Stability and instability of solitary waves for a system of coupled BBM equations

Abstract

We consider the system of two coupled generalized BBM equations where U = U(x, t) is an -valued function of the real variables x and t, c 0 is a non-negative parameter, A is a 2 × 2 real positive definite matrix, and ∇H is the gradient of a C 3 homogeneous function . Under suitable conditions on A and H we show that system (0.1) has solitary-wave solutions which are stable or unstable according to the variation of the speed c. Our results are obtained by methods developed by M. Grillakis, J. Shatah and W.A. Strauss (1987). Stability theory of solitary waves in the presence of symmetry I. Journal of Functional Analysis., 74, 160–197 and J.L. Bona, P.E. Souganidis and W.A. Strauss (1987). Stability and instability of solitary waves of Korteweg-de Vries type. Proceedings of the Royal Society of London, Series A, 411, 395–412 and P.E. Souganidis and W.A. Strauss (1990). Instability of a class of dispersive solitary waves. Proceedings of the Royal Society of Edinburgh, Section A, 114, 195–212 and are generalizations to the system (0.1) of previous results on stability and instability of solitary waves for the scalar generalized BBM equation.