AG Mathematische Physik, Evgeny Korotyaev (Northeast Normal University, Changchun, China): Spectral invariants for vector periodic NLS

Evgeny Korotyaev (Northeast Normal University, Changchun, China)

Spectral invariants for vector periodic NLS

Abstract. Firstly, we discuss the various properties of periodic Zkharov-Shabat operator,
associated with scaler periodic NLS. We describe the main results and techniques.
Secondly, we discuss first order operators with a periodic 3×3 matrix potential on the real
line. This operator is the Lax operator for the periodic vector NLS equation. The spectrum
of the operator covers the real line and it is union of the spectrum of multiplicity 3, separated
by intervals (gaps) of multiplicity 1. We prove the following:
· The corresponding 2 or 3-sheeted Riemann surface is described.
· Necessary and sufficient conditions are given when the Riemann surface is 2-sheeted.
In the case of the 2-sheeted Riemann surface the solution of vector NLS equation is determined
in terms of solutions of scalar NLS equations.
· One constructs an entire function, which is negative on the spectrum of multiplicity 3 and
is positive on its gaps.
· The conformal mapping of the upper half plane on the domain on the upper half plane is
constructed and the main properties are This conformal mapping has asymptotics at high
energy where coefficients are constants of motion. As a corollary we obtain the estimate of
potentials in terms of gap lengths.
· Finally the Borg type result is obtained.