L’équation de Szegö cubique

Gérard, Patrick; Grellier, Sandrine

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 2, 19 p.

[1] Bahouri, H., Gérard, P., Xu, C.J. : Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg, Journal d’Analyse Mathématique, 82, 93-118 (2000). | MR | Zbl

[2] Birnir, B., Kenig, C., Ponce, G., Svansted, N., Vega, L. : On the ill-posedness of the IVP for the generalized KdV and nonlinear Schrödinger equation. J. London Math. Soc. 53, 551-559 (1996). | MR | Zbl

[3] Bourgain, J. : Refinements of Strichartz’ inequality and applications to 2D NLS with critical nonlinearity, IMRN, 5, 253-283 (1998). | MR | Zbl

[4] Brezis, H., Gallouët, T. : Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4, 677-681 (1980). | MR | Zbl

[5] Burq, N., Gérard, P., Tzvetkov, N. : An instability property of the nonlinear Schrödinger equation on $S^d$. Math. Res. Lett., 9, 323–335 (2002). | MR | Zbl

[6] Burq, N., Gérard, P., Tzvetkov, N. : Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. Invent. math. 159, 187-223 (2005) | MR | Zbl

[7] Burq, N., Gérard, P., Tzvetkov, N. : Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations. Ann. Scient. Éc. Norm. Sup. 38, 255-301 (2005). | EuDML | Numdam | MR | Zbl

[8] Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T., : Weakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation, preprint, 2008, arXiv : 08081742v2 [math.AP].

[9] Gérard, P. : Nonlinear Schrödinger equations in inhomogeneous media : wellposednes and illposedness results. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006, European Mathematical Society. | MR | Zbl

[10] Kappeler, T., Pöschel, J. : KdV & KAM, A Series of Modern Surveys in Mathematics, vol. 45, Springer-Verlag, 2003. | MR

[11] Kuksin, S. B. : Analysis of Hamiltonian PDEs. Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000. | MR | Zbl

[12] Lax, P. : Integrals of Nonlinear equations of Evolution and Solitary Waves, Comm. Pure and Applied Math. 21, 467-490 (1968). | MR | Zbl

[13] Lax, P. : Periodic solutions of the the KdV equation. Comm. Pure Appl. Math. 28 , 141–188 (1975). | MR | Zbl

[14] Nikolski, N. K. : Operators, functions, and systems : an easy reading. Vol. 1. Hardy, Hankel, and Toeplitz. Translated from the French by Andreas Hartmann. Mathematical Surveys and Monographs, 92. American Mathematical Society, Providence, RI, 2002. | MR | Zbl

[15] Ogawa, T. : A proof of Trudinger’s inequality and its application to nonlinear Schrödinger equations. Nonlinear Anal. 14, 765–769 (1990). | MR | Zbl

[16] Peller, V. V. : Hankel operators and their applications. Springer Monographs in Mathematics. Springer-Verlag, New York, 2003. | MR | Zbl

[17] Rudin, W. : Real and Complex Analysis, Mac Graw Hill, Second edition, 1980. | MR | Zbl

[18] Tzvetkov, N. : A la frontière entre EDP semi et quasi linéaires, Mémoire d’habilitation à diriger les recherches, Université Paris-Sud, Orsay, 2003.

[19] Vladimirov, M. V. : On the solvability of a mixed problem for a nonlinear equation of Schrödinger type. Sov. math. Dokl. 29, 281-284 (1984). | MR | Zbl

[20] Weinstein, M. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 567–576 (1982/83). | MR | Zbl

[21] Weinstein, M. : Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 , 51–67 (1986). | MR | Zbl

[22] Yudovich, V. I. : Non-stationary flows of an ideal incompressible fluid. (Russian) Z. Vycisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963). | MR | Zbl

[23] Zakharov, V. E., Shabat, A. B. : Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics JETP 34 (1972), no. 1, 62–69. | MR