Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Instability of a class of dispersive solitary waves*

P. E. Souganidisa1 and W. A. Straussa1

a1 Division of Applied Mathematics, Department of Mathematics, and Lefschetz Center for Dynamical Systems, Brown University, Providence, RI02912, U.S.A.

Synopsis

This paper studies the stability and instability properties of solitary wave solutions φ(x – ct) of a general class of evolution equations of the form Muttf(u)x=0, which support weakly nonlinear dispersive waves. It turns out that, depending on their speed c and the relation between the dispersion (i.e. the order of the pseudodifferential operator) and the nonlinearity, travelling waves maybe stable or unstable. Sharp conditions to that effect are given.

(Received May 05 1989)

(Revised September 13 1989)

Footnotes

* Supported in part by National Science Foundation Grants DMS-86-01258, DMS-87-22331 and DMS-88-01208, ARO Grant DAAL-3-86-0074, DARPA Grant F49620-88-C-0129, ONR Grant N00014-83-K-0542 and the Sloan Foundation.