Two Hierarchies Of Nonlinear Evolution Equations And Their Integrability

With the aid of Lenard recursion equations,integrable hierarchies of nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is proposed,in which the first nontrivial member in the positive flows can be reduced to a new generalization of the Wadati-Konno-Ichikawa(WKI)equation.Further,a new generalization of the FokasLenells(FL)equation is derived from the negative flows.Resorting to these two Lax pairs and Riccati-type equations,the infinite conservation laws of these two corresponding equations are obtained.The Bi-Hamiltonian stuctures of this negative flows are constructed by using the trace identity.