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

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 $\beta_j > -1$, \(\gamma(\lambda)\) is a continuous, strictly positive function defined on $[\eta_1, \eta_2]$. The function \(\chi_c(\lambda)\) demonstrates a step-like behavior: it is given by \(\chi_c(\lambda) = 1\) for \(\lambda \in [\eta_1, \eta_0)\) and \(\chi_c(\lambda) = c^2\) for \(\lambda \in (\eta_0, \eta_2]\), 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 \(\eta_j\) and the singularity \(\eta_0\). At the endpoints \(\eta_j\), we employ a modified Bessel parametrix of the first kind. For the singularity \(\eta_0\), the parametrix selection depends on the reflection coefficient: the second kind of modified Bessel parametrix is used for \(r_0(\lambda)\), while a confluent hypergeometric parametrix is applied for \(r_c(\lambda)\).