Long time asymptotics of the Korteweg-de Vries equation
HTML articles powered by AMS MathViewer
- by Stephanos Venakides PDF
- Trans. Amer. Math. Soc. 293 (1986), 411-419 Request permission
Abstract:
We study the long time evolution of the solution to the Kortewegde Vries equation with initial data $\upsilon (x)$ which satisfy \[ \lim \limits _{x \to - \infty } \upsilon (x) = - 1,\qquad \lim \limits _{x \to + \infty } \upsilon (x) = 0\] We show that as $t \to \infty$ the step emits a wavetrain of solitons which asymptotically have twice the amplitude of the initial step. We derive a lower bound of the number of solitons separated at time $t$ for $t$ large.References
-
E. Ja. Hruslov, Asymptotics of the solution of the Cauchy problem for the Korteweg-de Vries equation with initial data of step type, Math USSR-Sb. 28 (1976).
A. Cohen and T. Kappeler, Scattering and inverse scattering for step-like potentials, The Schrödinger Equation, 1983, Preprint.
- Amy Cohen, Solutions of the Korteweg-de Vries equation with steplike initial profile, Comm. Partial Differential Equations 9 (1984), no. 8, 751–806. MR 748367, DOI 10.1080/03605308408820347
- V. Buslaev and V. Fomin, An inverse scattering problem for the one-dimensional Schrödinger equation on the entire axis, Vestnik Leningrad. Univ. 17 (1962), no. 1, 56–64 (Russian, with English summary). MR 0139371
- Stephanos Venakides, The zero dispersion limit of the Korteweg-de Vries equation for initial potentials with nontrivial reflection coefficient, Comm. Pure Appl. Math. 38 (1985), no. 2, 125–155. MR 780069, DOI 10.1002/cpa.3160380202 I, Kay and H. E. Moses, J. Appl. Phys. 27 (1956), 1503-1508.
- Peter D. Lax and C. David Levermore, The small dispersion limit of the Korteweg-de Vries equation. I, Comm. Pure Appl. Math. 36 (1983), no. 3, 253–290. MR 697466, DOI 10.1002/cpa.3160360302
- H. P. McKean, Theta functions, solitons, and singular curves, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977) Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 237–254. MR 535596
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 293 (1986), 411-419
- MSC: Primary 35B40; Secondary 35Q20
- DOI: https://doi.org/10.1090/S0002-9947-1986-0814929-0
- MathSciNet review: 814929