On the $\bar\partial$-problem and dressing method for the complex vector modified KdV equation

Starting from a $5\times 5$ local matrix $\bar\partial$-problem, we successfully use the $\bar\partial$-dressing method to derive a hierarchy of nonlinear evolution equations including the nonlinear Schrödinger equation as $n=2$, the vector modified Korteweg–de Vries equation as $n=3$, and the Lakshmanan–Porsezian–Danielvia equation as $n=4$ via introducing a suitable recursion operator $\Lambda^n$. In addition, we employ the $\bar\partial$-dressing method to find the $N$-soliton solutions of the vmKdV equation. Finally, the effects of each parameter on interactions between solitons are discussed, and the effects of the characteristic lines on the relative position of the waves are also analyzed. The method for controlling the propagation direction is presented in detail.