Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Explicit formula for the solution of the Szegö equation on the real line and applications
Pages: 607 - 649, Issue 3, November 2011

doi:10.3934/dcds.2011.31.607      Abstract        References        Full text (658.1K)           Related Articles

Oana Pocovnicu - Laboratoire de Mathématiques d’Orsay, Université Paris-Sud (XI), 91405, Orsay, France (email)

1 E.V. Abakumov, The inverse spectral problem for Hankel operators of finite rank, (Russian. English, Russian summary), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 217 (1994), Issled. po Linein. Oper. i Teor. Funktsii, 22, 5-15, 218; translation in J. Math. Sci. (New York), 85 (1997), 1759-1766.
2 M. Ablowitz, D. Kaup, A. Newell and H. Segur, The inverse scattering transform—-Fourier analysis for nonlinear problems, Studies in Appl. Math., 53 (1974), 249-315.       
3 V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989.       
4 J. Bourgain, On the Cauchy problem for periodic KdV-type equations, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), J. Fourier Anal. Appl. Special Issue, (1995), 17-86.       
5 J. Bourgain, Aspects of long time behavior of solutions of nonlinear Hamiltonian evolution equations, Geom. Funct. Anal., 5 (1995), 105-140.       
6 J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices, 6 (1996), 277-304.       
7 J. Bourgain, "Nonlinear Schrödinger Equations," Hyperbolic equations and frequency interactions (Park City, UT, 1995), 3-157, IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, 1999.
8 J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergodic Theory Dynam. Systems, 24 (2004), 1331-1357.       
9 J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39-113.       
10 P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251.       
11 W. Eckhaus and P. Schuur, The emergence of solitons of the Korteweg de Vries equation from arbitrary initial conditions, Math. Meth. Appl. Sci., 5 (1983), 97-116.       
12 E. Fiorani, G. Giachetta and G. Sardanashvily, The Liouville-Arnold-Nekhoroshev theorem for non-compact invariant manifolds, J. Phys. A, 36 (2003), L101-L107.       
13 E. Fiorani and G. Sardanashvily, Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds, J. Math. Phys., 48 (2007), 032901, 9 pp.       
14 C. S. Gardner, C. S. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.
15 P. Gérard and S. Grellier, Invariant tori for the cubic Szegö equation, to appear in Invent. Math., arXiv:1011.5479.
16 P. Gérard and S. Grellier, The cubic Szegö equation, Annales Scientifiques de l'Ecole Normale Supérieure, Paris, $4^e$ série, 43 (2010), 761-810.       
17 P. Gérard and S. Grellier, "L'Équation de Szegö Cubique," Séminaire X EDP, École Polytechnique, Palaiseau, 20 octobre 2008.       
18 Z. Hani, "Global and Dynamical Aspects of Nonlinear Schrödinger Equations on Compact Manifolds," Ph.D. thesis, UCLA, 2011.
19 R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1194.
20 S. B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal., 7 (1997), 338-363.       
21 P. Lax, Translation invariant spaces, Acta Math., 101 (1959), 163-178.       
22 P. Lax, Integral of nonlinear equations of evolution and solitary waves, Comm. Pure and Applied Math., 101 (1968), 467-490.       
23 P. Lax, "Linear Algebra," Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1997.       
24 S. V. Manakov, S. P. Novikov, L. P. Pitaevskii and V. E. Zakharov, "Theory of Solitons. The Inverse Scattering Method," Translated from the Russian, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984.       
25 Y. Martel and F. Merle, Description of two soliton collision for the quartic gKdV equations, to appear in Annals of Math., arXiv:0709.2672.
26 A. V. Megretskii, V. V. Peller and S. R. Treil, The inverse spectral problem for self-adjoint Hankel operators, Acta Math., 174 (1995), 241-309.       
27 N. K. Nikolskii, "Operators, Functions and Systems: An Easy Reading," Vol.I: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, 92, AMS, 2002.       
28 S. P. Novikov, "Theory of Solitons: The Inverse Scattering Method," Moscow: Nauka, 1980.       
29 V. V. Peller, "Hankel Operators and Their Applications," Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.       
30 O. Pocovnicu, Traveling waves for the cubic Szegö equation on the real line, to appear in Analysis and PDE, arXiv:1001.4037.
31 M. Reed and B. Simon, "Methods of Modern Mathematical Physics," Vol. I-IV, Academic Press, 1972-1978.       
32 T. Tao, Why are solitons stable?, Bulletin (New Series) of the American Mathematical Society, 46 (2009), 1-33.       
33 V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, 34 (1972), 62-69.       

Go to top