SYMMETRIES OF EQUATIONS q_(tt)=g(q,q_x,q_(xx),…) AND THE FORMAL COMPLETELY INTEGRABILITY OF BOUSSINESQ EQUATION

In this paper the symmetries of equations q_(tt)=g(q,q_1,q_2,…) are discussed, where q=q(x,t) and qi_=~iq/x~i. It is shown that if g=aq_s (q,…,q_r), a=const, s-r≥2, then any symmetry of the equation will be linear with respect to the term of highest order. Furthermore, if the equation can be reduced to a Hamiltonian equation, then pairs of its conserved densities are in involution. As an application of this result, the Boussinesq equation q_(tt)=q_4 6q_1q_2 q_2 is shown to be a formal completely integrable Hamiltonian equation.