## Symmetries of the nonlinear Schrödinger equation.(English)Zbl 1044.35088

Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum $$\cdots < \lambda^-_k \leq \lambda^+_k < \lambda^-_{k + 1} \leq \cdots$$ of a Zakharov-Shabat operator is symmetric, i.e. $$\lambda^\pm_k = - \lambda^\mp_{-k}$$ for all $$k$$, if and only if the sequence $$(\gamma_k)_{k\in \mathbb{Z} }$$ of gap lengths, $$\gamma_k:= \lambda^+_k - \lambda^-_k$$, is symmetric with respect to $$k=0$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37L20 Symmetries of infinite-dimensional dissipative dynamical systems 34A55 Inverse problems involving ordinary differential equations
Full Text: