Novel (3 + 1)-dimensional variable-coefficients Boussinesq-type equation: exploring integrability, Wronskian, and Grammian solutions

In this paper, we introduce a novel (3 + 1)-dimensional variable-coefficients Boussinesq-type equation. We analyze its integrability using the Painlevé test and the N -soliton solution, demonstrating that both tests yield identical conditions. Using the Hirota bilinear form of the equation, we derive Wronskian and Grammian determinant solutions utilizing Plücker relations and the Jacobi identity for determinants. In particular, we use elementary transformation and long wave limit to get the determinant expression of m th-order lump solutions from the 2 m th-order Wronskian determinant solutions. Furthermore, we reveal a variety of novel semi-rational solutions using the Hirota method and Grammian determinant techniques.