On the Periodic KdV Equation in Weighted Sobolev Spaces

Kappeler, Thomas; PöSchel, JüRgen

doi:10.1016/j.anihpc.2008.03.004

Kappeler, Thomas ; PöSchel, JüRgen

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 3, pp. 841-853

MR Zbl

DOI : 10.1016/j.anihpc.2008.03.004

@article{AIHPC_2009__26_3_841_0,
     author = {Kappeler, Thomas and P\"oSchel, J\"uRgen},
     title = {On the {Periodic} {KdV} {Equation} in {Weighted} {Sobolev} {Spaces}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {841--853},
     year = {2009},
     publisher = {Elsevier},
     volume = {26},
     number = {3},
     doi = {10.1016/j.anihpc.2008.03.004},
     mrnumber = {2526404},
     zbl = {1177.35199},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2008.03.004/}
}

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Kappeler, Thomas; PöSchel, JüRgen. On the Periodic KdV Equation in Weighted Sobolev Spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 3, pp. 841-853. doi: 10.1016/j.anihpc.2008.03.004

Bibliographie
Cité par

[1] Airault H., Mckean H., Moser J., Rational and Elliptic Solutions of the Korteweg-De Vries Equation and a Related Many-Body Problem, Comm. Pure Appl. Math 30 (1977) 95-148. | Zbl | MR

[2] Bättig D., Bloch A. M., Guillot J.-C., Kappeler T., On the Symplectic Structure of the Phase Space for Periodic KdV, Toda, and Defocusing NLS, Duke Math. J. 79 (1995) 549-604. | Zbl | MR

[3] Bättig D., Kappeler T., Mityagin B., On the Korteweg-De Vries Equation: Frequencies and Initial Value Problem, Pacific J. Math 181 (1997) 1-55. | Zbl | MR

[4] Bona J., Grujić Z., Kalisch H., Algebraic Lower Bounds for the Uniform Radius of Spatial Analyticity for the Generalized KdV Equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 783-797. | Zbl | MR | Numdam

[5] Bona J. L., Smith R., The Initial-Value Problem for the Korteweg-De Vries Equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975) 555-601. | Zbl | MR

[6] Bourgain J., Fourier Transform Restriction Phenomena for Certain Lattice Subsets and Applications to Nonlinear Evolution Equations, II: the KdV-Equation, Geom. Funct. Anal. 3 (1993) 209-262. | Zbl | MR

[7] Bourgain J., On the Cauchy Problem for Periodic KdV-Type Equations, J. Fourier Anal. Appl. Special Issue (1995) 17-86. | Zbl | MR

[8] Bourgain J., Periodic Korteweg-De Vries Equation With Measures as Initial Data, Selecta Math. (N.S.) 3 (1997) 115-159. | Zbl | MR

[9] Bourgain J., Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ., American Mathematical Society, Providence, RI, 1999. | Zbl | MR

[10] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Sharp Global Well-Posedness for KdV and Modified KdV on $R$ and $T$ , J. Amer. Math. Soc. 16 (2003) 705-749. | Zbl | MR

[11] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Local and Global Well-Posedness for Non-Linear Dispersive and Wave Equations, www.math.ucla.edu/~tao/Dispersive.

[12] Djakov P., Mityagin B., Smoothness of Schrödinger Operator Potential in the Case of Gevrey Type Asymptotics of the Gaps, J. Funct. Anal. 195 (2002) 89-128. | Zbl | MR

[13] Djakov P., Mityagin B., Spectral Triangles of Schrödinger Operators With Complex Potentials, Selecta Math. (N.S.) 9 (2003) 495-528. | Zbl | MR

[14] Djakov P., Mityagin B., Instability Zones of One-Dimensional Periodic Schrödinger and Dirac Operators, Uspekhi Mat. Nauk 61 (2006) 77-182, (in Russian). | Zbl | MR

[15] Gasymov M. G., Spectral Analysis of a Class of Second Order Nonselfadjoint Differential Operators, Funct. Anal. Appl. 14 (1980) 14-19. | Zbl | MR

[16] Grébert B., Kappeler T., Pöschel J., A Note on Gaps of Hill's Equation, Int. Math. Res. Not. 50 (2004) 2703-2717. | Zbl | MR

[17] Grujić Z., Kalisch H., Local Well-Posedness of the Generalized Korteweg-De Vries Equation in Spaces of Analytic Functions, Differential Integral Equations 15 (2002) 1325-1334. | Zbl | MR

[18] Kappeler T., Makarov M., On the Birkhoff Coordinates for KdV, Ann. Henri Poincaré 2 (2001) 807-856. | Zbl | MR

[19] Kappeler T., Mityagin B., Gap Estimates of the Spectrum of Hill's Equation and Action Variables for KdV, Trans. Amer. Math. Soc. 351 (1999) 619-646. | Zbl | MR

[20] Kappeler T., Mityagin B., Estimates for Periodic and Dirichlet Eigenvalues of the Schrödinger Operator, SIAM J. Math. Anal. 33 (2001) 113-152. | Zbl | MR

[21] Kappeler T., Pöschel J., KdV & KAM, Springer, Berlin, 2003. | MR

[22] Kappeler T., Topalov P., Global Wellposedness of KdV in $H^{- 1} (T, R)$ , Duke Math. J. 135 (2006) 327-360. | Zbl | MR

[23] Kenig C. E., Ponce G., Vega L., On the Cauchy Problem for the Korteweg-De Vries Equation in Sobolev Spaces of Negative Indices, Duke Math. J. 71 (1993) 1-20. | Zbl | MR

[24] Kenig C. E., Ponce G., Vega L., A Bilinear Estimate With Applications to the KdV Equation, J. Amer. Math. Soc. 9 (1996) 573-603. | Zbl | MR

[25] Kuksin S. B., Perturbation Theory for Quasiperiodic Solutions of Infinite-Dimensional Hamiltonian Systems, and Its Application to the Korteweg-De Vries Equation, Mat. Sb. 136 (1988), (in Russian). English translation in, Math. USSR Sb. 64 (1989) 397-413. | Zbl | MR

[26] Kuksin S. B., Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, vol. 1556, Springer, 1993. | Zbl | MR

[27] Kuksin S. B., A Kam-Theorem for Equations of the Korteweg-De Vries Type, Rev. Math. Phys. 10 (1998) 1-64. | Zbl | MR

[28] Kuksin S. B., Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000. | Zbl | MR

[29] Marčenko V. A., Ostrowskiĭ I. O., A Characterization of the Spectrum of Hill's Operator, Math. USSR Sb. 97 (1975) 493-554. | Zbl

[30] Polya G., Szegö G., Problems and Theorems in Analysis I, Springer, New York, 1976. | Zbl

[31] J. Pöschel, Hill's potential in weighted Sobolev spaces and their spectral gaps, Preprint, http://www.poschel.de/pbl.

[32] Pöschel J., Trubowitz E., Inverse Spectral Theory, Academic Press, Boston, 1987. | Zbl | MR

[33] Sjöberg A., On the Korteweg-De Vries Equation: Existence and Uniqueness, J. Math. Anal. Appl. 29 (1970) 569-579. | Zbl | MR

[34] Temam R., Sur Un Problème Non Linéaire, J. Math. Pures Appl. 48 (1969) 159-172. | Zbl | MR

[35] Tkachenko V., Characterization of Hill Operators With Analytic Potentials, Integral Equations Operator Theory 41 (2001) 360-380. | Zbl | MR

[36] Trubowitz E., The Inverse Problem for Periodic Potentials, Comm. Pure Appl. Math. 30 (1977) 321-342. | Zbl | MR

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