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)-6σq(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 $q_{+}=\hat{I}´q_{-}$, $\ddot{I}ƒ\hat{I}´=-1$. In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region $-6<\hat{I}¾<6$ with $\hat{I}¾=\frac{x}{t}$. In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for other solitonic regions $\hat{I}¾<-6$ and $\hat{I}¾>6$. 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 $\hat{I}¾<-6$, phase function $\hat{I}¸(z)$ has four stationary phase points on the $\mathbb{R}$. Correspondingly, $q(x,t)$ can be characterized with an $\mathcal{N}(\hat{I}›)$-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function $\mathrm{Im}\hat{I}½(\hat{I}{\P }_i)$. In the region $\hat{I}¾>6$, phase function $\hat{I}¸(z)$ has four stationary phase points on $i\mathbb{R}$, the corresponding asymptotic approximations can be characterized with an $\mathcal{N}(\hat{I}›)$-soliton with diverse residual error order $\mathcal{O}(t^{-1})$.

59 pages