The Stability of Solitary Waves on a Viscous Liquid of Finite Depth

The stability of unidirectional propagation of long waves on a viscous liquid, in a channel of finite depth H, satisfying the viscous counterpart of Korteweg-de Vries (KdV) equation whose solutions represent solitary waves, is explored. It is shown that the solutions of such solitary waves on a viscous liquid are stable. The proof, given here, depends on two nonlinear functionals which are invariant in time for the solutions. It is further shown that the viscous KdV equation possesses a variational principle. Also, considered here are two nonlinear invariants for the regularized equation, namely, the viscous Benjamin, Bona and Mahony (BBM) equation which has smoother mathematical properties. It has been proved that, in the absence of viscosity, both KdV and BBM equations share the same two nonlinear invariants, but in the presence of viscosity, these two equations have different nonlinear invariants.