Multi-soliton Solutions of the Gardner Equation Using Darboux Transformation

Abstract

AbstractThe Gardner equation is a particular version of the extended Korteweg-de Vries (KdV) equation which presents actually the same type of characters as the standard KdV model, but it extends the range of validity to a larger domain of the parameters of the wave motion for a dynamic system. The purpose of this article is to explore a new type of multi-soliton solutions for the Gardner equation. To confirm its integrability, first, we will construct the Lax pair of the Gardner equation using Ablowitz–Kaup–Newell–Segur (AKNS) approach and finally, derive the one-soliton and two-soliton wave solutions for the Gardner equation employing the Darboux transformation method (DTM). These solutions can be extended to the generalized multi-solitary for the Gardner equation by repetition of the transformation. Some numerical graphs of one-soliton and two-soliton solutions are drawn for a clear understanding of wave motion under Gardner equation.KeywordsGardner equationWave motionMulti-soliton solutionsLax pairDarboux transformation method