Vector form of Kundu–Eckhaus equation and its simplest solutions

Nowadays, new vector integrable models of nonlinear optics are actively investigated. This is motivated by a need to transmit more information per unit of time by using polarized waves. In our work we study one of such models and we construct an hierarchy of integrable vector nonlinear differential equations depending on the functional parameter $r$ by using a monodromy matrix. The first equation of this hierarchy for $r=\alpha(\mathbf{p}^t\mathbf{q})$ is a vector analogue of the Kundu–Eckhaus equation. As $\alpha=0$, the equations of this hierarchy turn into equations of the Manakov system hierarchy. Other values of the functional parameter $r$ correspond to other integrable nonlinear equations. New elliptic solutions to the vector analogue of the Kundu–Eckhaus and Manakov system are presented. We also give an example of a two-gap solution of these equations in the form of a solitary wave. We show that there exist linear transformations of solutions to the vector integrable nonlinear equations into other solutions to the same equations. This statement is true for many vector integrable nonlinear equations. In particular, this is true for multicomponent derivative nonlinear Schrödinger equations and for the Kulish-Sklyanin equation. Therefore, the corresponding Baker-Akhiezer function can be constructed from a spectral curve only up to a linear transformation. In conclusion, we show that the spectral curves of the finite-gap solutions of the Manakov system and the Kundu–Eckhaus vector equation are trigonal curves whose genus is twice the number of phases of the finite-gap solution, that is, in the finite-gap solutions of the Manakov system and the vector analogue of the Kundu–Eckhaus equation, only half of the phases contain the variables $t$, $z_1,\dots,z_n$. The second half of the phases depends on the parameters of the solutions.