A Vector Bilinear Framework for Soliton Dynamics in Coupled Modified KdV Systems

TL;DR

This paper develops a vector bilinear formalism for coupled mKdV systems, enabling explicit multi-soliton solutions and revealing new ground states in indefinite coupling regimes.

Contribution

It introduces a vector reformulation of Hirota's bilinear method for coupled mKdV systems, providing a unified framework for soliton solutions across different regimes.

Findings

Constructed explicit multi-soliton solutions in vector form

Confirmed three-soliton condition at the vector level

Discovered nontrivial ground states in indefinite coupling regimes

Abstract

We investigate the integrable structure and soliton dynamics of a coupled modified Korteweg-de Vries (cmKdV) system with a real symmetric coupling matrix. We introduce a vector reformulation of Hirota's bilinear formalism in which both the bilinear equations and their solutions are expressed directly at the vector level, rather than through a component-wise construction. This formulation preserves the intrinsic structure of the coupled system and provides a compact framework for multi-component nonlinear wave dynamics. Within this approach, we construct explicit one-, two-, and three-soliton solutions in closed vector form and recover the three-soliton condition directly at the vector level, confirming consistency with integrability. The method enables a unified treatment of focusing, defocusing, and mixed-sign regimes. In particular, for indefinite coupling, it reveals the existence of nontrivial vector ground states, leading to soliton solutions on non-zero backgrounds. These results highlight the structural advantages of the vector bilinear approach and open perspectives for the study of more general nonlinear excitations in multi-component integrable systems.