Abstract
We prove that there exist potentials so that near them the focusing non-linear Schrödinger equation on the torus does not admit local Birkhoff coordinates. The proof is based on the construction of a local normal form of the linearization of the equation at such potentials.
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Notes
- 1.
In what follows we will restrict our attention to the real space \(i H^s_r\), \(s\in \mathbb {R}\).
- 2.
Here ω(α k, β k) = ω(α −k, β −k) = 1 while all other skew-symmetric products between these vectors vanish.
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Acknowledgements
Both authors would like to thank the Fields Institute and its staff for the excellent working conditions and the organizers of the workshop “Inverse Scattering and Dispersive PDEs in One Space Dimension” for their efforts to make it such a successful and pleasant event. The second author also acknowledges the support of the Institute of Mathematics of the BAS.
T.K. is partially supported by the Swiss National Science Foundation. P.T. is partially supported by the Simons Foundation, Award #526907.
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Kappeler, T., Topalov, P. (2019). On the Nonexistence of Local, Gauge-Invariant Birkhoff Coordinates for the Focusing NLS Equation. In: Miller, P., Perry, P., Saut, JC., Sulem, C. (eds) Nonlinear Dispersive Partial Differential Equations and Inverse Scattering. Fields Institute Communications, vol 83. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9806-7_7
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