Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass

Abstract

In this paper we prove the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed L 2 -norm, and the stability of associated solitary waves for two classes of coupled nonlinear dispersive equations. The first problem here describes the nonlinear interaction between two Schrödinger type short waves and a generalized Korteweg-de Vries type long wave and the second problem describes the nonlinear interaction of two generalized Kortewegde Vries type long waves with a common Schrödinger type short wave. The results here extend many of the previously obtained results for two-component coupled Schrödinger-Korteweg-de Vries systems. Mathematics Subject Classification. 35Q53, 35Q55, 35B35, 35B65, 35A15.1 2 3 + (R) in the non-resonant case. In [15], Corcho and Linares improved the local well-posedness result obtained in [28] to a larger region of the Sobolev indices. Recently, Wu [30] obtained the best local well-posedness result for the (1 + 1)-component NLS-KdV system in the resonant case. Our aim here is to obtain analogous results to the full system of equations (1.1) and (1.2), considering general power nonlinearities, in the Sobolev spaces of the form H s × H s × H k and H s × H k × H k , respectively.

Existence and stability of standing waves for coupled nonlinear Hartree type equations

“…M takes the following form Proof of Lemma 4.1. We use ideas from [8,9,10]. Since M 1 + T 1 > 0, we have the following possibilities: M 1 > 0 and T 1 > 0; or M 1 = 0 and T 1 > 0; or M 1 > 0 and T 1 = 0.…”