Analytical non-autonomous wave solitons for the dispersive cubic-quintic Gross–Pitaevskii equation and the interactions

• A variable coefficients Schrödinger equation with nonlinear terms is considered.
• We report several families of novel non-autonomous wave solutions.
• We consider the numerical simulation of two solitons collision for the NLSE.

A high-order dispersive cubic-quintic Gross–Pitaevskii (HDCQGP) equation (a generalized variable coefficients nonlinear Schrödinger equation with the third and fourth-order and the cubic-quintic nonlinear terms) is considered, and is transformed into a standard cubic-quintic nonlinear Schrödinger equation (NLSE). By using the generalized tanh-function method, we study exact solutions of the HDCQGP equation with time-modulated potential and nonlinearity. In particular, based on the similarity transformation, we report several families of non-autonomous wave solutions of the HDCQGP equation with snaking behaviors and different amplitude surfaces. At last, we consider the numerical simulation of two solitons collision for the NLSE with different parameters. These results may raise the possibility of relative experiments and potential applications.