Strange tori of the derivative nonlinear Schrödinger equation
Derivative nonlinear Schrödinger equation \(i q_t = q_{xx} - i (| q| ^2 q)_x\) is investigated along the lines of the book [Chaos in partial differential equations, Somerville, MA: International Press (2003; Zbl 1043.35001)] by the same author. Under periodic boundary conditions, Floquet theory is applicable to the Lax equation. As the main result of the paper, tori and their whiskers generated by means of iterated Darboux transformation are shown to exhibit unusual properties: One level set of the Floquet discriminant can contain disconnected tori of different dimensions. Diffusion (variation of invariants) along the tori under small perturbations of the equation is also studied. Finally, the open problem of invariant manifolds is discussed.
derivative nonlinear Schrödinger equation
scientific article; zbMATH DE number 5159814