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

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 $c \neq 1$. 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 $r_c(\lambda)$.