Well-posedness and scattering for the KP-II equation in a critical space

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 26, Issue 3, 2009, Pages 917-941

  Hadac, Martin ^a   Herr, Sebastian ^a   Koch, Herbert ^a  

^a UNIVERSITY OF BONN   (Germany)

Author keywords

Bilinear estimates; Bounded p variation; Kadomtsev Petviashvili II equation; Scale invariant space; Scattering; Well posedness

Indexed keywords

SIGNAL FILTERING AND PREDICTION;

BILINEAR ESTIMATES; BOUNDED P-VARIATION; KADOMTSEV-PETVIASHVILI-II EQUATION; SCALE INVARIANT SPACE; WELL-POSEDNESS;

SCATTERING;

EID: 67349238733     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2008.04.002     Document Type: Article

References (24)

1. Biondini G., and Chakravarty S. Elastic and inelastic line-soliton solutions of the Kadomtsev-Petviashvili II equation. Math. Comput. Simulation 74 2-3 (2007) 237-250
2. Bourgain J. On the Cauchy problem for the Kadomtsev-Petviashvili equation. Geom. Funct. Anal. 3 4 (1993) 315-341
3. Christ M., Colliander J., and Tao T. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125 6 (2003) 1235-1293
4. Dickey L.A. Soliton Equations and Hamiltonian Systems. Advanced Series in Mathematical Physics. second ed. vol. 26 (2003), World Scientific Publishing Co. Inc., River Edge, NJ
5. Federer H. Geometric Measure Theory. Reprint of the 1969 edition. Classics in Mathematics (1996), Springer-Verlag, Berlin
6. Ginibre J., Tsutsumi Y., and Velo G. On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151 2 (1997) 384-436
7. M. Hadac, Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations, Trans. Amer. Math. Soc., in press
8. Isaza J. P., and Mejía L. J. Global solution for the Kadomtsev-Petviashvili equation (KPII) in anisotropic Sobolev spaces of negative indices. Electron. J. Differential Equations 68 (2003) 12 pp. (electronic)
9. Kadomtsev B.B., and Petviashvili V.I. On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15 (1970) 539-541
10. Kenig C.E., and Ziesler S.N. Local well posedness for modified Kadomstev-Petviashvili equations. Differential Integral Equations 18 10 (2005) 1111-1146
11. Kiselev O.M. Asymptotics of solutions of multidimensional integrable equations and their perturbations. J. Math. Sci., New York 138 6 (2006) 6067-6230 Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya, vol. 11, Differential Equations, 2004
12. Koch H., and Tataru D. Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math. 58 2 (2005) 217-284
13. Koch H., and Tataru D. A priori bounds for the 1D cubic NLS in negative Sobolev spaces. Internat. Math. Res. Notices 2007 (2007) article ID rnm053, 36 pages
14. Martel Y., and Merle F. Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18 1 (2005) 55-80
15. Peetre J. New Thoughts on Besov Spaces. Duke University Mathematics Series. I (1976), Mathematics Department, Duke University, Durham 305 p
16. F. Rousset, N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. I. H. Poincaré - AN (2008), in press, doi: 10.1016/j.anihpc.2007.09.006
17. Saut J.-C. Remarks on the generalized Kadomtsev-Petviashvili equations. Indiana Univ. Math. J. 42 3 (1993) 1011-1026
18. Schul'man E.I., and Zakharov V.E. Degenerative dispersion laws, motion invariants and kinetic equations. Physica D 1 2 (1980) 192-202
19. Takaoka H. Well-posedness for the Kadomtsev-Petviashvili II equation. Adv. Differential Equations 5 10-12 (2000) 1421-1443
20. Takaoka H., and Tzvetkov N. On the local regularity of the Kadomtsev-Petviashvili-II equation. Internat. Math. Res. Notices 2001 2 (2001) 77-114
21. Tao T. Scattering for the quartic generalised Korteweg-de Vries equation. J. Differential Equations 232 2 (2007) 623-651
22. Tzvetkov N. On the Cauchy problem for Kadomtsev-Petviashvili equation. Comm. Partial Differential Equations 24 7-8 (1999) 1367-1397
23. Ukai S. Local solutions of the Kadomtsev-Petviashvili equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 2 (1989) 193-209
24. Wiener N. The quadratic variation of a function and its Fourier coefficients. In: Masani P.R. (Ed). Collected Works with Commentaries. Volume II: Generalized Harmonic Analysis and Tauberian Theory; Classical Harmonic and Complex Analysis. Mathematicians of Our Time vol. 15 (1979), The MIT Press, Cambridge, MA - London (1924), XIII, 969 p