Small solutions to nonlinear Schrödinger equations

Abstract It is shown that the initial value problem for the nonlinear Schrodinger equations ∂ t u = i Δ u + P ( u , ∇ x u , u ¯ , ∇ x u ¯ ) , t ∈ ℝ , x ∈ ℝ n , where Ρ (.) is a polynomial having no constant or linear terms, is locally well posed for a class of “small” data u 0 . The main ingredients in the proof are new estimates describing the smoothing effect of Kato type for the group { e i t Δ } − ∞ ∞ . This method extends to systems and other dispersive models.

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