In this paper we consider the Degasperis-Procesi equation, which is an approximation to the incompressible Euler equation in shallow water regime. First we provide the existence of solitary wave solutions for the original DP equation and the general theory of geometric singular perturbation. Then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Degasperis-Procesi equation is transformed into the slow-fast system by using the traveling wave transformation. It is proved that the perturbed equation also has a homoclinic orbit, which corresponds to a solitary wave solution of the delayed Degasperis-Procesi equation.

中文翻译：

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具有分布时滞的Degasperis-Procesi方程的几何奇异摄动分析

在本文中，我们考虑了Degasperis-Procesi方程，它是浅水状态下不可压缩的Euler方程的近似值。首先，我们为原始的DP方程和几何奇异摄动的一般理论提供了孤立波解的存在。然后，利用几何奇异摄动理论和不变流形理论，证明了具有特殊局部时滞卷积核和特殊非局部时滞卷积核的方程的孤波解的存在性。根据孤立波与同斜轨道之间的关系，通过行波变换将Degasperis-Procesi方程转化为慢速-快速系统。证明了摄动方程还具有一个同斜轨道，