Bäcklund transformation, Lax pair and dynamic behaviour of exact solutions for a (3+1)-dimensional nonlinear equation

In this article, we focuss primarily on a (3+1)-dimensional nonlinear equation in fluids and the dynamic behaviour of the lump solution and new interaction solutions of the equation. First, the bilinear form of the equation is investigated using the Bell polynomials and the Hirota bilinear method. The Bell-polynomial-type Backlund transformation of the equation and the corresponding Lax pair are obtained by using the transformation technique. Based on the obtained bilinear form, we use the positive quadratic function method to obtain the lump solutions of the equation and study its dynamic behaviour in detail. On the other hand, we constructed two types of interaction solutions, lump-soliton and lump-periodic, and combined the images to vividly show the interaction phenomenon. Furthermore, the homoclinic breather solutions of the equation are constructed using the homoclinic test method. Finally, we hope that the solutions we obtained can explain some nonlinear phenomena in the fluid mechanics and shallow water wave fields.