N-Fold Darboux Transformation for the Nonlocal Nonlinear Schrodinger(NNLS)Equation with the Self-Induced PT-Symmetric Potential

The Darboux transformation (DT) method is studied in a lot of local equations, but there are few of work to solve nonlocal equations by DT. In this letter, we solve the nonlocal nonlinear Schrödinger equation (NNLSE) with the self-induced PT-symmetric potential by DT. Then the N-fold DT of NNLSE is derived with the help of the gauge transformation between the Lax pairs. Then we derive some novel exact solutions including the bright soliton, breather wave soliton. In particularly, the dynamic features of one-soliton, two-soliton, three-soliton solutions and the elastic interactions between the two solitons are displayed.

Some space-time fractional bright–dark solitons and propagation manipulations for a fractional Gross–Pitaevskii equation with an external potential

Some nonautonomous bright–dark solitons(NBDSs)and nonautonomous controllable behaviors in the conformable space-time fractional Gross–Pitaevskii(FGP)equation with some external potentials are derived.We consider the relations between the space-time FGP equation and the fractional nonlinear Schr?dinger equation and analyze the properties of the obtained equation with group velocity dispersion and spatiotemporal dispersion.Then,some constraint conditions of the valid soliton solutions are given.Furthermore,we consider the effect ofαandβin NBDSs of the space-time FGP equation.Some fractional spatial–temporal controlling wave prolong phenomena are considered,and some different propagation dynamics are generated via the different parametersαandβ.We study novel shape bright soliton solution,novel‘h’-shape dark soliton and some interactions of nonautonomous bright–dark solitons.The reported results of some novel interactions are considered,which can explain some models of the electrical and optical fields.

COUPLED NONLOCAL NONLINEAR SCHR?DINGER EQUATION AND N-SOLITON SOLUTION FORMULA WITH DARBOUX TRANSFORMATION

Ablowitz and Musslimani proposed some new nonlocal nonlinear integrable equations including the nonlocal integrable nonlinear Schr?dinger equation. In this paper, we investigate the Darboux transformation of coupled nonlocal nonlinear Schr?dinger(CNNLS) equation with a spectral problem. Starting from a special Lax pairs, the CNNLS equation is constructed. Then, we obtain the one-, two-and N-soliton solution formulas of the CNNLS equation with N-fold Darboux transformation. Based on the obtained solutions, the propagation and interaction structures of these multi-solitons are shown, the evolution structures of the one-dark and one-bright solitons are exhibited with N = 1,and the overtaking elastic interactions among the two-dark and two-bright solitons are considered with N = 2. The obtained results are different from those of the solutions of the local nonlinear equations. Some different propagation phenomena can also be produced through manipulating multi-soliton waves.The results in this paper might be helpful for understanding some physical phenomena described in plasmas.

A NEW DISCRETE INTEGRABLE COUPLING SYSTEM AND ITS HAMILTONIAN STRUCTURE FOR THE MODIFIED TODA LATTICE HIERARCHY

We present a new discrete integrable coupling system by using the matrix Lax pair U, V C s/（4）. A novel spectral problem of modified Toda lattice soliton hierarchy is considered. Then, a new discrete integrable coupling equation hierarchy is obtained through the method of the enlarged Lax pair. Finally, we obtain the Hamiltonian structure of the integrable coupling system of the soliton equation hierarchy using the matrix-form trace identity. This discrete integrable coupling system includes a kind of a modified Toda lattice hierarchy.