Orbital stability vs. scattering in the cubic-quintic Schrödinger equation

• Published in: Reviews in mathematical physics Vol. 33; no. 3; p. 2150004
• Main Authors: Carles, Rémi, Sparber, Christof
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.04.2021; World Scientific Publishing Co. Pte., Ltd; World Scientific Publishing
• Subjects: Analysis of PDEs; Cauchy problems; Defocusing; Mathematical Physics; Mathematics; Nonlinearity; Orbital stability; Schrodinger equation; Solitary waves; scattering theory; Nonlinear Schrödinger equation; solitary waves; orbital stability
• ISSN: 0129-055X; 1793-6659; 0129-055X
• DOI: 10.1142/S0129055X21500045

We consider the cubic-quintic nonlinear Schrödinger equation of up to three space dimensions. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is already known that all solitons are orbitally stable. In dimension two, we show that if the initial data belong to the conformal space, and have at most the mass of the ground state of the cubic two-dimensional Schrödinger equation, then the solution is asymptotically linear. For larger mass, solitary wave solutions exist, and we review several results on their stability. Finally, in dimension three, relying on previous results from other authors, we show that solitons may or may not be orbitally stable.