This paper is devoted to the existence of traveling, solitary and periodic waves for the perturbed generalized KdV by applying geometric singular perturbation, differential manifold theory and the regular perturbation analysis of Hamiltonian systems. Under the assumptions that the distributed delay kernel is the strong general one and the average delay is sufficiently small, traveling, solitary and periodic waves are shown to exist in the perturbed system. It is further proved that the wave speed is decreasing by analyzing the ratio of Abelian integrals, and we analyze these functions by using the theory of analytic functions and algebraic geometry. Moreover, the upper and lower bounds of the limit wave speed are presented. The relationship between wavelength and wave speed of traveling waves is also established.

通过应用几何奇异摄动，微分流形理论和汉密尔顿系统的正规摄动分析，研究了摄动广义KdV的行波，孤波和周期波的存在。在分布式时延核是强通用核且平均时延足够小的假设下，扰动系统中存在行波，孤波和周期波。通过分析阿贝尔积分的比率进一步证明了波速正在减小，并且我们使用解析函数和代数几何理论对这些函数进行了分析。此外，给出了极限波速的上限和下限。还建立了行波的波长和波速之间的关系。