2025/02/04 by Guoqiang Zhang, Weifang Weng, Zhang, Guoqiang +3
Chemistry · Mathematics · #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph) #Optics (physics.optics) #Pattern Formation and Solitons (nlin.PS) #Spectroscopy and Laser Applications
paper · pdf · doi:10.48550/arxiv.2502.02261
openalex publication_date 2025/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 → ∞. 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 [η1, η2]: r0(λ) = (λ- η1)β1(η2 - λ)β2|λ- η0|β0γ(λ), rc(λ) = (λ- η1)β1(η2 - λ)β2χc(λ)γ(λ), where 0 < η1 < η0 < η2 and βj > -1 (j = 0, 1, 2), γ(λ) is continuous and positive on [η1, η2], with an analytic extension to a neighborhood of this interval, χc(λ) = 1 for λ∈ [η1, η0) and χc(λ) = c2 for λ∈ (η0, η2], where c>0 with c ≠ 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 ηj for j = 1, 2, and singularity η0. At the endpoints, we employ the Airy parametrix and the first type of modified Bessel parametrix. At the singularity η0, we use the second type of modified Bessel parametrix for r0 and confluent hypergeometric parametrix for rc(λ).