Genus two KdV soliton gases and their long-time asymptotics

This paper employs the Riemann-Hilbert problem and nonlinear steepest descent method of Deift-Zhou to provide a comprehensive analysis of the asymptotic behavior of the genus two Korteweg-de Vries soliton gases. It is demonstrated that the genus two soliton gas is related to the two-phase Riemann-Theta function as $x \to +\infty $ , and approaches zero as $x \to -\infty $ . Additionally, the long-time asymptotic behavior of this genus two soliton gas can be categorized into five distinct regions in the x - t plane, which from left to right are quiescent region, 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 to the multiphase Riemann-Theta function. A general discussion on the case of arbitrary genus N soliton gas is also presented.