On the periodic KdV equation in weighted Sobolev spaces
Thomas Kappeler; Jürgen Pöschel
Annales de l'I.H.P. Analyse non linéaire (2009)
- Volume: 26, Issue: 3, page 841-853
- ISSN: 0294-1449
Access Full Article
topHow to cite
topKappeler, Thomas, and Pöschel, Jürgen. "On the periodic KdV equation in weighted Sobolev spaces." Annales de l'I.H.P. Analyse non linéaire 26.3 (2009): 841-853. <http://eudml.org/doc/78869>.
@article{Kappeler2009,
author = {Kappeler, Thomas, Pöschel, Jürgen},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {periodic KdV equation; well posedness; weighted Sobolev spaces},
language = {eng},
number = {3},
pages = {841-853},
publisher = {Elsevier},
title = {On the periodic KdV equation in weighted Sobolev spaces},
url = {http://eudml.org/doc/78869},
volume = {26},
year = {2009},
}
TY - JOUR
AU - Kappeler, Thomas
AU - Pöschel, Jürgen
TI - On the periodic KdV equation in weighted Sobolev spaces
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2009
PB - Elsevier
VL - 26
IS - 3
SP - 841
EP - 853
LA - eng
KW - periodic KdV equation; well posedness; weighted Sobolev spaces
UR - http://eudml.org/doc/78869
ER -
References
top- [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. Math30 (1977) 95-148. Zbl0338.35024MR649926
- [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. Zbl0855.58035MR1355177
- [3] Bättig D., Kappeler T., Mityagin B., On the Korteweg–de Vries equation: Frequencies and initial value problem, Pacific J. Math181 (1997) 1-55. Zbl0899.35096MR1491035
- [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 Linéaire22 (2005) 783-797. Zbl1095.35039MR2172859
- [5] Bona J.L., Smith R., The initial-value problem for the Korteweg–de Vries equation, Philos. Trans. Roy. Soc. London Ser. A278 (1975) 555-601. Zbl0306.35027MR385355
- [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. Zbl0787.35098MR1215780
- [7] Bourgain J., On the Cauchy problem for periodic KdV-type equations, J. Fourier Anal. Appl.Special Issue (1995) 17-86. Zbl0891.35137MR1364878
- [8] Bourgain J., Periodic Korteweg–de Vries equation with measures as initial data, Selecta Math. (N.S.)3 (1997) 115-159. Zbl0891.35138MR1466164
- [9] Bourgain J., Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ., American Mathematical Society, Providence, RI, 1999. Zbl0933.35178MR1691575
- [10] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Sharp global well-posedness for KdV and modified KdV on and , J. Amer. Math. Soc.16 (2003) 705-749. Zbl1025.35025MR1969209
- [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. Zbl1178.35345
- [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. Zbl1037.34080MR1934354
- [13] Djakov P., Mityagin B., Spectral triangles of Schrödinger operators with complex potentials, Selecta Math. (N.S.)9 (2003) 495-528. Zbl1088.34072MR2031750
- [14] Djakov P., Mityagin B., Instability zones of one-dimensional periodic Schrödinger and Dirac operators, Uspekhi Mat. Nauk61 (2006) 77-182, (in Russian). Zbl1128.47041MR2279044
- [15] Gasymov M.G., Spectral analysis of a class of second order nonselfadjoint differential operators, Funct. Anal. Appl.14 (1980) 14-19. Zbl0574.34012MR565091
- [16] Grébert B., Kappeler T., Pöschel J., A note on gaps of Hill's equation, Int. Math. Res. Not.50 (2004) 2703-2717. Zbl1082.34072MR2127368
- [17] Grujić Z., Kalisch H., Local well-posedness of the generalized Korteweg–de Vries equation in spaces of analytic functions, Differential Integral Equations15 (2002) 1325-1334. Zbl1031.35124MR1920689
- [18] Kappeler T., Makarov M., On the Birkhoff coordinates for KdV, Ann. Henri Poincaré2 (2001) 807-856. Zbl1017.76015MR1869523
- [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. Zbl0924.58074MR1473448
- [20] Kappeler T., Mityagin B., Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator, SIAM J. Math. Anal.33 (2001) 113-152. Zbl1097.34553MR1857991
- [21] Kappeler T., Pöschel J., KdV & KAM, Springer, Berlin, 2003. MR1997070
- [22] Kappeler T., Topalov P., Global wellposedness of KdV in , Duke Math. J.135 (2006) 327-360. Zbl1106.35081MR2267286
- [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. Zbl0787.35090MR1230283
- [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. Zbl0848.35114MR1329387
- [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. Zbl0678.58037MR959490
- [26] Kuksin S.B., Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, vol. 1556, Springer, 1993. Zbl0784.58028MR1290785
- [27] Kuksin S.B., A Kam-theorem for equations of the Korteweg–de Vries type, Rev. Math. Phys.10 (1998) 1-64. Zbl0920.35135MR1754991
- [28] Kuksin S.B., Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000. Zbl0960.35001MR1857574
- [29] Marčenko V.A., Ostrowskiĭ I.O., A characterization of the spectrum of Hill's operator, Math. USSR Sb.97 (1975) 493-554. Zbl0343.34016
- [30] Polya G., Szegö G., Problems and Theorems in Analysis I, Springer, New York, 1976. Zbl0338.00001
- [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. Zbl0623.34001MR894477
- [33] Sjöberg A., On the Korteweg–de Vries equation: Existence and uniqueness, J. Math. Anal. Appl.29 (1970) 569-579. Zbl0179.43101MR410135
- [34] Temam R., Sur un problème non linéaire, J. Math. Pures Appl.48 (1969) 159-172. Zbl0187.03902MR261183
- [35] Tkachenko V., Characterization of Hill operators with analytic potentials, Integral Equations Operator Theory41 (2001) 360-380. Zbl0994.34015MR1853676
- [36] Trubowitz E., The inverse problem for periodic potentials, Comm. Pure Appl. Math.30 (1977) 321-342. Zbl0403.34022MR430403
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.