Genus two KdV soliton gases and their long-time asymptotics

TL;DR

This paper analyzes the long-time asymptotics of high-genus KdV soliton gases using Riemann-Hilbert problems, revealing five distinct asymptotic regions and introducing a novel method for high-genus Riemann surfaces.

Contribution

It introduces a new approach to analyze high-genus soliton gases and categorizes their long-time behavior into five distinct regions.

Findings

Two-genus soliton gas relates to two-phase Riemann-Theta functions as x→+∞.

The asymptotic behavior divides into five regions with different wave characteristics.

A new method for solving high-genus Riemann surface problems is developed.

Abstract

This paper employs the Riemann-Hilbert problem to provide a comprehensive analysis of the asymptotic behavior of the high-genus Korteweg-de Vries soliton gases. It is demonstrated that the two-genus soliton gas is related to the two-phase Riemann-Theta function as \(x \to +\infty\), and approaches to zero as \(x \to -\infty\). Additionally, the long-time asymptotic behavior of this two-genus soliton gas can be categorized into five distinct regions in the \(x\)-\(t\) plane, which from left to right are rapidly decay, modulated one-phase wave, unmodulated one-phase wave, modulated two-phase wave, and unmodulated two-phase wave. Moreover, an innovative method is introduced to solve the model problem associated with the high-genus Riemann surface, leading to the determination of the leading terms, which is also related with the multi-phase Riemann-Theta function. A general discussion on the case of arbitrary \(N\)-genus soliton gas is also presented.