We consider the interaction of solitary waves in a model involving the well-known ϕ ⁴ Klein–Gordon theory, but now bearing both Laplacian and biharmonic terms with different prefactors. As a result of the competition of the respective linear operators, we obtain three distinct cases as we vary the model parameters. In the first the biharmonic effect dominates, yielding an oscillatory inter-wave interaction; in the third the harmonic effect prevails yielding exponential interactions, while we find an intriguing linearly modulated exponential effect in the critical second case, separating the above two regimes. For each case, we calculate the force between the kink and antikink when initially separated with sufficient distance. Being able to write the acceleration as a function of the separation distance, and its corresponding ordinary differential equation, we test the corresponding predictions, finding very good agreement, where appropriate, with the corresponding partial differential equation results. Where the two findings differ, we explain the source of disparities. Finally, we offer a first glimpse of the interplay of harmonic and biharmonic effects on the results of kink–antikink collisions and the corresponding single- and multi-bounce windows.

我们在一个涉及众所周知的ϕ ⁴的模型中考虑孤立波的相互作用Klein-Gordon 理论，但现在同时包含具有不同前置因子的拉普拉斯项和双调和项。由于各个线性算子的竞争，我们在改变模型参数时获得了三种不同的情况。在第一种情况下，双谐波效应占主导地位，产生振荡的波间相互作用；在第三种情况下，谐波效应占优势，产生指数相互作用，而我们在关键的第二种情况下发现了有趣的线性调制指数效应，将上述两种机制分开。对于每种情况，当最初以足够的距离分开时，我们计算扭结和反扭结之间的力。能够将加速度写成分离距离的函数，及其相应的常微分方程，我们测试相应的预测，发现非常一致，在适当的情况下，带有相应的偏微分方程结果。在这两个发现不同的地方，我们解释了差异的来源。最后，我们初步了解了谐波和双谐波效应对扭结-反扭结碰撞结果以及相应的单次和多次反弹窗口的相互作用。