On the stability of internal waves

The extended KdV equation u(t) + uu(x) + alpha u(2)u(x) + u(xxx) = 0 is widely used as a model describing internal waves in ideal fluids. The equation admits a family of negative and positive solitary waves Phi(c). These solitary waves exhibit the typical broadening effect seen in internal waves. It is shown here that all solitary-wave solutions of the extended KdV equation are orbitally stable. The proof of stability is based on the general theory of Grillakis et al (1987 J. Funct. Anal. 74 160) for equations of the form u(t) = JE'(u) which have two conserved integrals E(u) and V(u). A spectral analysis of the linear operator L-c = E ''(Phi(c))+ cV ''(Phi(c)) reduces the question of orbital stability to the question of whether the scalar function d(c) = E(Phi(c)) + cV (Phi(c)) is convex. To prove the stability, an explicit calculation showing the convexity is performed.