A Soliton Hierarchy Associated With A Energy-dependent 3×3 Matrix Spectral Problem And Their Hamiltonian Structures

By introducing a energy-dependent 3×3 matrix spectral problem, we construct soliton hier-archy as follows. First, we construct a loop algebra and introduce the concept rank, then take asolution of V_x = [U, V] by using the homogeneous rank convention; second, we search for aΔ_n∈(?)such that for V(n)=(λⁿV)₊+Δ_n, we could get equation hierarchy U_t-V_<sup>(n)+[U,V⁽ⁿ⁾]=0.Furthermore the Liouville integrability and Hamiltinian structure of equation hierarchy could besuccessfully estabilished by making use of the trace identity.