N $N$ -Soliton Matrix mKdV Solutions: Some Special Solutions Revisited

In this article, a general solution formula is derived for the $d\times d$-matrix modified Korteweg–de Vries equation. Then, a solution class corresponding to special parameter choices is examined in detail. Roughly, this class can be described as $N$-solitons (in the sense of Goncharenko) with common phase matrix. It turns out that such a solution even takes values in a commutative subalgebra of the $d\times d$-matrices. We arrive at a rich picture of possibilities for generalized 1-solitons and at visual patterns of $N$-solitons which combine nonlinear with linear features. The impact of the phase matrix is visualized in computer plots.