Asymptotic stability manifolds for solitons in the generalized Good Boussinesq equation

Abstract

We consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space H 1 ˆL2 . This model has solitary waves with speeds ´1 ă c ă 1. When |c| approaches 1, Bona and Sachs showed orbital stability of such waves. It is well-known from a work of Liu that for small speeds solitary waves are unstable. In this paper we consider in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space. Contents 1. Introduction 1.1. Setting 1.2. Standing waves 1.3. Main results 1.4. Idea of the proof Organization of this paper Acknowledgments 2. A virial identity for the (gGB) system 2.1. Decomposition of the solution in a vicinity of the soliton 2.2. Notation for virial argument 2.3. Virial estimate 2.4. End of Proposition 2.1 3. Transformed problem and second virial estimates 3.1. The transformed problem 3.2. Virial functional for the transformed problem 3.3. Proof of Proposition 3.1: first computations 3.4. First technical estimates 3.5. Controlling error and nonlinear terms 3.6. End of proof of Proposition 3.1 4. Gain of derivatives via transfer estimates 4.1. A virial estimate related to Mptq 4.2. Second set of technical estimates 4.3. Start of proof of Proposition 4.2 4.4. Controlling error terms 4.5. Controlling nonlinear terms 4.6. End of proof Proposition 4.2 5. A second transfer estimate 5.1. A virial identity for N ptq 5.2. Start of proof of Proposition 5.2 5.3. Control of nonlinear terms