The three-component coupled time-varying coefficient complex mKdV equation via the $$\bar{\partial}$$-dressing method

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.