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- Title
ASYMPTOTIC BEHAVIOR FOR A QUADRATIC NONLINEAR SCHRÖDINGER EQUATION.
- Authors
HAYASHI, NAKAO; NAUMKIN, PAVEL I.
- Abstract
We study the initial-value problem for the quadratic nonlinear Schrödinger equation iut + 1/2 uxx = ∂xū², x ∈ ∇, t > 1, u(1, x) = u1(x), x ∈ ∇. 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 |x| ≤ C√t by the self-similar solution 1/√t MS(x/√t) such that the total mass 1/√t ∫ ∇ MS(x/√t)dx = ∫∇ u1(x)dx, and in the far region |x| > √t the asymptotic behavior of solutions has rapidly oscillating structure similar to that of the cubic nonlinear Schrödinger equations.
- Subjects
ASYMPTOTIC theory in partial differential equations; SCHRODINGER equation; QUADRATIC equations; CAUCHY problem; NONLINEAR differential equations
- Publication
Electronic Journal of Differential Equations, 2008, Vol 2008, p1
- ISSN
1550-6150
- Publication type
Article