Related papers: Soliton asymptotics for the KdV shock problem of l…
This paper discusses some general aspects and techniques associated with the long-time asymptotics of steplike solutions of the Korteweg--de Vries (KdV) equation via vector Riemann--Hilbert problems. We also elaborate on an ill-posedness of…
We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of shock waves for the Korteweg-de Vries (KdV) equation in the soliton region. In particular, we improve the results…
This work investigates the long-time asymptotic behaviors of the solution to the KdV equation with delta function initial profiles in different regions, employing the Riemann-Hilbert formulation and Deift-Zhou nonlinear steepest descent…
We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory…
The soliton resolution for the focusing modified Korteweg-de vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation…
We analytically study the large-time asymptotics of the solution of the defocusing modified Korteweg-de Vries (mKdV) equation under a symmetric non-vanishing background, which supports the emergence of solitons. It is demonstrated that the…
We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann--Hilbert problem which we…
In this paper we study the asymptotics of solutions to the Korteweg--de Vries equation with steplike initial data, which lead to shock waves in the region between the asymptotically constant region and the soliton region, as $t \rightarrow…
We address the problem of long-time asymptotics for the solutions of the Korteweg-de Vries equation under low regularity assumptions. We consider rapidly decreasing initial data admitting only a finite number of moments. For the so-called…
We investigate the long-time asymptotics for the solutions to the Cauchy problem of defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data. The present paper is the subsequent work of our previous paper…
We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless…
In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the $I$-method and spectral analysis following Pego and Weinstein, we are…
In this paper, we reconsider the well-known result of Pego-Weinstein \cite{MR1289328} that soliton solutions to the Korteweg-deVries equation are asymptotically stable in exponentially weighted spaces. In this work, we recreate this result…
The Hirota equation is one of the integrable higher-order extensions of the nonlinear Schr\"odinger equation, and can describe the ultra-short optical pulse propagation in the form $iq_t+\alpha(q_{xx}+ 2|q|^2q)+i\beta (q_{xxx}+…
We systematically investigate the long-time asymptotics for the $N_{\infty}$-soliton solution to the KdV equation in the different regions with the aid of the Riemann-Hilbert (RH) problems with two types of generalized reflection…
We describe an approach to construct multi-soliton asymptotic solutions for non-integrable equations. The general idea is realized in the case of three waves and for the KdV-type equation with nonlinearity $u^4$. A brief review of…
The KdV-Sawada-Kotera equation has single-, two- and three-soliton solutions. However, it is not known yet whether it has N-soliton solutions for any N. Viewing it as a perturbed KdV equation, the asymptotic expansion of the solution is…
This work investigates the long-time asymptotics of solution to defocusing modified Korteweg-de Vries equation with a class of step initial data. A rigorous asymptotic analysis is conducted on the associated Riemann-Hilbert problem by…
We consider the modified Korteveg de Vriez equation on the whole line. Initial data is real and step-like, i.e. $q(x,0)=0$ for $x\geq0$ and $q(x,0)=c$ for $x<0$, where c is arbitrary real number. The goal of this paper is to study the…
Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann--Hilbert problems and the Dbar approach, the long-time asymptotic behavior of solutions to the fifth-order modified Korteweg-de Vries equation on the…