Generalization Of AKNS System And Some Applications

This paper includes two parts. The ?rst part focuses on several nonautonomous nonlinear mathematical physics equations about soliton management, constructs a general non-isospectral AKNS system, in which the spectral parameter is determined by an ordinary di?erential equation with high order polynomial nonlinearity. The second part generalize the AKNS hierarchy determined by the semisimple matrix Lie algebra to the AKNS hierarchy determined by a non-semisimple matrix Lie algebra,and investigate a kind of AKNS integrable couplings.In the ?rst part of this paper, we introduce integrable nonautonomous nonlinear Schr¨odinger equations, which includes the local and the nonlocal case, and obtain PT symmetric local and nonlocal nonautonomous Gross-Pitaevskii equations. Based on this, we present the in?nite conservation laws of the nonautonomous nonlinear Schr¨odinger equation; construct their Darboux transformation and present the determinant representation of the N-fold soliton-like solutions; and construct the connection between the nonlocal nonlinear Schr¨odinger equation and the nonautonomous Heisenberg spin chain equation with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. Furthermore, we apply these results to the soliton management in Bose-Einstein condensates and nonlinear optics, the dynamics in ferromagnetic magnetism, and obtain some new physical results. For the PT symmetric local and nonlocal nonautonomous Gross-Pitaevskii equations, we discussed their inverse scattering transformation. We ?nd that unlike the local case,the PT-symmetry of the nonlocal Gross-Pitaevskii equation allows two di?erent choices of the symmetry reductions of the eigenfunctions which guarantee that it has two di?erent kinds of inverse scattering solutions.In the second part of this paper, we construct a kind of non-semisimple matrix Lie algebra, and introduce a novel hierarchy of integrable couplings for the AKNS system. Moreover, we discuss the bi-Hamiltonian structures of the resulted coupling system, and the Liouville integrability has been proved. We also propose a formulation of Darboux transformation for the couplings and present the exact one-soliton-like solution for the integrable couplings of the second and third order AKNS equations. In addition, we exploit AKNS integrable couplings to bi-integrable couplings of the AKNS system, and generalize the Darboux transformation to the bi-integrable couplings of the AKNS system.