Conservation laws and integral relations for the Boussinesq equation

Abstract

We are concerned with conservation laws and integral relations associated with rational solutions of the Boussinesq equation, a soliton equation solvable by inverse scattering which was first introduced by Boussinesq in 1871. The rational solutions are logarithmic derivatives of a polynomial, are algebraically decaying and have a similar appearance to rogue-wave solutions of the focusing nonlinear Schr\"{o}dinger equation. For these rational solutions the constants of motion associated with the conserved quantities are zero and they have some interesting integral relations which depend on the total degree of the associated polynomial