Conservation laws, bright matter wave solitons and modulational instability of nonlinear Schrödinger equation with time-dependent nonlinearity

In this paper, we consider a general form of nonlinear Schrödinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schrödinger equation is identified by admitting an infinite number of conservation laws. Using the Darboux transformation method, we obtain some explicit bright multi-soliton solutions in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.

► The complete integrability of such nonlinear Schrödinger equation is confirmed by admitting an infinite number of conservation laws. ► The explicit bright multi-soliton solutions are constructed via algebraic iterative algorithm. ► The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. ► The modulational instability of solutions is analyzed in the presence of small perturbation.