Global existence of small solutions to a relativistic nonlinear Schrödinger equation. (English) Zbl 0948.81025

de Bouard, Anne; Hayashi, Nakao; Saut, Jean-Claude

Commun. Math. Phys. 189, No. 1, 73-105 (1997).

Summary: We study the Cauchy problem associated to a nonlinear Schrödinger equation modelling the self-channeling of a high power, ultra-short laser pulse in matter. The new nonlinear terms arise from relativistic effects and from the ponderomotive force. We prove global existence and uniqueness of small solutions in transverse space dimensions 2 and 3, and local existence without any smallness condition in transverse space dimension 1.

MSC:

81V80	Quantum optics
35Q55	NLS equations (nonlinear Schrödinger equations)
35B60	Continuation and prolongation of solutions to PDEs

Keywords:

Cauchy problem; nonlinear Schrödinger equation; self-channeling; high power, ultra-short laser pulse