Asymptotic Behaviour for Schrodinger Equations with a Quadratic Nonlinearity in One-Space Dimension
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We consider the Cauchy problem for the Schrödinger equation with a quadratic nonlinearity in one space dimension iut + 1/2 uxx = t-α |ux|2, u(0, x) = u0(x), where α ∈ (0, 1). From the heuristic point of view, solutions to this problem should have a quasilinear character when α ∈ (1/2, 1). We show in this paper that the solutions do not have a quasilinear character for all α ∈ [1/2, 1) if the initial data u0 ∈ H3,0 ∩ H2,2 are small, then the solution has a slow time decay such as t-α/2. For α ∈ (0,1/2), if we assume that the initial data u0 are analytic and small, then the small time decay occurs.
CitationHayashi, N., & Naumkin, P. I. (2001). Asymptotic behaviour for Schrodinger equations with a quadratic nonlinearity in one-space dimension. Electronic Journal of Differential Equations, 2001(54), pp. 1-18.
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