Bi-Hamiltonian Structures Of Some CH Type Equations

Firstly, we obtain bi-Hamiltonian structures of the Geng-Xue equation and a three-component CH type equation which was also proposed by Geng and Xue. Then we recover the bi-Hamiltonian structure of Novikov equation by the Dirac reduction for those of Geng-Xue equation. Infinitely conserved quantities either can be obtained by bi-Hamiltonian structure, or can be drived by the Riccati equation based on Lax pair. Secondly, we relate Geng-Xue equation to a first negative flow of modified Boussinesq hierarchy by a reciprocal transformation. At last, we consider a3×3spectral problem which contains almost all known3×3spectral problems of the CH type equations, and obtain bi-Hamiltonian structure of a four-component CH type equation corresponding to the3×3spectral problem. Then we study some reductions of the3×3spectral problem, and workout bi-Hamiltonian structures of the reduced equations and their infinite conserved quantities.