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Local well posedness of nonlinear Schrödinger equations. (English) Zbl 0874.35112

In this note I describe some recent work, done jointly with Gustav Ponce and Luis Vega, on nonlinear Schrödinger equations of the form \[ i{\partial u\over\partial t}+ \Delta u+F(u,\overline u,\nabla_xu,\nabla_x\overline u)=0,\quad u(x,0)=u_0(x), \] where \(x\in \mathbb{R}^n\), \(t\in [0,T]\). Here \(F:\mathbb{C}^{2n+2} \to\mathbb{C}\) is a polynomial having no constant or linear terms. We are interested in establishing local well posedness results, and global well posedness results, with data in Sobolev spaces.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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