Solitary and Periodic Waves of a New Kind

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences Vol. 354; no. 1713; pp. 1775 - 1806
• Main Author: Benjamin, Thomas Brooke
• Format: Journal Article
• Language: English
• Published: London The Royal Society 15.07.1996
• Subjects: Approximation; Eigenfunctions; Eigenvalues; Existence theorems; Fourier transformations; Linear transformations; Nontrivial solutions; Perturbation theory; Solitons; Waves
• ISSN: 1364-503X; 1471-2962
• DOI: 10.1098/rsta.1996.0078

Various aspects of an unusual evolutionary equation are analysed. This pseudo-differential equation arises as an approximate model for nonlinear dispersive waves in a two-fluid system where the interface is subject to capillarity and the depth of the upper fluid is much smaller than the depth of the lower, more dense fluid. The character of solitary and periodic waves of permanent form is shown to depend primarily on a parameter $\gamma $ = $\frac{1}{2}\alpha $/($\beta $C)$^{1/2}\in $ (0, 1), where $\alpha $ and $\beta $ are constants representing two different types of dispersion and C > 0 represents wave velocity relative to the velocity of infinitesimal long waves. The asymptotic properties of solitary waves are examined first, being demonstrated to differ markedly whenever $\gamma $ > 0 from those when $\gamma $ = 0. When $\gamma $ is close to 1, moreover, solitary waves have protracted oscillations at their outskirts. In all cases $\gamma \in $ (0, 1), solitary-wave solutions of the original equation are not single signed; but their Fourier transforms are positive functions, and this property is made the basis of existence theories using positive-operator methods. Periodic solutions are proved to exist by consideration of the nonlinear equation for their Fourier series, which is posed in a cone of positive sequences from the space l$_{2}$. Then solitary-wave solutions are treated by a comparable strategy, which also relies on Leray-Schauder degree theory applied to a positive-operator equation but must circumvent the difficulty that the operator in question is not compact. Finally, the existence and stability of solitary waves is shown to be inferable by comparatively simple means, with use of the implicit-function theorem, in the case that $\gamma $ is sufficiently small.