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New families of solitons with compact support for Boussinesq-like \(B(m,n)\) equations with fully nonlinear dispersion. (English) Zbl 1038.35082
Summary: Studying special solitons with compact support is of important significance in soliton theory. There exists much good work in the area of usual solitons, but there appears very little in the way of compacton solutions. In this paper, the Boussinesq-like equations with fully nonlinear dispersion, \(B(m,n)\) equations, \(u_{tt}=(u^n)_{xx}+(u^m)_{xxxx}\), which were introduced by the author to understand the role of nonlinear dispersion in pattern formation [Z. Y. Yan, Commun. Theor. Phys. 36, 385–390 (2001)], are investigated again. New soliton solutions with compact support are found which have the following remarkable properties: They collide elastically, but unlike the usual solitons, they have compact support. With the aid of Maple, the two special cases, \(B(2,2)\) equation and \(B(3,3)\) equation, are chosen to illustrate the concrete scheme of the decomposition method in the \(B(m,n)\) equation. In addition, two new general compacton solutions of the \(B(m,m)\) equation are also found.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B25 Solitary waves for incompressible inviscid fluids
Software:
Maple
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