Rigorous analysis of large-space and long-time asymptotics for the short-pulse soliton gases
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2025
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| author | Zhang, Guoqiang Weng, Weifang Yan, Zhenya |
| author_facet | Zhang, Guoqiang Weng, Weifang Yan, Zhenya |
| contents | 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 $r_c(λ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_02261 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigorous analysis of large-space and long-time asymptotics for the short-pulse soliton gases Zhang, Guoqiang Weng, Weifang Yan, Zhenya Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs Pattern Formation and Solitons Optics 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 $r_c(λ)$. |
| title | Rigorous analysis of large-space and long-time asymptotics for the short-pulse soliton gases |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs Pattern Formation and Solitons Optics |
| url | https://arxiv.org/abs/2502.02261 |