Existence of Solitary‐Wave‐Type Solutions for the Delayed NKdV‐2 Equation

ABSTRACT

In this paper, we provide a rigorous analysis of solitary‐wave‐type solutions for the second member in the negative‐order Korteweg‐de Vries equations, called the NKdV‐2 equation. This system represents a significant nonlocal generalization of the classical KdV hierarchy. The existence of solitary wave solutions for the transformed NKdV‐2 system is established for local, nonlocal, and peakon‐type nonsmooth spatial nonlocal convolution kernels (). To the best of our knowledge, this is the first systematic persistence analysis for the NKdV‐2 traveling‐wave system under such local and nonlocal perturbations, with the ‐case providing the most distinctive new result. Consequently, the original NKdV‐2 equation admits solitary‐wave‐type solutions corresponding to these perturbed homoclinic structures. These works extend the usual solitary wave analysis to a class of nonlocal integrable systems as well as providing new insights into their dynamical behavior under delay perturbations.