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Large-time asymptotic behaviour of a step for the Benjamin–Bona–Mahony–Burgers equation

Published online by Cambridge University Press:  14 November 2011

P. I. Naumkin
P. I. Naumkin
Affiliation:
Department of Computational Mathematics and Cybernetics, Moscow State University, Moscow 119 899, Russia
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Abstract

The asymptotic behaviour for large-time values of solutions of the initial-boundary-value problem for the Benjamin–Bona–Mahony–Burgers (BBMB) equation is studied. The solution of the initial-boundary-value problem is proved to converge to the standing wave as t → ∞ uniformly with respect to x ∈ R1. The estimate obtained of the time-decay rate of the remainder appears to depend on the rate of decay at infinity of the deviation of initial data from the standing wave. Due to the known expansion formulae with respect to the eigenfunctions associated to the stationary Schrodinger operator with the standing wave as a potential, it is possible to find the second term of the large-time asymptotic expansion of the solution to the initial-boundary-value problem for the BBMB equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 1 , 1996 , pp. 1 - 18
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Amick, C. J., Bona, J. L. and Schonbek, M. E.. Decay of solutions of some nonlinear wave equations. J. Differential Equations 81 (1989), 1–49.Google Scholar
2Benjamin, T. B.. Lectures in nonlinear wave motion. Lectures in Appl. Math. 15 (1974), 3–47.Google Scholar
3Benjamin, T. B., Bona, J. L. and Mahony, J. J.. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 111 (1972), 47–78.Google Scholar
4Bona, J. L.. Model equations for waves in nonlinear dispersive systems. In Proceedings of the International Congress of Mathematicians. Helsinki, 1978, 887–94 (Academia Scientiarum Fennica, Hungary, 1980).Google Scholar
5Bona, J. L. and Schonbek, M. E.. Travelling wave solutions to the Korteweg-de Vries-Burgers equation. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 207–26.Google Scholar
6Bona, J. L. and Smith, R. W.. The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555–604.Google Scholar
7Broer, L. J. F.. Approximate equations for long water waves. Appl. Sci. Res. 31 (1975), 337–95.Google Scholar
8Cole, J. D.. On a quasilinear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9 (1950), 225–36.Google Scholar
9Hopf, E.. The partial differential equation u1 + νux = νuxx. Comm. Pure Appl. Math. 3 (1950), 201–30.Google Scholar
10Il'in, A. M. and Olei'nik, O. A.. Behaviour of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time. Dokl. Akad. Nauk SSSR 120 (1958), 25–8; Amer. Math. Soc. Transl. 42 (1964), 19-23.Google Scholar
11Kolmogoroff, A., Petrovsky, I. and Piscounoff, N.. Etude de l'equations de la diffusion avec croiseance de la quantite de matiere et son application a un probleme biologique. Bull. Univ. Moscow. Ser. Int. Sect. A 1 (1937), 1–25.Google Scholar
12Naumkin, P. I. and Shishmarev, I. A.. The step-decay problem for the Korteweg-de Vries-Burgers equation. Fund. Anal. Appl. 25 (1991), 16–25.Google Scholar
13Naumkin, P. I. and Shishmarev, I. A.. On the decay of step for the Korteweg-de Vries-Burgers equation. Fund. Anal. Appl. 26 (1992), 148–51.Google Scholar
14Naumkin, P. I. and Shishmarev, I. A.. Nonlinear Nonlocal Equations in the Theory of Waves, Translations of Mathematical Monographs 133 (Providence, R.I.: American Mathematical Society, 1994).CrossRefGoogle Scholar
15Peregrine, D. H.. Calculations of the development of an undular bore. J. Fluid Mech. 25 (1966), 321–30.Google Scholar
16Titchmarsh, E. C.. Eigenfunction Expansions associated with Second-order Differential Equations (Oxford: Clarendon Press, 1946).Google Scholar
17Weyl, H.. Über gewöhnliche lineare Differentialgleichungen mit singularen Stellen und ihre Eigenfunktionen. Göttinger Nachr. (1909), 37–64.Google Scholar