Long-time asymptotic analysis of the Korteweg–de Vries equation via the dbar steepest descent method: the soliton region

Abstract

We address the problem of long-time asymptotics for the solutions of the Korteweg-de Vries equation under low regularity assumptions. We consider decreasing initial data admitting only a finite number of moments. For the so-called "soliton region", an improved asymptotic estimate is provided, in comparison with the one in [8]. Our analysis is based on the dbar steepest descent method proposed by P

Long-time Asymptotic Behavior of the coupled dispersive AB system in Low Regularity Spaces

“…Since then, more and more scholars have paid attention to the nonlinear descent method to study the long-time asymptotic behavior of the solution for the initial value problem of integrable systems, and many equations have been studied by this method [14][15][16][17][18][19][20][21]. Later, McLaughlin and Miller extended the classical Deift-Zhou steepest descent method to Dbar steepest descent method, which was successfully used to study the asymptotic stability of NLS multiple soliton solutions [23][24][25], and the long-time proximity of KdV equation and NLS equation [26][27][28][29][30]. TheDbar steepest descent method is to transform the discontinuous part on the jump line into the form of Dbar problem, rather than the analysis of the asymptotic properties of orthogonal polynomials in singular integrals on the jump line.…”

The long-time asymptotic behaviors of the solutions for the coupled dispersive AB system with weighted Sobolev initial data

“…Hereafter, the method was also developed rigorously to study the defocusing NLS under essentially minimal regularity assumptions on finite mass initial data [26] and the defocusing NLS with finite density initial data [27]. With the continuous development of the ∂ steepest descent method, more and more work has been studied, including KdV equation [28], focusing NLS equation [29], short-pluse equation [30], focusing Fokas-Lenells equation [31], modified Camassa-Holm equation [32], Wadati-Konno-Ichikawa [33] and other works [34,35,36,37].…”

Long-time asymptotic behavior of the nonlocal nonlinear Schrödinger equation with initial potential in weighted sobolev space

“…The dbar-steepest descent method, developed from the Deift-Zhou steepest descent method, is very powerful in analyzing the long-time asymptotic behavior with potential in weighted Sobolev space [15][16][17][18][19]. Cuccagna et al also use it to analyze the asymptotic stability for the soliton solutions of the NLS equation [20,21].…”