Existence of bound and ground states for a system of coupled nonlinear Schrödinger–KdV equations

Abstract

Presented by Haïm BrézisWe prove the existence of bound and ground states for a system of coupled nonlinear Schrödinger-Korteweg-de Vries equations, depending on the size of the coupling coefficient.

Ground States for a Nonlinear Schrödinger System with Sublinear Coupling Terms

“…This has been done for d = 2 equations for instance in [2,6,12,19], and for d ≥ 2 in [18,20,21], among others. For some recent results in this directions concerning a Schrödinger-KdV system, see also [7,8] .…”

A higher order system of some coupled nonlinear Schrödinger and Korteweg-de Vries equations

“…Furthermore, using the idea performed in [8] we will prove that there is no loss of mass at infinity for µ n (x) := u 2 n (x) + v 2 n (x), where u n = (u n , v n ), i.e, there exist R, C > 0 such that…”

“…This system appears in phenomena of interactions between short and long dispersive waves, arising in fluid mechanics, such as the interactions of capillary -gravity water waves [16]. Indeed, f represents the short-wave, while g stands for the long-wave; see references [2,8,9,10,14] for further details on similar system. Moreover, the interaction between long and short waves appears in magnetised plasma [15], [19] and in many physical phenomena as well, such that Bose-Einstein condensates [6].…”

“…Moreover, the interaction between long and short waves appears in magnetised plasma [15], [19] and in many physical phenomena as well, such that Bose-Einstein condensates [6]. The solutions studied in papers [8,9] (see also [11,12]) are taken as solitary traveling waves, i.e.,…”

New Optical Solitons And Modulation Instability Analysis of Generalized Coupled Nonlinear Schrodinger-KdV System

“…It is known that, due to the effect of nonlinearity and dispersion, the coupled NLS-KdV system usually possesses such kind of solutions. The existence of solutions for the coupled system of NLS-KdV has been studied in [5,6]. Furthermore, several stability theories have been used to prove the stability of solitary wave solutions of this system [7][8][9].…”