WKB analysis for nonlinear Schrödinger equations with potential

Details WKB analysis for nonlinear Schrödinger equations with potential WKB analysis for nonlinear Schrödinger equations with potential

2006-01-25

arxiv.org/abs/math/0601611v2

Comm. Math. Phys. 269 (2007) 195-221

Mathematics

Analysis of PDEs

25 pages, 11pt, a4. Appendix withdrawn, due to some inconsistencies

Scientific paper

10.1007/s00220-006-0077-2

We justify the WKB analysis for the semiclassical nonlinear Schr\"{o}dinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. Grenier; we also have to restrict to nonlinearities which are defocusing and cubic at the origin, but besides subquadratic potentials, we consider initial phases which may be unbounded. For this, we construct solutions for some compressible Euler equations with unbounded source term and unbounded initial velocity.

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