Integrability classification and exact solutions to generalized variable-coefficient nonlinear evolution equation

This paper is concerned with the generalized variable-coefficient nonlinear evolution equation (vc-NLEE). The complete integrability classification is presented, and the integrable conditions for the generalized variable-coefficient equations are obtained by the Painlevé analysis. Then, the exact explicit solutions to these vc-NLEEs are investigated by the truncated expansion method, and the Lax pairs (LP) of the vc-NLEEs are constructed in terms of the integrable conditions.