Matrix generalisation of the modified Korteweg-de Vries equation

Abstract

The matrix-modified Korteweg-de Vries equation (MMKdV) Phi t+3( Phi , Phi xx)-6 Phi Phi x Phi + Phi xxx=0 is studied by means of the inverse scattering problem. The Lax L-A pair is found; the direct and the inverse scattering problems for the L-operator are investigated; and the time evolution of the scattering data is traced. The simplest solutions of the MMKdV equation and the infinite series of integrals of motion are constructed.