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

TL;DR

This work rigorously analyzes the large-space and long-time asymptotics of soliton gases for the short-pulse equation by formulating a 2-coefficient Riemann-Hilbert problem in the N→∞ limit. It develops a tailored Deift-Zhou steepest descent framework, featuring a piecewise $g$-function and local parametrices (Airy, modified Bessel of both types, and confluent hypergeometric) to handle endpoint and interior singularities in the spectral data. The authors derive region-wise asymptotics for the initial data $u(x,0)$ and the evolving field $u(x,t)$, providing explicit leading-order formulas in terms of theta-functions and elliptic integrals, along with precise error estimates. The results extend soliton-gas analysis beyond KdV/NLS to the short-pulse setting and lay groundwork for multi-singularity and more general reflection-data configurations, with potential implications for wave-transport in ultra-short pulse regimes.

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 $\left[η_1, η_2\right]$: $r_0(λ) = \left(λ- η_1\right)^{β_1}\left(η_2 - λ\right)^{β_2}|λ- η_0|^{β_0}γ(λ)$, $r_c(λ) = \left(λ- η_1\right)^{β_1}\left(η_2 - λ\right)^{β_2}χ_c(λ)γ(λ)$, where $0 < η_1 < η_0 < η_2$ and $β_j > -1$ ($j = 0, 1, 2$), $γ(λ)$ is continuous and positive on $\left[η_1, η_2\right]$, with an analytic extension to a neighborhood of this interval, $χ_c(λ) = 1$ for $λ\in \left[η_1, η_0\right)$ and $χ_c(λ) = c^2$ for $λ\in \left(η_0, η_2\right]$, 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 $η_j$ for $j = 1, 2$, and singularity $η_0$. At the endpoints, we employ the Airy parametrix and the first type of modified Bessel parametrix. At the singularity $η_0$, we use the second type of modified Bessel parametrix for $r_0$ and confluent hypergeometric parametrix for $r_c(λ)$.

Figures (11)

• Figure 1: Left: Jump contours for Airyparametrix $M^{\mathrm{mB}}$; Right: Jump contours for the first type of modified Bessel parametrix $M^{\mathrm{Ai}}$.
• Figure 2: Left: Jump contour for modified Bessel parametrix $M^{\mathrm{mb}}$; Right: Jump contour for Confluent Hypergeometric parametrix $M^{\mathrm{CH}}$.
• Figure 3: Sign charts for $\Re(\theta)$ with $\xi = -1$ (Left) and $\xi = 0$ (Right): $\Re(\theta) > 0$ in grey regions, and $\Re(\theta) < 0$ in white regions.
• Figure 4: Contour deformation by opening lenses for $r=r_0, r_c$ with $\beta_0\ne 0$
• Figure 5: Contour deformation by opening lenses for $r=r_0$ with $\beta_0=0$.

Theorems & Definitions (15)

• Theorem 1: Large-$\hat{x}$ asymptotics for the initial value $u(x, 0)$ of the soliton gas
• Theorem 2: Long-time asymptotics for the soliton gas $u(x, t)$
• Remark 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3: Small norm estimate
• proof
• Proposition 4: Small Norm Estimate in the Region $\xi \in (\xi_0, -\eta_2^{-2})$