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

Abstract

We consider the cubic-quintic nonlinear Schr{ö}dinger equation in space dimension up to three. 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 Schr{ö}dinger 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.

Orbital stability clarified

References in corpus (2)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

Cited by in corpus (10)

• On global minimizers for a mass constrained problem
• Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode
• Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach
• Sharp scattering threshold for the cubic-quintic NLS in the focusing-focusing regime
• Stabilizing solitons of the cubic-quintic nonlinear Schrödinger equation by frequency-dependent linear gain-loss and delayed Raman response
• On ground states for the 2D Schrodinger equation with combined nonlinearities and harmonic potential
• On an intercritical log-modified nonlinear Schrodinger equation in two spatial dimensions
• Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions
• Collision of two solitons for $1d$ Nonlinear Schrodinger Equation with the same mass
• Normalized solutions with positive energies for a coercive problem and application to the cubic-quintic nonlinear Schrödinger equation