Abstract
The authors show how to obtain the full asymptotic expansion for solutions of integrable wave equations to all orders, as t→∞. The method is rigorous and systematic and does not rely on an a priori ansatz for the form of the solution.
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[BC] Beals R., Coifman, R.R.: Scattering and Inverse, Scattering for First Order Systems. Comm. Pure Appl. Math.37, 39–90 (1984)
[BSa] Buslaev V.S., Sukhanov V.V.: Asymptotic Behavior of Solutions of the Korteweg-de Vries Equation. Proc. Sci. Seminar, LOMI120, 32–50 (1982) (Russian); J. Sov. Math.34, 1905–1920 (1986) (in English)
[BSb] Buslaev V.S., Sukhanov V.V.: On the Asymptotic Behavior ast→∞ of the Solutions of the Equationψ xx+u(x, t)ψ+(λ/4)ψ With Potentialu Satisfying the Korteweg-de Vries Equation, I. Prob. Math. Phys.10, M. Birman, ed., 70–102, (1982) (in Russian); Sel. Math. Sov.4, 225–248 (1985) (in English); II. Proc. Sci. Seminar LOMI138, 8–32 (1984) (in Russian); J. Sov. Math.32, 426–446 (1986) (in English); III. Prob. Math. Phys.11, M. Birman, ed., 78–113, (1986) (in Russian)
[DIZ] Deift P. A., Its, A.R., Zhou, X.: Long-time Asymptotics for Integrable Nonlinear Wave Equations. Important developments in Soliton theory. Fokas, A.S., Zakharov, V.E. (eds.) Berlin, Heidelberg, New York: Springer, 1993
[DZ1] Deift P.A., Zhou, X.: A Steepest Descent Method for Oscillatory Riemann-Hilbert Problems. Asymptotics for the MKdV equation. Ann. of Math.137, 295–368 (1993)
[DZ2] Deift P.A., Zhou X.: Long-time Asymptotics for the Autocorrelation Function of the Transverse Ising Chain at the Critical Magnetic Field. To appear in the Proceedings of the Nato Advanced Research Workshop: Singular Limits of Dispersive Waves, Lyons, France, 8–12 July, 1992
[DZ3] Deift P.A., Zhou X.: Asymptotics for the Painlevè II equation. Announcement in Adv. Stud. Pure Math.23, 17–26 (1994). Full paper to appear in Comm. Pure Appl. Math 1994
[DVZ] Deift, P.A., Venakides, S., Zhou, X.: The Collisionless Shock Region for the Long-time Behavior of Solutions of the KdV equation. Comm. Pure Appl. Math.47, 199–206 (1994)
[K] Kamvissis, S.: Long-time Behavior of the Toda Lattice Under Initial Data Decaying at Infinity. Commun. Math. Phys.153, 479–519 (1993)
[N] Novokshenov, V.Y.: Asymptotics ast→∞ of the Solution of the Cauchy, Problem for the Nonlinear Schrödinger Equation. Sov. Math. Dokl.,21, 529–533 (1980)
[SA1] Segur, H., Ablowitz, M.J.: Asymptotics Solutions and Conservation Laws for the Non-Linear Schrödinger equation. Part I, J. Math. Phys.17, 710–713 (1976)
[SA2] Segur, H., Ablowitz, M.J.: Asymptotic solutions of the Korteweg de Vries equation. Stud. Appl. Math.57(1), 13–44 (1977)
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Deift, P.A., Zhou, X. Long-time asymptotics for integrable systems. Higher order theory. Commun.Math. Phys. 165, 175–191 (1994). https://doi.org/10.1007/BF02099741
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DOI: https://doi.org/10.1007/BF02099741