Determinant solitons and rational lump waves in a (3+1)-dimensional extended Korteweg-de Vries equation

Abstract

In this paper, we study a (3 + 1)-dimensional extended Korteweg-de Vries equation and present a unified analysis of its soliton and lump wave dynamics using two complementary analytical approaches. First, within Hirota's bilinear framework, we establish a Wronskian formulation for the equation and derive general n-soliton solutions in determinant form. This construction extends classical Wronskian results to a higher-dimensional KdV-type system and reveals rich interaction structures, including resonant Y-type solitons and mixed soliton patterns. Second, using a method independent of the Wronskian technique, we construct rational solutions corresponding to fundamental and higher-order lump waves through a reduced bilinear equation and polynomial ansatz. Generalized lump wave solutions with free parameters are obtained, allowing controlled manipulation of peak locations. These results provide the first systematic construction of both multi-soliton and high-order lump wave solutions for this extended equation and demonstrate the versatility of bilinear methods in high-dimensional integrable systems.