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Global existence of small solutions to a relativistic nonlinear Schrödinger equation. (English) Zbl 0948.81025
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:
81V80Applications of quantum theory to quantum optics
35Q55NLS-like (nonlinear Schrödinger) equations
35B60Continuation of solutions of PDE