arXiv 2501.034932025-01-07nlin.SImath-phmath.APmath.MPphysics.optics

关于 sine-Gordon 方程中扭结孤子气体的大空间与长时间渐近行为

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

Guoqiang Zhang, Weifang Weng, Zhenya Yan

备注：: 54 pages, 11 figures

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AI总结：

本文研究 sine-Gordon 方程中扭结孤子气体的大空间、长时间渐近，基于 Riemann–Hilbert 与 Deift–Zhou 方法处理两类广义反射系数，并构造相应局部参数矩阵。

摘要：

我们对 sine-Gordon（sG）方程中扭结孤子气体的大空间和长时间渐近行为进行了全面分析，解决了近期工作 [Phys. Rev. E 109 (2024) 061001] 所强调的一个重要开放问题。我们关注在 Riemann–Hilbert 框架内建模、并由两类广义反射系数刻画的扭结孤子气体；每类系数均定义在区间 $[η_1, η_2]$ 上：$r_0(λ) = (λ- η_1)^{β_1} (η_2 - λ)^{β_2} |λ- η_0|^{β_0} γ(λ)$ 与 $r_c(λ) = (λ- η_1)^{β_1} (η_2 - λ)^{β_2} χ_c(λ) γ(λ)$，其中 $0 < η_1 < η_0 < η_2$、$β_j > -1$，而 $γ(λ)$ 是定义在 $[η_1, η_2]$ 上的连续严格正函数。函数 $χ_c(λ)$ 表现出阶梯状行为：当 $λ\in [η_1, η_0)$ 时 $χ_c(λ)=1$，当 $λ\in (η_0, η_2]$ 时 $χ_c(λ)=c^2$，其中 $c$ 是不同于一的正常数。为严格推导渐近结果，我们利用 Deift–Zhou 最速下降法。该方法的核心组成部分是为共轭过程构造适当的 $g$-函数。与 KdV 方程不同，sG 给 $g$-函数表述带来了独特挑战，尤其是原点处的奇异性。Riemann–Hilbert 问题还要求在端点 $η_j$ 和奇点 $η_0$ 附近精心构造局部参数矩阵。在端点 $η_j$ 处，我们采用第一类修正 Bessel 参数矩阵。对于奇点 $η_0$，参数矩阵的选择取决于反射系数：对 $r_0(λ)$ 使用第二类修正 Bessel 参数矩阵，而对 $r_c(λ)$ 应用合流超几何参数矩阵。

英文摘要：

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 $[η_1, η_2]$: $r_0(λ) = (λ- η_1)^{β_1} (η_2 - λ)^{β_2} |λ- η_0|^{β_0} γ(λ)$ and $r_c(λ) = (λ- η_1)^{β_1} (η_2 - λ)^{β_2} χ_c(λ) γ(λ)$, where $0 < η_1 < η_0 < η_2$ and $β_j > -1$, $γ(λ)$ is a continuous, strictly positive function defined on $[η_1, η_2]$. 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 $r_c(λ)$.

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