On Large-Space and Long-Time Asymptotic Behaviors of Kink-Soliton Gases in the Sine-Gordon Equation
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912178620071936 |
|---|---|
| author | Zhang, Guoqiang Weng, Weifang Yan, Zhenya |
| author_facet | Zhang, Guoqiang Weng, Weifang Yan, Zhenya |
| contents | 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 \(r_c(λ)\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03493 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Large-Space and Long-Time Asymptotic Behaviors of Kink-Soliton Gases in the Sine-Gordon Equation Zhang, Guoqiang Weng, Weifang Yan, Zhenya Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs Optics 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 \(r_c(λ)\). |
| title | On Large-Space and Long-Time Asymptotic Behaviors of Kink-Soliton Gases in the Sine-Gordon Equation |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs Optics |
| url | https://arxiv.org/abs/2501.03493 |