On the complete integrability of an equation having solitons but not known to have a Lax pair

Abstract

It is usually assumed that a system having N-soliton solutions is completely integrable. Here we have analyzed a set of equations occuring in case of capillary gravity waves. Though the system under discussion has N-soliton solutions, it has yet to be shown that the system is completely integrable. No Lax pair is known for the system. Here we show that the system is not completely integrable in the sense of Ablowitz et al.