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Register a new userLong time asymptotics for the defocusing mKdV equation with finite density initial data in different solitonic regions
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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.
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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. Based on the Rieamnn-Hilbert problem characterization for the Cauchy problem and the $overline{partial}$-nonlinear steepest descent method, we find qualitatively different long time asymptotic forms for the Sasa-Satsuma equation in three solitonic space-time regions: (1) For the region $x<0, |x/t|=mathcal{O}(1)$, the long time asymptotic is given by $$q(x,t)=u_{sol}(x,t| sigma_{d}(mathcal{I})) + t^{-1/2} h + mathcal{O} (t^{-3/4}). $$ in which the leading term is $N(I)$ solitons, the second term the second $t^{-1/2}$ order term is soliton-radiation interactions and the third term is a residual error from a $overlinepartial$ equation. (2) For the region $ x>0, |x/t|=mathcal{O}(1)$, the long time asymptotic is given by $$ u(x,t)= u_{sol}(x,t| sigma_{d}(mathcal{I})) + mathcal{O}(t^{-1}).$$ in which the leading term is $N(I)$ solitons, the second term is a residual error from a $overlinepartial$ equation. (3) For the region $ |x/t^{1/3}|=mathcal{O}(1)$, the Painleve asymptotic is found by $$ u(x,t)= frac{1}{t^{1/3}} u_{P} left(frac{x}{t^{1/3}} right) + mathcal{O} left(t^{2/(3p)-1/2} right), qquad 4<p < infty.$$ in which the leading term is a solution to a modified Painleve $mathrm{II}$ equation, the second term is a residual error from a $overlinepartial$ equation.
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 asymptotic behavior of the solution $q(x,t)$ in space-time soliton region. Based on the resulting asymptotic behavior, the asymptotic approximation of the WKI equation is characterized with the soliton term confirmed by $N(I)$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ leading order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$. Our results also confirm the soliton resolution conjecture for the WKI equation with finite density initial data.
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} Recently, for the space-time region $|x/(2t)|<1$ without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the $N$-soliton solutions for the NLS equation by using the $bar{partial}$ generalization of the nonlinear steepest descent method. Their asymptotic result is the form begin{align} q(x,t)= T(infty)^{-2} q^{sol,N}(x,t) + mathcal{O}(t^{-1 }). end{align} However, for the space-time region $ |x/(2t)|>1$, there will be two stationary points appearing on the jump contour, the corresponding long-time asymptotics is still unknown. In this paper, for the region $|x/(2t)|>1, x/t=mathcal{O}(1)$, we found a different asymptotic expansion $$ q(x,t)= e^{-ialpha(infty)} left( q_{sol}(x,t;sigma_d^{(out)}) +t^{-1/2} h(x,t) right)+mathcal{O}left(t^{-3/4}right),$$ whose leading term is $N$-soliton solutions; the second $t^{-1/2}$ order term is soliton-soliton and soliton-radiation interactions; and the third term $mathcal{O}(t^{-3/4})$ is a residual error from a $overlinepartial$-equation. Additionally, the asymptotic stability property for the N-soliton solutions of the defocusing NLS equation sufficiently is obtained.
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 Riemann-Hilbert problem. Based on the initial value condition, the original Riemann-Hilbert problem is constructed to express the solution of the Wadati-Konno-Ichikawa equation. Through a series of deformations, the original RH problem is transformed into a model RH problem, from which the long-time asymptotic solution of the equation is obtained explicitly.
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.
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