Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on ${\mathbb{R}}^3$

Abstract

We consider the cubic-quintic nonlinear Schrödinger equation: \[ i\partial_t u = -Δu - |u|^2u + |u|^4u. \] In the first part of the paper, we analyze the one-parameter family of ground-state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo--Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region $\mathcal{R}$ of the mass/energy plane where the virial is positive. We show this region is partially bounded by solitons but also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology.

References in corpus (4)

• Travelling waves for the Gross-Pitaevskii equation II
• Blowup and Scattering problems for the Nonlinear Schrödinger equations
• Existence of a ground state and blow-up problem for a nonlinear Schrodinger equation with critical growth
• Quintic NLS in the exterior of a strictly convex obstacle

Cited by in corpus (4)

• The focusing cubic NLS with inverse-square potential in three space dimensions
• Global dynamics above the ground state energy for the combined power type nonlinear Schrodinger equations with energy critical growth at low frequencies
• The energy-critical NLS with inverse-square potential
• The final-state problem for the cubic-quintic NLS with non-vanishing boundary conditions