Invariant tori for the Schrödinger equation in the Heisenberg Ferromagnetic chain

Abstract

In this paper, we consider the nonlinear Heisenberg Ferromagnetic chain equation $$\mathrm{i}{u}_t+u_{\mathit{xx}}-\frac{2\overline{u}}{1+{|u|}^2}{u}_x^2=0$$

under Dirichlet boundary conditions. By Taylor formula, the nonlinear Heisenberg Ferromagnetic chain equation can be described by the nonlinear Schrödinger type equation. Using an infinite dimensional KAM theorem for reversible system, we prove the existence of many n-dimensional invariant tori under sufficiently small perturbation and thus many time quasi-periodic solutions for the above equation.

Keywords:

• Heisenberg Ferromagnetic chain
• KAM theorem
• Schrödinger equation
• quasi-periodic solution
• normal form

AMS Subject Classifications:

• 37K55
• 35Q55

Notes

No potential conflict of interest was reported by the authors.

2 Here we only prove that the Heisenberg FM equation is not Hamiltonian with symplectic structure $J=\text{i}.$ Maybe it is Hamiltonian with other symplectic form.

Additional information

Funding

This work was supported by NNSFC [11401041], [11601036]; the University Science and Technology Foundation of Shandong Province Education Department [J14LI54]; Binzhou University [2013Y02], [BZXYL1401].