Collective variable approach to soliton dynamics in the Kuramoto–Sivashinsky equation for nonlinear transport and pattern formation

Abstract

In the present article, the collective variable (CV) approach is implemented to investigate soliton dynamics associated with the Kuramoto–Sivashinsky equation, a nonlinear fourth-order partial differential equation frequently arising in nonlinear transport phenomena, dissipative systems, and pattern formation processes. The proposed framework decomposes the governing model into soliton and residual components and employs Gaussian ansatz to formulate a reduced-order dynamical system. Six important collective variables, namely amplitude, temporal position, width, chirp, frequency, and phase, are systematically derived and analyzed. The implementation of the collective variable framework transforms the infinite-dimensional nonlinear system into a finite-dimensional representation, thereby reducing computational complexity while preserving essential dynamical characteristics. Analytical expressions governing the evolution of collective variables are obtained and graphical investigations are conducted to interpret their temporal behavior. The obtained findings demonstrate that the proposed methodology efficiently captures the evolution of nonlinear wave structures and provides a computationally efficient alternative framework for studying nonlinear transport and pattern formation systems. The developed framework may further be extended to other complex nonlinear partial differential equations arising in physics and engineering applications.