Spectral curves and vector integrable nonlinear equations

Studies of the two-component vector nonlinear Schrödinger equation, the Kundu-Eckhaus vector equation and the Gerdjikov-Ivanov vector equation have shown that the spectral curves of multiphase solutions of these equations have unusual properties. In particular,

• These equations are invariant with respect to orthogonal transformations of solutions. And the spectral curves of multiphase solutions are also invariant with respect to orthogonal transformations of solutions. I.e., it is impossible to know the direction of the wave vector from the spectral curve.
• The procedure for constructing the simplest nontrivial solutions of these equations showed that the equation for the length of the vector appears first. Then, from the additional relations, an equation follows that determines the dependence of the direction of the vector on its length. I.e., the solution of the equation is determined not so much by the dynamics of its components as by the dynamics of the length of the vector and its direction.
• For all vector equations, there are parameter values at which the direction of the vector is fixed. I.e., the evolution of the vector is reduced to the evolution of the length of the vector. In these cases, the spectral curve splits into separate components and the evolution of the vector is determined by a curve of a smaller kind than in the case when the direction of the vector is not fixed.

We have studied examples illustrating these provisions.