Bilinear auto-Backlund transformations, breather and mixed lump-kink solutions for a (3+1)-dimensional integrable fourth-order nonlinear equation in a fluid

In this paper, we investigate a (3+1)-dimensional integrable fourth-order nonlinear equation that can model both the right- and left-going waves in a fluid. Based on the Hirota method, we derive three sets of the bilinear auto-Backlund transformations along with some analytic solutions. Through the extended homoclinic test approach, we obtain some breather solutions. We find that the breather propagates steadily along a straight line, with one hole and one peak in each period. Graphical investigation indicates that the coefficients in that equation affect the location and shape of the breather. Moreover, we construct some mixed lump-kink solutions. Fusion and fission between a lump wave and a kink soliton are analyzed graphically. The solutions addressed in this paper may be applied to mimic some complex waves in fluids.

Automated summary

In this paper, a (3+1)-dimensional integrable fourth-order nonlinear equation is investigated to model both right- and left-going waves in fluids. Three sets of bilinear auto-Backlund transformations and some analytic solutions are derived using the Hirota method. Breather solutions are obtained through the extended homoclinic test approach, exhibiting steady propagation with one hole and one peak in each period. Graphical investigation shows that the coefficients in the equation impact the breather's location and shape. Additionally, mixed lump-kink solutions are constructed, and fusion and fission between a lump wave and a kink soliton are analyzed graphically.