×

Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation. (English) Zbl 0596.35109

The generalizations of the equations in the title \[ (1)\quad u_ t+u_ x+(F(u))_ x+u_{xxx}=0\quad and\quad (2)\quad u_ t+u_ x+(F(u))_ x-u_{xxt}=0 \] are considered. The main result is the following analogue of a result of Strauss for the equation (1). Theorem: Let \(F: R\to R\) be a \(C^{\infty}\) function such that \(| F'(s)| =O(| s|^{6+\epsilon})\) as \(s\to 0\) for some \(\epsilon >0\). Then there exists a number \(\delta_ F>0\) such that if I.V.P. for (2) at the initial function \(u(x,0)=\phi (x)\) for which \(\phi \in C^ 2_ b\cap N\) and \(| \phi |_ N<\delta_ F\), then the solution satisfies \(| u(x,t)| \leq A(1+t)^{-1/3}\) for all \(x\in R\) and \(t\geq 0\), where A is independent of x and t.
Reviewer: V.Kostova

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
81U30 Dispersion theory, dispersion relations arising in quantum theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Albert, J. Bona, and D. B. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, to appear.; J. Albert, J. Bona, and D. B. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, to appear. · Zbl 0634.35079
[2] Benjamin, T. B., The stability of solitary waves, (Proc. Roy. Soc. London Ser. A, 328 (1972)), 153
[3] Benjamin, T. B.; Bona, J. L.; Bose, D. K., Solitary-wave solutions for some model equations for waves in nonlinear dispersive media, (Germain, P.; Nayroles, B., Applications of Methods of Functional Analysis to problems in Mechanics. Applications of Methods of Functional Analysis to problems in Mechanics, Lecture Notes in Mathematics, Vol. 503 (1976), Springer-Verlag: Springer-Verlag New York/Berlin), 207 · Zbl 0346.76012
[4] Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272, 47 (1972) · Zbl 0229.35013
[5] Bona, J. L., On the stability of solitary waves, (Proc. Roy. Soc. London Ser. A, 344 (1975)), 363 · Zbl 0328.76016
[6] Bona, J. L., On solitary waves and their role in the evolution of long waves, (Amann, H.; Bazley, N.; Kirchgassner, K., Applications of Nonlinear Analysis in the Physical Sciences (1981), Pitman: Pitman New York), 183-205
[7] Bona, J. L.; Bose, D. K., Solitary-wave solutions for unidirectional wave equations having general forms of nonlinearity and dispersion, (Fluid Mechanics Research Institute Report 99 (1978), University of Essex)
[8] Bona, J. L.; Smith, R., The initial-value problem for the Korteweg-deVries equation, Philos. Trans. Roy. Soc. London Ser. A, 278, 555 (1975) · Zbl 0306.35027
[9] Fornberg, B.; Whitham, G. B., A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. Roy. Soc. London Ser. A, 289, 373-404 (1978) · Zbl 0384.65049
[10] Kato, T., On the Korteweg-deVries equation, Manuscripta Math., 28, 89-99 (1979) · Zbl 0415.35070
[11] Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, (Studies in Applied Mathematics. Studies in Applied Mathematics, Advances in Mathematics Supplementary Studies, Vol. 9 (1983), Academic Press: Academic Press New York), 93
[12] Korteweg, D. J.; de Vries, G., On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary wave, Philos. Magazine, 39, 422 (1895) · JFM 26.0881.02
[13] Olver, F. W.J, Asymptotics and Special Functions (1974), Academic Press: Academic Press New York · Zbl 0303.41035
[14] Strauss, W. A., Dispersion of low-energy waves for two conservative equations, Arch. Rational Mech. Anal., 55, 86 (1974) · Zbl 0289.35048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.