Antidark solitons and soliton molecules in a (3+1)-dimensional nonlinear evolution equation

We investigate a (3 + 1)-dimensional nonlinear evolution equation which is a higher-dimensional generalization of the Korteweg-de Vries equation. On the basis of the decomposition approach, theN-antidark soliton solution on a finite background is constructed by using the Darboux transformation together with the limit technique. The asymptotic analysis for theN-antidark soliton solution is performed, and the collision between multiple antidark solitons is proved to be elastic. Under the velocity resonant mechanism, the antidark soliton molecules on the (x, t), (y, t), (y, z) and (t, z) planes are found instead of the (x, y) and (x, z) planes. Based on the three- and the four-antidark soliton solutions, the elastic collision between a soliton molecule and a common soliton and the elastic collision between two soliton molecules are analytically demonstrated, respectively. These results may be useful for the study of soliton molecules in hydrodynamics and nonlinear optics.