On a Hierarchy of Vector Derivative Nonlinear Schrödinger Equations

Abstract

We propose a new hierarchy of the vector derivative nonlinear Schrödinger equations and consider the simplest multiphase solutions of this hierarchy. The study of the simplest solutions of these equations led to the following results. First, the three-leaf spectral curves Γ={(μ,λ)} of the simplest multiphase solutions have a quite simple symmetry. They are invariant with respect to holomorphic involution τ. The type of this involution depends on the genus of the spectral curve. Or the involution has the form τ:(μ,λ)→(μ,−λ), or τ:(μ,λ)→(−μ,−λ). The presence of symmetry leads to the fact that the dynamics of the solution is determined not by the entire spectral curve Γ, but by its factor Γ/τ, which has a smaller genus. Secondly, it turned out that the dynamics of the two-component vector p=(p1,p2)t is determined, first of all, by the dynamics of its length |p|. Independent equations determine the dependence of the direction of the vector p from its length. In cases where the direction of the vector p is fixed, the corresponding spectral curve splits into separate components. In conclusion, we note that, as in the case of the Manakov system, the equation of the spectral curve is invariant with respect to the orthogonal transformation of the vector solutions. I.e., the solution can be found from the spectral curve up to the orthogonal transformation. This fact indicates that the spectral curve does not depend on the individual components of the solution, but on their symmetric functions. Thus, the spectral data of multiphase solutions have two symmetries. These symmetries make it difficult to reconstruct signals from their spectral data. The work contains examples illustrating these statements.