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Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. (English) Zbl 1038.35113

The authors obtain long-time asymptotics for solutions \(q(x,t)\) of the defocusing nonlinear Schrödinger equation for initial data with sufficient smoothness and decay as \(t\) approaches to infinity. They describe a new technique which produces an error estimate of order \(O(t^{-((1/2)-k)})\) for any \(0<k<1/4\), under the assumption that the initial data \(q_0\) lies in a weighted Sobolev space.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35Q15 Riemann-Hilbert problems in context of PDEs
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[1] Ablowitz, Studies in Appl Math 53 pp 249– (1974) · Zbl 0408.35068
[2] Beals, Comm Pure Appl Math 37 pp 39– (1984)
[3] ; Factorization of matrix functions and singular integral operators. Operator Theory: Advances and Applications, 3. Birkhäuser, Basel-Boston, 1981.
[4] Chang, Comm Pure Appl Math 53 pp 590– (2000)
[5] Cazenave, Nonlinear Anal 14 pp 807– (1990)
[6] ; ; Long-time asymptotics for integrable nonlinear wave equations. Important developments in soliton theory, 181-204. Springer Series in Nonlinear Dynamics. Springer, Berlin, 1993.
[7] Deift, Ann of Math (2) 137 pp 295– (1993)
[8] ; Long-time behavior of the nonfocusing nonlinear Schrödinger equation ? a case study. New Series: Lectures in Math Sciences, 5. University of Tokyo, 1994.
[9] Deift, Comm Math Phys 165 pp 175– (1994)
[10] Deift, Math Res Lett 4 pp 761– (1997) · Zbl 0890.35138
[11] Deift, Acta Math
[12] ; Uniform Lp estimates for solutions of Riemann-Hilbert problems depending on external parameters. In preparation.
[13] Theory of Hp spaces. Pure and Applied Mathematics, 38. Academic, New York-London, 1970.
[14] ; Hamiltonian methods in the theory of solitons. Springer Series in Soviet Mathematics. Springer, Berlin, 1987. · Zbl 1111.37001
[15] Ginibre, Ann Inst H Poincaré Anal Non Linéaire 2 pp 309– (1985)
[16] ; ; Inequalities. 2d ed. Cambridge University Press, Cambridge, 1952.
[17] Zhou, Comm Pure Appl Math 51 pp 697– (1998)
[18] Zakharov, Soviet Physics JETP 34 pp 62– (1972)
[19] Zakharov, Soviet Physics JETP 44 pp 106– (1976)
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