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

Guoqiang Zhang, Weifang Weng, Zhenya Yan

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 $[η_1, η_2]$: $r_0(λ) = (λ- η_1)^{β_1} (η_2 - λ)^{β_2} |λ- η_0|^{β_0} γ(λ)$ 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 \(r_c(λ)\).

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 : and , where and , \(γ(λ)\) is a continuous, strictly positive function defined on . 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 , with 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 -function for the conjugation process. Unlike in the KdV equation, the sG presents unique challenges for -function formulation, particularly concerning the singularity at the origin. The Riemann-Hilbert problem also requires carefully constructed local parametrices near endpoints and the singularity . At the endpoints , we employ a modified Bessel parametrix of the first kind. For the singularity , 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 \(r_c(λ)\).
Paper Structure (23 sections, 8 theorems, 157 equations, 11 figures)

This paper contains 23 sections, 8 theorems, 157 equations, 11 figures.

Table of Contents

  1. Introduction and Main results
  2. Backgrounds
  3. The main results
  4. Solvable Riemann-Hilbert models
  5. Airy parametrix $M^{\mathrm{Ai}}(\zeta)$
  6. The first type of modified Bessel parametrix $M^{\mathrm{mB}}\left(\zeta; \beta\right)$
  7. The second type of modified Bessel parametrix $M^{\mathrm{mb}}(\zeta; \beta)$
  8. Confluent hypergeometric parametrix $M^{\mathrm{CH}}(\zeta; \kappa)$
  9. A generalized $g$-function construction
  10. Riemann-Hilbert problem for the sine-Gordon kink-soliton gas
  11. Large-$x$ asymptotics for the initial value $u(x, 0)$ of the sine-Gordon kink-soliton gas
  12. Riemann-Hilbert problem for $T(\lambda; x, 0)$
  13. Riemann-Hilbert problem for $S(\lambda; x, 0)$
  14. Riemann-Hilbert problem for $E(\lambda; x, 0)$
  15. Large-$x$ asymptotics of the initial values $u(x, 0)$ as $x\to-\infty$
  16. ...and 8 more sections

Key Result

Theorem 1

For these two types of generalized reflection coefficients $r=r_0$ and $r=r_c$, the large-$x$ asymptotics for the initial value $u(x, 0)$ of the kink-soliton gas are established as follows.

Figures (11)

  • Figure 1: Left: Jump contours for Airy parametrix $M^{\mathrm{mB}}$; Right: Jump contours for the first type of modified Bessel parametrix $M^{\mathrm{Ai}}$.
  • Figure 2: Left: Jump contour for modified Bessel parametrix $M^{\mathrm{mb}}$; Right: Jump contour for Confluent Hypergeometric parametrix $M^{\mathrm{CH}}$.
  • Figure 3: Sign charts for $\Re\left(\theta\right)$ with $\xi=-1$(Left) and $\xi=0$(Right): $\Re\left(\theta\right)>0$ in greay regions and $\Re\left(\theta\right)<0$ in white regions
  • Figure 4: Contour deformation by opening lenses for $r=r_0, r_c$ with $\beta_0\ne 0$.
  • Figure 5: Contour deformation by opening lenses for $r=r_0$ with $\beta_0=0$
  • ...and 6 more figures

Theorems & Definitions (15)

  • Theorem 1: Large-$x$ asymptotics for the initial value $u(x, 0)$ of the kink-soliton gas
  • Theorem 2: Long-time asymptotics for the kink-soliton gas $u(x, t)$
  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3: Small norm estimate
  • proof
  • Proposition 4: Small norm estimate in the region $\xi\in\left(\xi_0, -\eta_2^{-2}\right)$
  • ...and 5 more