Stable standing waves for the two-dimensional Klein–Gordon–Schrödinger system

Abstract We study the orbital stability of standing waves for the Klein–Gordon–Schrödinger system in two spatial dimensions. It is proved that the standing wave is stable if the frequency is sufficiently small. To prove this, we obtain the uniqueness of ground state and investigate the spectrum of the appropriate linearized operator by using the perturbation method developed by Genoud and Stuart and Lin and Wei. Then we apply to our system the general theory of Grillakis, Shatah and Strauss.

[1]  J. Shatah,et al.  Instability of nonlinear bound states , 1985 .

[2]  TAI-CHIA LIN,et al.  Orbital Stability of Bound States of Semiclassical Nonlinear Schrödinger Equations with Critical Nonlinearity , 2008, SIAM J. Math. Anal..

[3]  Alan Weinstein,et al.  Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential , 1986 .

[4]  M. Weinstein.  Nonlinear Schrödinger equations and sharp interpolation estimates , 1983 .

[5]  P. Lions,et al.  Orbital stability of standing waves for some nonlinear Schrödinger equations , 1982 .

[6]  Alain Bachelot.  Problème de Cauchy pour des systèmes hyperboliques semi‐linéaires , 1984, Annales de l'Institut Henri Poincaré C, Analyse non linéaire.

[8]  E. Ferreira.  On the interaction of the elementary particles , 1961 .

[9]  Takafumi Akahori.  Global solutions of the wave-Schr\"odinger system below $L^{2}$ , 2006 .

[10]  Wolf von Wahl,et al.  On the global strong solutions of coupled Klein-Gordon-Schrödinger equations , 1987 .

[11]  W. Ni,et al.  Locating the peaks of least energy solutions to a semilinear Neumann problem , 1993 .

[12]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

[13]  I. Iliev,et al.  Stability and instability of solitary waves for one-dimensional singular Schrödinger equations , 1993, Differential and Integral Equations.

[14]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

[15]  W. Rother,et al.  Nonlinear scalar field equations , 1992, Differential and Integral Equations.

[16]  Masahito Ohta,et al.  Stability of standing waves for the Klein–Gordon–Schrödinger system , 2010 .

[17]  Nikolaos Tzirakis,et al.  Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schr\ , 2006, math/0603595.

[18]  Jong Yeoul Park,et al.  On Coupled Klein-Gordon-Schrödinger Equations with Acoustic Boundary Conditions , 2010 .

[19]  F. Genoud,et al.  Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves , 2008 .

[20]  J. Shatah,et al.  Stability theory of solitary waves in the presence of symmetry, II☆ , 1990 .

[21]  T. Cazenave.  Semilinear Schrodinger Equations , 2003 .

[22]  Reika Fukuizumi.  Stability of standing waves for nonlinear Schrödinger equations with critical power nonlinearity and potentials , 2005, Advances in Differential Equations.

[23]  W. Strauss,et al.  Nonlinear bound states outside an insulated sphere , 1994 .