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Nash-Moser methods for the solution of quasilinear Schrödinger equations. (English) Zbl 0935.35153
The authors study the nonlinear Schrödinger equation \[ iu_t=-\Delta u+K(\Delta|u|^2)u,\quad u(0,x)=\phi(x),\quad (x\in\mathbb{R}^n). \] Applying some new techniques by M. Poppenberg on Nash-Moser type implicit function theorems for Fréchet spaces, the local existence, uniqueness and continuous dependence of smooth solutions in space dimension \(n=1\), is proved. The basic function space \(H^\infty\) is used.
The method consists in finding an appropriate linearization of the equation, and proving that this linear Schrödinger equation admits a strongly continuous evolution operator which provides the necessary a priori estimates for any derivative.
Reviewer: L.Vazquez (Madrid)

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35F25 Initial value problems for nonlinear first-order PDEs
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