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 Painleve 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 Pl & uuml;cker relations and the Jacobi identity for determinants. In particular, we use elementary transformation and long wave limit to get the determinant expression of mth-order lump solutions from the 2mth-order Wronskian determinant solutions. Furthermore, we reveal a variety of novel semi-rational solutions using the Hirota method and Grammian determinant techniques.