On the instability for the cubic nonlinear Schrodinger equation

Abstract

We study the flow map associated to the cubic Schrodinger equation in space dimension at least three. We consider initial data of arbitrary size in $H^s$, where $0<s<s_c$, $s_c$ the critical index, and perturbations in $H^\si$, where $\si<s_c$ is independent of $s$. We show an instability mechanism in some Sobolev spaces of order smaller than $s$. The analysis relies on two features of super-critical geometric optics: creation of oscillation, and ghost effect.

4 pages

References in corpus (2)

• Ill-posedness for nonlinear Schrodinger and wave equations
• Geometric optics and instability for semi-classical Schrodinger equations

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• Finite time blowup for a supercritical defocusing nonlinear Schrödinger system
• Critical exponents line, scattering thoeries for a weighted gradient system of semilinear Schrodinger equations