Study on extensions of (modified) Korteweg–de Vries equations: Painlevé integrability and multiple soliton solutions in fluid mediums

Abstract

This work develops two higher-dimensional extensions for both Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. We investigate the Painlevé integrability of each couple of the aforementioned two models. We show that the Painlevé integrability fails for one equation of each couple but holds true for the x-derivative of this model. We examine multiple soliton solutions for the integrable extensions of these two models by utilizing the bilinear form. The outcomes will contribute to a deep understanding of the propagation mechanism of the propagation and interaction of multi-solitons in a variety of nonlinear media, including sea waves, optical fibers, and plasma physics.