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[A]Ablowitz, M. J., Applications of slowly varying nonlinear dispersive wave theories.Stud. Appl. Math., 50 (1971), 329–344.
[AKNS]Ablowitz, M. J., Kaup, D. J., Newell, A. C. &Segur, H. The inverse scattering transform—Fourier analysis for nonlinear problems.Stud. Appl. Math.. 53 (1974), 249–315.
[AS]Ablowitz, M. J. &Segur, H. Solitons and the Inverse Scattering Transform SIAM Stud. Appl. Math., 4. SIAM, Philadelphia, PA, 1981.
[BC]Beals, R. &Coifman, R. R., Scattering and inverse scattering for first order systems.Comm. Pure Appl. Math., 37 (1984), 39–90.
[Br]Bronski, J. C., Nonlinear scattering and analyticity properties of solitons.J. Nonlinear Sci., 8 (1998), 161–182.
[CG]Clancey, K. &Gohberg, I.,Factorization of Matrix Functions and Singular Integral Operators, Oper. Theory: Adv. Appl., 3, Birkhäuser, Basel-Boston, MA, 1981.
[Cr]Craig, W., KAM theory in infinite dimensions, inDynamical Systems and Probabilistic Methods in Partial Differential Equations (Berkeley, CA 1994), pp. 31–46. Lectures in Appl. Math., 31 Amer. Math. Soc., Providence, RI, 1996.
[CrW]Craig, W. &Wayne, C. E. Newton's method and periodic solutions of nonlinear wave equations.Comm. Pure Appl. Math., 46 (1993), 1409–1498.
[DIZ]Deift, P., Its, A., &Zhou, X., Long-time asymptotics for integrable nonlinear wave equations, inImportant Developments in Soliton Theory 1989–1990, pp. 181–204. Springer-Verlag Berlin, 1993.
[DKMVZ]Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. &Zhou, X., Strong asymptotics of orthogonal polynomials with respect to exponential weights.Comm. Pure Appl. Math., 52 (1999), 1491–1552.
[Du]Duren, P. L.,Theory of H p Spaces, Pure Appl. Math., 38. Academic Press, New York-London, 1970.
[DZ1]Deift, P. &Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation.Ann. of Math (2), 137 (1993), 295–368
[DZ2]—Long-Time Behaviour of the Non-Focusing Nonlinear Schrödinger Equation. A Case Study. New Series: Lectures in Math. Sciences., 5. University of Tokyo. Tokyo, 1994.
[DZ3]— Near integrable systems on the line. A case study—perturbation theory of the defocusing nonlinear Schrödinger equation.Math. Res. Lett., 4 (1997), 761–772.
[DZ4]Deift, P. & Zhou, X. Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Preprint, 2002,ArXiv:math.AP/0206222.
[DZ5]Deift, P. & Zhou, X. AprioriL p estimates for solutions of Riemann-Hilbert problems. Preprint, 2002.ar Xiv:math. CA/0206224.
[DZW]Deift, P. & Zhou, X. An extended web version of this paper, posted on http://www.ml.kva.se/publications/acta/webarticles/deift.
[FaT]Faddeev, L. &Takhtajan, L.,Hamiltonian Methods in the Theory of Solitons. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1987.
[FL]Fokas, A. S. &Liu, Q. M. Asymptotic integrability of water waves.Phys. Rev. Lett, 77 (1996), 2347–2351.
[GV1]Ginibre, J. &Velo, G. On a class of nonlinear Schrödinger equations, III. Special theories in dimensions 1,2 and 3.Ann. Inst. H. Poincaré Phys. Théor., 28 (1978), 287–316.
[GV2]— Scattering theory in the energy space for a class of nonlinear Schrödinger equations.J. Math. Pures Appl. (9), 64 (1985), 363–401.
[H]Hartman, P.,Ordinary Differential Equations, Wiley & Sons, New York, 1964.
[HLP]Hardy, G. H., Littlewood, J. E. &Pólya, G.,Inequalities, 2nd edition. Cambridge Univ. Press Cambridge, 1952.
[HN]Hayashi, N. &Naumkin, P. I., Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations.Amer. J. Math., 120 (1998), 369–389.
[K1]Kaup, D. J., A perturbation expansion for the Zakharov-Shabat inverse scattering transform.SIAM J. Appl. Math., 31 (1976), 121–133.
