A KAM Theorem for Two Dimensional Completely Resonant Reversible Schrödinger Systems

Abstract

In this paper, we prove an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional reversible Schrödinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we obtain the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schrödinger systems on two dimensional torus.

Appendix

Suppose vector field $X(\theta , I, z, {\bar{z}})$ is defined on $D_\rho (r,s)=\{y=(\theta , I, z, {\bar{z}}):|Im \theta |<r,\, |I|<s,\, \Vert z\Vert _{\rho }<s,\, \Vert {\bar{z}}\Vert _{\rho }<s\}.$

Definition 6.1

(Reversible vector field) Suppose S is an involution map: $S^2=id.$ Vector field X is called reversible with respect to S (or S-reversible), if

$$\begin{aligned} DS\cdot X=- X\circ S, \end{aligned}$$

i.e.,

$$\begin{aligned} (DS(y))X(y)=-X(S(y)),\,y\in D_\rho (r,s), \end{aligned}$$

where DS is the tangent map of S.

Definition 6.2

Suppose S is an involution map: $S^2=id.$ Vector field X is called invariant with respect to S (or S-invariant), if

$$\begin{aligned} DS\cdot X=X\circ S. \end{aligned}$$

Definition 6.3

A transformation $\Phi$ is called invariant with respect to above involution S (or S-invariant), if $\Phi \circ S=S\circ \Phi .$

Lemma 6.1

• (1) If X and Y are both S-reversible (or S-invariant), then [X, Y] is S-invariant.

• (2) If X is S-reversible, Y is S-invariant and the transformation $\Phi$ is S-invariant, then [X, Y] and $\Phi ^*{X}$ are both S-reversible. In particular, the flow $\phi _{Y} ^t$ of Y are S-invariant, thus $(\phi _{Y} ^t)^*{X}$ is S-reversible.

Lemma 6.2

(Cauchy’s inequality, [14]) Let $0<\delta <r.$ For an analytic function $f(\theta , I, z, {\bar{z}})$ on $D_\rho (r,s),$

$$\begin{aligned}&\left\| \frac{\partial f}{\partial \theta _b}\Vert _{D_\rho (r-\delta ,s)}\le \frac{c}{\delta }\right\| f\Vert _{D_\rho (r,s)},\\&\left\| \frac{\partial f}{\partial I_b}\right\| _{D_\rho (r,s/2)}\le \frac{c}{s}\Vert f\Vert _{D_\rho (r,s)}, \end{aligned}$$

and

$$\begin{aligned} \left\| \frac{\partial f}{\partial z^\sigma _i}\right\| _{D_\rho (r,s/2)}\le \frac{c}{s}\Vert f\Vert _{D_\rho (r,s)}\mathrm {e}^{\rho |i|},\,\sigma =\pm . \end{aligned}$$