A note on the Painlevé test for nonlinear variable-coefficient PDEs

For a generalized nonlinear PDEs with variable coefficients, it is not Painlevé integrable unless the variable coefficients satisfy certain constraint conditions. In this note a generalized algorithm is proposed for the Painlevé test of nonlinear variable-coefficient PDEs. For the three steps of Painlevé test, i.e. leading order analysis, resonance determination and verification of resonant conditions, the analysis of parametric constraints is similar to those of nonlinear PDEs with constant coefficients given in my previous work. The main difference lies in the coefficients of Laurent series should have proper dependence according to the types of variable coefficients. By this generalized algorithm, several important nonlinear variable-coefficient PDEs, including KdV equation, mKdV equation, KP equation, NLS equation and higher-order NLS equation are studied and, in addition to rederiving all known P-integrable conditions, some new P-integrable models are obtained with the assistance of Maple.