On the role of quadratic oscillations in nonlinear Schrodinger equations

Details On the role of quadratic oscillations in nonlinear Schrodinger equations On the role of quadratic oscillations in nonlinear Schrodinger equations

2002-12-12

arxiv.org/abs/math/0212171v2

Journal of Functional Analysis, 203/2 (2003), 453-493

Mathematics

Analysis of PDEs

29 pages. References added/updated, some typos fixed, more explanations

Scientific paper

We consider a nonlinear semi-classical Schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrodinger equation with harmonic potential.

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