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

Grébert, Benoît; Kappeler, Thomas

Bull. Soc. Math. Fr. 130, No. 4, 603-618 (2002).

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

Keywords:

NLS equation; Zakharov-Shabat operators; action-angle variables; symmetries