Integrability conditions for Boussinesq type systems

TL;DR

This paper develops integrability conditions for Boussinesq type systems by constructing canonical densities through an extended diagonalisation method, aiding classification of such equations.

Contribution

It introduces a method to derive integrability conditions for Boussinesq systems using generalized diagonalisation, expanding the symmetry approach.

Findings

Derived canonical densities for Boussinesq systems.

Established strong constraints on the form of equations for integrability.

Extended the formal diagonalisation technique to new classes of equations.

Abstract

The symmetry approach to the classification of evolution integrable partial differential equations (see, for example \cite{MikShaSok91}) produces an infinite series of functions, defined in terms of the right hand side, that are conserved densities of any equation having infinitely many infinitesimal symmetries. For instance, the function $\frac{\partial f}{\partial u_{x}}$ has to be a conserved density of any integrable equation of the KdV type $u_t=u_{xxx}+f(u,u_x)$. This fact imposes very strong conditions on the form of the function $f$. In this paper we construct similar canonical densities for equations of the Boussinesq type. In order to do that, we write the equations as evolution systems and generalise the formal diagonalisation procedure proposed in \cite{MSY} to these systems.