On Large-Space and Long-Time Asymptotic Behaviors of Kink-Soliton Gases in the Sine-Gordon Equation

Abstract

We conduct a comprehensive analysis of the large-space and long-time asymptotics of kink-soliton gases in the sine-Gordon (sG) equation, addressing an important open problem highlighted in the recent work [Phys. Rev. E 109 (2024) 061001]. We focus on kink-soliton gases modeled within a Riemann-Hilbert framework and characterized by two types of generalized reflection coefficients, each defined on the interval $[{\eta }_1,{\eta }_2]$: $r_0(\lambda )=(\lambda -{\eta }_1{)}^{{\beta }_1}({\eta }_2-\lambda {)}^{{\beta }_2}|\lambda -{\eta }_0{|}^{{\beta }_0}\gamma (\lambda )$ and $r_c(\lambda )=(\lambda -{\eta }_1{)}^{{\beta }_1}({\eta }_2-\lambda {)}^{{\beta }_2}{\chi }_c(\lambda )\gamma (\lambda )$, where $0<{\eta }_1<{\eta }_0<{\eta }_2$ and ${\beta }_j>-1$, \(γ(λ)\) is a continuous, strictly positive function defined on $[{\eta }_1,{\eta }_2]$. The function \(χ_c(λ)\) demonstrates a step-like behavior: it is given by \(χ_c(λ) = 1\) for \(λ\in [η_1, η_0)\) and \(χ_c(λ) = c^2\) for \(λ\in (η_0, η_2]\), with \(c\) as a positive constant distinct from one. To rigorously derive the asymptotic results, we leverage the Deift-Zhou steepest descent method. A central component of this approach is constructing an appropriate \(g\)-function for the conjugation process. Unlike in the KdV equation, the sG presents unique challenges for \(g\)-function formulation, particularly concerning the singularity at the origin. The Riemann-Hilbert problem also requires carefully constructed local parametrices near endpoints \(η_j\) and the singularity \(η_0\). At the endpoints \(η_j\), we employ a modified Bessel parametrix of the first kind. For the singularity \(η_0\), the parametrix selection depends on the reflection coefficient: the second kind of modified Bessel parametrix is used for \(r_0(λ)\), while a confluent hypergeometric parametrix is applied for \(r_c(λ)\).

54 pages, 11 figures