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Space-time estimates for null gauge forms and nonlinear Schrödinger equations. (English) Zbl 1008.35070

Summary: We consider the Cauchy problem for the nonlinear Schrödinger equation in one space dimension with interaction satisfying null gauge condition. We prove the local well-posedness of the problem in the Sobolev space \(H^{1/2}\). The method depends on the nonlinear gauge transformation and on sharp smoothing estimates for the null gauge form.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics