Genus two KdV soliton gases and their long-time asymptotics

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.