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We investigate the long time asymptotics for the Cauchy problem of the defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data in different solitonic regions begin{align*} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, quad (x,t)inmathbb{R}times mathbb{R}^{+}, &q(x,0)=q_{0}(x), quad lim_{xrightarrowpminfty}q_{0}(x)=pm 1, end{align*} where $q_0mp 1in H^{4,4}(mathbb{R})$.Based on the spectral analysis of the Lax pair, we express the solution of the mKdV equation in terms of a Riemann-Hilbert problem. In our previous article, we have obtained long time asymptotics and soliton resolutions for the mKdV equation in the solitonic region $xiin(-6,-2)$ with $xi=frac{x}{t}$.In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for the solitonic region $xiin(-varpi,-6)cup(-2,varpi)$ with $ 6 < varpi<infty$ being an arbitrary constant.For $-varpi<xi<-6$, there exist four stationary phase points on jump contour, and the asymptotic approximations can be characterized with an $N$-soliton on discrete spectrums and a leading order term $mathcal{O}(t^{-1/2})$ on continuous spectrum up to a residual error order $mathcal{O}(t^{-3/4})$. For $-2<xi<varpi$, the leading term of asymptotic expansion is described by the soliton solution and the error order $mathcal{O}(t^{-1})$ comes from a $bar{partial}$-problem. Additionally, asymptotic stability can be obtained.
The Sasa-Satsuma equation with $3 times 3 $ Lax representation is one of the integrable extensions of the nonlinear Schr{o}dinger equation. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying initial data.
In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data. Employing the $bar{partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the long time asympto
We consider the Cauchy problem for the defocusing Schr$ddot{text{o}}$dinger (NLS) equation with finite density initial data begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, onumber &q(x,0)=q_0(x), quad lim_{x to pm infty}q_0(x)=pm 1. onumber end{align} Rece
In this work, we investigate the long-time asymptotic behavior of the Wadati-Konno-Ichikawa equation with initial data belonging to Schwartz space at infinity by using the nonlinear steepest descent method of Deift and Zhou for the oscillatory Rieman
The purpose of this article is to give a streamlined and self-contained treatment of the long-time asymptotics of the Toda lattice for decaying initial data in the soliton and in the similarity region via the method of nonlinear steepest descent.