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.

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Notes

  1. 1.

    The norm of vector valued function \(G:D_\rho (r,s)\times {\mathcal {O}}\rightarrow {\mathbb {C}}^b\), \(b<\infty ,\) is defined as \(\Vert G\Vert _{D_\rho (r,s)\times {\mathcal {O}}}=\mathop {\sum }\nolimits ^b_{a=1}\Vert G_a\Vert _{D_\rho (r,s)\times {\mathcal {O}}}\).

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Acknowledgements

We would like to thank anonymous referee for helping to improve this paper. The research was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11971012). Z. Lou was supported by NSFC (Grant No. 11901291) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190395). Y. Sun was supported by the China Scholarship Council (CSC) (Grant No. 202006190134).

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Correspondence to Jiansheng Geng.

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Appendix

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. (1)

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

  2. (2)

    If X is S-reversible, Y is S-invariant and the transformation \(\Phi \) is S-invariant, then [XY] 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

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

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Geng, J., Lou, Z. & Sun, Y. A KAM Theorem for Two Dimensional Completely Resonant Reversible Schrödinger Systems. J Dyn Diff Equat (2021). https://doi.org/10.1007/s10884-021-09941-z

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Keywords

  • KAM
  • NLS
  • Quasi-periodic solution
  • Reversible vector field
  • Birkhoff normal form

Mathematics Subject Classification

  • 37K55
  • 35B15