[K2]— Second-order perturbations for solitons in optical fibers.Phys. Rev. A, 44 (1991), 4582.
[Ka]Kappeler, T., Solutions to the Korteweg-de Vries equation with irregular initial profile.Comm. Partial Differential Equations. 11 (1986), 927–945.
[KGSV]Kivshar, Y. S., Gredeskul, S. A., Sanchez, A. &Vazques, L., Localization decay induced by strong nonlinearity in disordered system.Phys. Rev. Lett. L, 64 (1990), 1693.
[KM]Karpman, V. I., &Maslov, E. M., Structure of tails produced under the action perturbations on solitons.Soviet Phys. JETP, 48 (1978), 252–259.
[KM]Kaup, D. J. &Newell, A. C., Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory.Proc. Roy. Soc. London Ser. A, 361 (1978), 413–446.
[Ko]Kodama, Y., On integrable systems with higher order corrections.Phys. Lett. A., 107 (1985), 245–249.
[Ku1]Kuksin, S., Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum.Funct. Anal. Appl., 21 (1987), 192–205.
[Ku2]—Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Lecture Notes in Math., 1556, Springer-Verlag, Berlin, 1993.
[MKS]McKean, H. P. &Shatah, J., The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form.Comm. Pure Appl. Math., 44 (1991), 1067–1080.
[MKS]McLaughlin, D. W. &Scott, A. C. Perturbation analysis of fluxon dynamics.Phys. Rev. A., 18 (1978), 1652–1680.
[MMT]Majda, A. J., McLaughlin, D. W. &Tabak, E. G., A one-dimensional model for dispersive wave turbulence.J. Nonlinear Sci., 7 (1997), 9–44.
[Mo1]Moser, J., Finitely many mass points on the line under the influence of an exponential potential—an integrable system, inDynamical Systems, Theory and Applications. (Seattle, WA, 1974). pp. 467–497. Lecture Notes in Phys., 38. Springer-Verlag, Berlin, 1975.
[Mo2]— A rapidly convergent iteration method and non-linear differential equations, II.Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 499–535.
[N]Nikolenko, N. V., The method of Poincaré normal forms in problems of integrability of equations of evolution type.Russian math. Surveys, 41 (1986), 63–114.
[O]Ozawa, T. Long range scattering for nonlinear Schrödinger equations in one space dimension.Comm. Math. Phys. 139 (1991), 479–493.
[P]Poincaré, H., Sur les propriétés des fonctions définies par les équations aux différences partielles (Thèses présentées à la Faculté des Sciences de Paris, 1879), inEuvres, tome I., pp. IL-CXXXII Gauthier-Villars, Paris, 1928.
[RS]Reed, M. &Simon, B. Methods of Modern Mathematical Physics, III. Scattering Theory. Academic Press, New York-London, 1979.
[Si]Siegel, C. L., Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung.Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. Math.-Phys.-Chem. Abt. (1952), 21–30.
[St]Strauss, W. A., Dispersion of low-energy waves for two conservative equations.Arch. Rational. Mech. Anal., 55 (1974), 86–92.
[W]Whitham, G. B.,Linear and Nonlinear Waves. Wiley-Interscience, New York-London-Sidney, 1974.
[Z1]Zhou, X. L 2-Sobolev space bijectivity of the scattering and inverse scattering transforms.Comm. Pure Appl. Math. 51 (1998), 697–731.
[Z2]— Strong regularizing effect of integrable systems.Comm. Partial Differential Equations, 22 (1997), 503–526.
[Za]Zakharov, V. E.,Kolmogorov Spectra in Weak Turbulence Problems. Handbook Plasma Phys., Vol. 2, 1984.
[ZaM]Zakharov, V. E. &Manakov, S. V., Asymptotic behavior of nonlinear wave systems integrated by the inverse scattering method.Soviet Phys. JETP, 44 (1976), 106–112.
[ZaS]Zakharov, V. E. &Shabat, A. B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media.Soviet Phys. JETP, 34 (1972), 62–69.
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In memory of Jürgen Moser
A more detailed, extended version of this paper is posted on http://www.ml.kva.se/publications/acta/webarticles/deift. Throughout this paper we refer to the web version as [DZW].
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Deift, P., Zhou, X. Perturbation theory for infinite-dimensional integrable systems on the line. A case study. Acta Math. 188, 163–262 (2002). https://doi.org/10.1007/BF02392683
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DOI: https://doi.org/10.1007/BF02392683