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.
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