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 $[{\eta }_1,{\eta }_2]$: $r_0(\lambda )=(\lambda -{\eta }_1{)}^{{\beta }_1}({\eta }_2-\lambda {)}^{{\beta }_2}\mathrm{\mid }\lambda -{\eta }_0{\mathrm{\mid }}^{{\beta }_0}\gamma (\lambda )$ 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 $r_c(\lambda )$.