An Infinite Number Of Conservation Laws And Hamilton Structure Of The Boussinesq Equations

Based on a spectral problem of the Boussinesq type equations, a new Lax integrable system is obtained. Then, by selecting a new Lenard sequence Gn, a symplectic operator is deduced which proves that the soliton hierarchy has the Bi-Hamilton structures and Liouville integrability. According to the combina-tion of the characteristic function that satisfied with Ricatti equation, infinite conservation laws of the evolution equations are obtained. Finally, by the method of striving for the functional derivative under some constraint conditions, a cor-respondence between conserved density and Hamiltonians of the Boussinesq equations is acquired.