We study the initial-value problem for the quadratic nonlinear Schrödinger equation iut + 1/2uxx= δxū2, x ∈ ℝ, t > 1, u(1, x) = u1(x), x ∈ ℝ. For small initial data u1 ∈ H2,2 we prove that there exists a unique global solution u ∈ C([1, ∞); H2,2) of this Cauchy problem. Moreover we show that the large time asymptotic behavior of the solution is defined in the region ΙxΙ ≤ C√t by the self-similar solution 1/√tMS(x/ √t) such that the total mass 1/√t ∫ℝ MS(x/√t)dx = ∫ℝ u1(x)dx, and in the far region ΙxΙ > √t the asymptotic behavior of solutions has rapidly oscillating structure similar to that of the cubic nonlinear Schrödinger equations.
|ジャーナル||Electronic Journal of Differential Equations|
|出版ステータス||Published - 2008 2 1|
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