Abstract
In this paper we focus on the application of the \(\bar{\partial}\) -dressing method to the three-component coupled time-varying coefficient complex mKdV equation. Based upon a \((4 \times 4)\) -matrix \(\bar{\partial}\) -problem and two linear equations of the spectral transformation matrix, we derive the Lax pair and infinitely many conservation laws for the three-component coupled time-varying coefficient complex mKdV equation. Besides, we construct a hierarchy of the three-component coupled time-varying coefficient complex mKdV equation with a source term by making use of the recursion operator. We derive symmetry conditions of the spectral transformation matrix. We establish \(N\) -solution solutions and multi-pole solutions for the three-component coupled time-varying coefficient complex mKdV equation and express them in compact forms based on an explicit spectral transformation matrix.
