The general Degasperis–Procesi equation (gDP) describes the evolution of the water surface in a unidirectional shallow water approximation. We propose a finite‐difference scheme for this equation that preserves some conservation and balance laws. In addition, the stability of the scheme and the convergence of numerical solutions to exact solutions for solitons are proved. Numerical experiments confirm the theoretical conclusions. For essentially nonintegrable versions of the gDP equation, it is shown that solitons and antisolitons collide almost elastically: they retain their shape after interaction, but a small “tail”, the so‐called “radiation”, appears.

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一般Degasperis–Procesi方程的光滑解的有限差分格式

一般的Degasperis–Procesi方程（gDP）描述了单向浅水近似中水面的演变。我们为该方程式提出了一个有限差分方案，该方案保留了一些守恒和平衡定律。此外，证明了该方案的稳定性以及数值解与孤子精确解的收敛性。数值实验证实了理论结论。对于gDP方程的本质上不可积分的形式，表明孤子和反孤子几乎发生弹性碰撞：它们在相互作用后保持其形状，但是会出现一个小的“尾巴”，即所谓的“辐射”。