Multi‐Hump Solutions of Singularly Perturbed Korteweg‐de Vries Equation

ABSTRACT

The paper studies a singularly perturbed Korteweg‐de Vries (KdV) equation  for , where is a small parameter and are real constants. The equation arises in the study of surface waves in finite‐depth water with small surface tension on the free surface. It is known that the equation admits homoclinic solutions with small oscillatory tails at infinity, called generalized solitary‐wave solutions (or generalized one‐hump solutions). Here, the multi‐hump solutions are formally constructed. First, two‐hump solutions are derived by drawing on ideas and methods from rigorous existence studies of two‐hump solutions for various equations, including the singularly perturbed KdV equation. Three‐hump solutions are then formally obtained via a matching procedure, although a rigorous mathematical justification for the existence of three‐hump or other multi‐hump solutions remains open. Finally, the ideas and methods for the formal construction of multi‐hump solutions with arbitrary number of humps are discussed.