Integrable nonlinear perturbed hierarchies of NLS-mKdV equation and soliton solutions

Abstract

We propose three spectral problems for NLS-mKdV equation by combining three integrable coupling ways. Then we obtain three nonlinear perturbation terms to derive three integrable nonlinear perturbed hierarchies of the NLS-mKdV equation. We proved the Lax integrability of the integrable nonlinear perturbed hierarchies. On the basis of a special orthogonal group, we prove the Liouville integrability of a third-order integrable nonlinear perturbed hierarchy of NLS-mKdV equation by deriving its bi-Hamiltonian structures. We build three Darboux matrices for constructing the Darboux transformations of the first two equations. As applications of the Darboux transformation, we present explicit solutions of these equations, three-dimensional plots,  and density profiles the evolution of solitary waves.