Abstract
We prove a generalization of Gromov’s symplectic nonsqueezing theorem for the case of Hilbert spaces. Our approach is based on filling almost complex Hilbert spaces by complex discs partially extending Gromov’s results on existence of J-complex curves. We apply our result to the flow of the discrete nonlinear Schrödinger equation.
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References
A. Abbondandolo and P. Majer, A non-squeezing theorem for convex symplectic images of the Hilbert ball. Preprint, arXiv:1405.3200, 2014.
Antoncev S.N., Monakhov V.N.: The Riemann-Hilbert boundary value problem with discontinuous boundary conditions for quasilinear elliptic systems of equations. Sov. Math. Dokl. 8, 868–870 (1967)
K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton, NJ, 2009.
Aubin J.-P.: Un thèoréme de compacitè. C. R. Acad. Sci. Paris 256, 5042–5044 (1963)
M. Audin and J. Lafontaine (eds.), Holomorphic Curves in Symplectic Geometry. Progress in Mathematics 117, Birkhäuser, Basel, 1994.
Bourgain J.: Approximation of solutions of the cubic nonlinear Schrödinger equations by finite-dimensional equations and nonsqueezing properties. Int. Math. Res. Not. IMRN 2, 79–90 (1994)
J. Bourgain: Aspects of long time behaviour of solutions of nonlinear hamiltonian evolution equations. Geom. Funct. Anal. 5, 105–140 (1995)
D. Burkholder, Martingales and singular integrals in Banach spaces. In: Handbook of the Geometry of Banach Spaces, Vol. I, North-Holland, Amsterdam, 2001, 233–269.
E. M. Chirka, Complex Analytic Sets. Kluwer, Dordrecht, 1989.
Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.: Symplectic nonsqueezing of the Korteweg-de Vries flow. Acta Math. 195, 197–252 (2005)
Eilbeck J.C., Lomdahl P.S., Scott A.C.: The discrete self-trapping equation. Phys. D 16, 318–338 (1985)
O. Fabert, Infinite-dimensional symplectic non-squeezing using the nonstandard analysis. Preprint, arXiv 1501.05905, 2015.
L. Grafakos, Classical Fourier Analysis. Springer, New York, 2008.
Gromov M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)
Herz C.: Theory of p-spaces with an application to convolution operators. Trans. Amer. Math. Soc. 154, 69–82 (1971)
Kuksin S.: Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDE's. Comm. Math. Phys. 167, 531–552 (1995)
S. Kuksin, Analysis of Hamiltonian PDEs. Oxford University Press, Oxford, 2000.
V. Monakhov, Boundary Value Problems with Free Boundary for Elliptic Systems of Equations. Translations of Mathematical Monographs 57, Amer. Math. Soc., Providence, RI, 1983.
D. Roumégoux: A symplectic non-squeezing theorem for BBM equation. Dyn. Partial Differ. Equ. 7, 289–305 (2010)
Sukhov A., Tumanov A.: Filling hypersurfaces by discs in almost complex manifolds of dimension 2. Indiana Univ. Math. J. 57, 509–544 (2008)
Sukhov A., Tumanov A.: Gromov’s non-squeezing theorem and Beltrami type equation. Comm. Partial Differential Equations 39, 1898–1905 (2014)
Sukhov A., Tumanov A.: Pseudoholomorphic discs and symplectic structures in Hilbert space. Contemp. Math. 662, 23–49 (2016)
I. N. Vekua, Generalized Analytic Functions. Pergamon, London, 1962.
K. Yoshida, Functional Analysis. Springer, New York, 1995.
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Sukhov, A., Tumanov, A. Symplectic nonsqueezing in Hilbert space and discrete Schrödinger equations. J. Fixed Point Theory Appl. 18, 867–888 (2016). https://doi.org/10.1007/s11784-016-0318-8
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DOI: https://doi.org/10.1007/s11784-016-0318-8