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

Abstract

In this paper, we introduce a novel (3+1)-dimensional variable-coefficients Boussinesq-type equation. We
analyze its integrability using the Painlev´e 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¨ucker 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.