture remains unresolved, marking the need to generalize existing results and methods to these more challenging regimes.

References

In fact, in the soliton gas community one is often interested in statistical quantities (density of states, effective velocity, collision rate, etc.) for left step-type KdV solutions of the form produced by Theorem \ref{MainThm} with q(x,t)=0 (zero background) and specific absolutely continuous measures dσ≥0 supported on intervals [−a², −b²] with b>0. The inclusion of q(x,t)≠0 (nonzero backgrounds) and b=0 (small solitons) into this picture is yet to be fully understood.

— Continuous binary Darboux transformation as an abstract framework for KdV soliton gases (2512.12495 - Rybkin, 13 Dec 2025) in Section 6 (Conclusion)