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

journal contribution
posted on 23.03.2021, 18:50 by R Carles, C Sparber
© 2021 World Scientific Publishing Company. 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.

Funding

CAREER: Adiabatic theory for nonlinear Schrodinger equations with applications in complex quantum systems

Directorate for Mathematical & Physical Sciences

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Publisher Statement

Electronic version of an article published as Carles, R.Sparber, C. (2020). Orbital stability vs. scattering in the cubic-quintic Schrödinger equation. Reviews in Mathematical Physics, 2150004-. https://doi.org/10.1142/S0129055X21500045 ©World Scientific Publishing Company https://www.worldscientific.com/worldscinet/rmp

Citation

Carles, R.Sparber, C. (2020). Orbital stability vs. scattering in the cubic-quintic Schrödinger equation. Reviews in Mathematical Physics, 2150004-. https://doi.org/10.1142/S0129055X21500045

Publisher

World Scientific Pub Co Pte Lt

Language

en

issn

0129-055X

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