Long time asymptotic behavior for the nonlocal mKdV equation in space-time solitonic regions-II

Abstract

We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions \begin{align*} &q_t(x,t)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, &q(x,0)=q_{0}(x),\ \ \lim_{x\to \pm\infty} q_{0}(x)=q_{\pm}, \end{align*} where $|q_{\pm}|=1$ and $q_{+}=\delta q_{-}$, $\sigma\delta=-1$. In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region $-6<\xi<6$ with $\xi=\frac{x}{t}$. In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for other solitonic regions $\xi<-6$ and $\xi>6$. Based on the Riemann-Hilbert problem of the the Cauchy problem, further using the $\bar{\partial}$ steepest descent method, we derive different long time asymptotic expansions of the solution $q(x,t)$ in above two different space-time solitonic regions. In the region $\xi<-6$, phase function $\theta(z)$ has four stationary phase points on the $\mathbb{R}$. Correspondingly, $q(x,t)$ can be characterized with an $\mathcal{N}(\Lambda)$-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function ${\rm Im}\nu(\zeta_i)$. In the region $\xi>6$, phase function $\theta(z)$ has four stationary phase points on $i\mathbb{R}$, the corresponding asymptotic approximations can be characterized with an $\mathcal{N}(\Lambda)$-soliton with diverse residual error order $\mathcal{O}(t^{-1})$.