Soliton Shielding of the focusing modified KdV equation

TL;DR

This paper investigates soliton gas solutions of the modified KdV equation, revealing a phenomenon called soliton shielding where spectral data simplifies under certain geometric and analytic conditions.

Contribution

It introduces the concept of soliton shielding in the mKdV equation and characterizes spectral data reduction for specific domains and soliton densities.

Findings

Soliton gas reduces to finite solitons in quadrature domains with analytic density.

In elliptical domains, spectral data simplifies to a segment between foci.

Initial data exhibits step-like oscillations with elliptic function behavior at infinity.

Abstract

We consider soliton gas solutions of the modified Korteweg-de Vries (mKdV) equation, where the point spectrum of the condensate is located within a bounded domain in the upper half-plane. We first demonstrate that when the domain is a quadrature and the soliton density is an analytic function, the corresponding deterministic soliton gas coincides with a finite number of solitons, which we call this effect soliton shielding. When the domain is an ellipse and the soliton density is analytic, the corresponding deterministic soliton gas reduces the spectral data to the segment joining the foci. The initial datum of this Cauchy problem is asymptotically step-like oscillatory, described by a periodic elliptic function as \( x \to +\infty \), and it vanishes exponentially fast as \( x \to -\infty \).