Long-Time Asymptotics for Solutions of the NLS Equation with Initial Data in a Weighted Sobolev Space

Communications on Pure and Applied Mathematics

Volumn 56, Issue 8, 2003, Pages 1029-1077

  Deift, Percy ^a   Zhou, Xin ^b  

^a NEW YORK UNIVERSITY   (United States)

^b DUKE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038336602     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.3034     Document Type: Article

References (22)

1. Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H. The inverse scattering transform-Fourier analysis for nonlinear problems. Studies in Appl. Math. 53 (1974), no. 4, 249-315.
2. Beals, R.; Coifman, R. R. Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math. 37 (1984), no. 1, 39-90.
3. S. Nonlinear Anal. 14 (1990), no. 10, 807-836.
4. Chang, N.-H.; Shatah, J.; Uhlenbeck, K. Schrödinger maps. Comm. Pure Appl. Math. 53 (2000), no. 5, 590-602.
5. Clancey, K. F.; Gohberg, I. Factorization of matrix functions and singular integral operators. Operator Theory: Advances and Applications, 3. Birkhäuser, Basel-Boston, 1981.
6. Deift, P. A.; Itṡ, A. R.; Zhou, X. 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, P.; Zhou, X. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. of Math. (2) 137 (1993), no. 2, 295-368.
8. Deift, P. A.; Zhou, X. Long-time asymptotics for integrable systems. Higher order theory. Comm. Math. Phys. 165 (1994), no. 1, 175-191.
9. Deift, P. A.; Zhou, X. Long-time behavior of the non-focusing nonlinear Schrödinger equation-a case study. New Series: Lectures in Mathematical Sciences, 5. University of Tokyo, 1994.
10. Deift, P.; Zhou, X. Near integrable systems on the line. A case study. Perturbation theory of the defocusing nonlinear Schrödinger equation. Math. Res. Lett. 4 (1997), no. 5, 761-772.
11. Deift, P.; Zhou, X. Perturbation theory for infinite dimensional integrable systems on the line. A case study. Acta Math. 188 (2002), no. 2, 163-262; also see http://www.ml.kva.se/publications/acta/webarticles/deift.
12. Deift, P.; Zhou, X. Perturbation theory for infinite dimensional integrable systems on the line. A case study. Acta Math. 188 (2002), no. 2, 163-262; also see http://www.ml.kva.se/publications/acta/webarticles/deift.
13. Deift, P.; Zhou, X. A priori LP estimates for solutions of Riemann-Hilbert problems. Int. Math. Res. Not. 2002, no. 40, 2121-2154.
14. p spaces. Pure and Applied Mathematics, 38. Academic, New York-London, 1970.
15. Faddeev, L. D.; Takhtajan, L. A. Hamiltonian methods in the theory of solitons. Springer Series in Soviet Mathematics. Springer, Berlin, 1987.
16. Ginibre, J.; Velo, G. The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309-327.
17. Hardy, G. H.; Littlewood, J. E.; Pólya, G. Inequalities. 2nd ed. Cambridge University Press, Cambridge, 1952.
18. Zakharov, V. E.; Manakov, S. V. Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method. Soviet Physics JETP 44 (1976), no. 1, 106-112; translated from Ž. Èksper. Teoret. Fiz. 71 (1976), no. 1, 203-215.
19. Zakharov, V. E.; Manakov, S. V. Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method. Soviet Physics JETP 44 (1976), no. 1, 106-112; translated from Ž. Èksper. Teoret. Fiz. 71 (1976), no. 1, 203-215.
20. Zakharov, V. E.; Shabat, A. B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics JETP 34 (1972), no. 1, 62-69; translated from Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118-134.
21. Zakharov, V. E.; Shabat, A. B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics JETP 34 (1972), no. 1, 62-69; translated from Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118-134.
22. 2-Sobolev space bijectivity of the scattering and inverse scattering transforms. Comm. Pure Appl. Math. 51 (1998), no. 7, 697-731.