Non-autonomous exact solutions and dynamic behaviors for the variable coefficient nonlinear Schrödinger equation with external potential

The nonlinear Schr & ouml;dinger(NLS) equation represents a nonlinear dynamical system, which is usually used to describe nonlinear waves in deep water, self-focusing of intense lasers contained in electrolytes, and so on. The exact solutions of the variable coefficient nonlinear Schr & ouml;dinger equation with external potential are considered. The variable coefficient nonlinear Schr & ouml;dinger equation is transformed into a constant coefficient nonlinear equation by using the similarity transformation method, the exact solutions of the constant coefficient nonlinear equation are generated by using the homogeneous balance method. We obtain one-soliton, two-soliton, kink type soliton, bright soliton, parabolic soliton and rogue wave solutions. A 'stepped' type soliton solution is obtained in this paper, which is a novel type solution and different from the solutions of most nonlinear Schr & ouml;dinger equations. Some special dynamic behaviors of solitons of the variable coefficient nonlinear Schr & ouml;dinger equation with external potential are obtained via selecting some free functions. We found that the numerical simulation result is consistent with the exact solution through illustrating the time evolution of bright soliton solution.