Rigorous analysis of large-space and long-time asymptotics for the short-pulse soliton gases

Abstract

We rigorously analyze the asymptotics of soliton gases to the short-pulse (SP) equation. The soliton gas is formulated in terms of a RH problem, which is derived from the RH problems of the $N$-soliton solutions with $N\to \infty$. Building on prior work in the study of the KdV soliton gas and orthogonal polynomials with Jacobi-type weights, we extend the reflection coefficient to two generalized forms on the interval $\left[{\eta }_1,{\eta }_2\right]$: $r_0(\lambda )={\left(\lambda -{\eta }_1\right)}^{{\beta }_1}{\left({\eta }_2-\lambda \right)}^{{\beta }_2}|\lambda -{\eta }_0{|}^{{\beta }_0}\gamma (\lambda )$, $r_c(\lambda )={\left(\lambda -{\eta }_1\right)}^{{\beta }_1}{\left({\eta }_2-\lambda \right)}^{{\beta }_2}{\chi }_c(\lambda )\gamma (\lambda )$, where $0<{\eta }_1<{\eta }_0<{\eta }_2$ and ${\beta }_j>-1$ ($j=0,1,2$), $\gamma (\lambda )$ is continuous and positive on $\left[{\eta }_1,{\eta }_2\right]$, with an analytic extension to a neighborhood of this interval, ${\chi }_c(\lambda )=1$ for $\lambda \in \left[{\eta }_1,{\eta }_0\right)$ and ${\chi }_c(\lambda )=c^2$ for $\lambda \in \left({\eta }_0,{\eta }_2\right]$, where $c>0$ with $c\ne 1$. The asymptotic analysis is performed using the steepest descent method. A key aspect of the analysis is the construction of the $g$-function. To address the singularity at the origin, we introduce an innovative piecewise definition of $g$-function. To establish the order of the error term, we construct local parametrices near ${\eta }_j$ for $j=1,2$, and singularity ${\eta }_0$. At the endpoints, we employ the Airy parametrix and the first type of modified Bessel parametrix. At the singularity ${\eta }_0$, we use the second type of modified Bessel parametrix for $r_0$ and confluent hypergeometric parametrix for $r_c(\lambda )$.

55 pages, 11 figures