Nonlinear waves on the Weierstrass elliptic function periodic background for the (4+1)-dimensional Korteweg-de Vries–Calogero–Bogoyavlenskii–Schiff equation

In this paper, the eigenfunctions for the linear spectral problem, which can be served as the Lax pair of the (4+1)-dimensional Korteweg-de Vries–Calogero–Bogoyavlenskii–Schiff (KdV–CBS) equation, with the Weierstrass function as the external potential are studied by the Lamé function method, from which various nonlinear wave solutions on the Weierstrass function periodic background for the KdV–CBS equation are obtained by the Darboux transformation (DT) method. At the same time, the degenerated nonlinear wave solutions are investigated by using the properties of Weierstrass function and gDT method. Furthermore, the evolution and nonlinear dynamics for various nonlinear wave are also discussed. These findings may be useful for explaining and predicting some nonlinear phenomena in related physical fields, such as physical ocean and hydrodynamics.