Searching for Painlevé integrable conditions of nonlinear PDEs with constant parameters using symbolic computation

Based on the Kruskal simplification for the WTC method, a generalized algorithm is devised to establish P-integrable (Painlevé integrable) conditions for nonlinear PDEs with multiple constant parameters. The generalized algorithm fully considers the impact of parameter coefficients upon every step of the Painlevé test. For parametric constraints obtained in the resonance analysis and verification of resonant conditions, the original equations should be regarded as a new system and repeating the test again. For illustration, we apply the generalized algorithm to coupled Schrödinger–Boussinesq equations. Based on the generalized algorithm, a Maple package SPIC is presented, which attributes to derive P-integrable conditions for given nonlinear PDEs with general forms. The higher order water wave equation and coupled KdV equations are selected to illustrate the effectiveness of our package. As a result, some P-integrable conditions are in agreement with the known results, in addition, several new P-integrable models are first given.