In this paper, we show that the coupled modified Kd V equations possess rich mathematical structures and some remarkable properties. The connections between the system and skew orthogonal polynomials,convergence acceleration algorithms and Laurent property are discussed in detail.
A novel hierarchy of integrable nonlinear evolution equations related to the combined Ablowitz–Kaup–Newell–Segur（AKNS） and Wadati–Konno–Ichikawa（WKI） spectral problems is proposed,from which the Lax pair for a corresponding negative flow and its infinite many conservation laws are obtained.Furthermore,a reduction of this hierarchy is discussed,by which a generalized sinh-Gordon equation is derived on the basis of its negative flow.
We present a matrix coupled dispersionless(CD)system.A Lax pair for the matrix CD system is proposed and Darboux transformation is constructed on the solutions of the matrix CD system and the associated Lax pair.We express an N soliton formula for the solutions of the matrix CD system in terms of quasideterminants.By using properties of the quasideterminants,we obtain some exact solutions,including bright and dark-type solitons,rogue wave and breather solutions of the matrix CD system.Furthermore,it has been shown that the solutions of the matrix CD system are expressed in terms of solutions to the usual CD system,sine-Gordon equation and Maxwell-Bloch system.
