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, the N-antidark soliton solution on a finite background is constructed by using the Darboux transformation together with the limit technique. The asymptotic analysis for the N-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.

我们研究了（3 +1）维非线性演化方程，它是Korteweg-de Vries方程的高维概括。在分解方法的基础上，利用Darboux变换和极限技术构造了有限背景下的N-反暗孤子解。进行了N-反暗孤子解的渐近分析，证明了多个反暗孤子之间的碰撞是弹性的。在速度共振机制下，（x，  t），（y，  t），（y，  z）和（t，  z上的反暗孤子分子找到）平面，而不是（x，  y）和（x，  z）平面。基于三反暗孤子解和四反暗孤子解，分别分析了孤子分子与普通孤子之间的弹性碰撞以及两个孤子分子之间的弹性碰撞。这些结果可能对研究流体动力学和非线性光学中的孤子分子很有用。