The cubic Szegő equation

Gérard, Patrick; Grellier, Sandrine

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 5, pp. 761-810.

On considère l’équation hamiltonienne suivante sur l’espace de Hardy du cercle

$$i{\partial }_{t}u={\Pi \left(\right|u|}^{2}u),$$

où $\Pi$ désigne le projecteur de Szegő. Cette équation est un cas modèle d’équation sans aucune propriété dispersive. On établit qu’elle admet une paire de Lax et une infinité de lois de conservation en involution, et qu’elle peut être approchée par une suite de systèmes hamiltoniens de dimension finie complètement intégrables. Néanmoins, on met en évidence des phénomènes d’instabilité illustrant la dégénérescence de cette structure complètement intégrable. Enfin, on caractérise les ondes progressives de ce système.

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle,

$$i{\partial }_{t}u={\Pi \left(\right|u|}^{2}u),$$

where $\Pi$ is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating the degeneracy of this completely integrable structure. We also classify the traveling waves for this system.

Classification : 35B15, 37K10, 47B35
Keywords: nonlinear schrödinger equations, integrable hamiltonian systems, Lax pairs, Hankel operators
Mot clés : Équations de schrödinger non linéaires, systèmes hamiltoniens intégrables, paires de Lax, opérateurs de Hankel

[1] V. I. Arnold, Mathematical methods of classical mechanics, Springer, 1978. | MR | Zbl

[2] B. Birnir, C. E. Kenig, G. Ponce, N. Svanstedt & L. Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. 53 (1996), 551-559. | MR | Zbl

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

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

[5] N. Burq, P. Gérard & N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), 569-605. | Zbl

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

[7] N. Burq, P. Gérard & N. Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. École Norm. Sup. 38 (2005), 255-301. | Zbl

[8] N. Burq, P. Gérard & N. Tzvetkov, High frequency solutions of the nonlinear Schrödinger equation on surfaces, Quart. Appl. Math. 68 (2010), 61-71. | Zbl

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

[10] P. Gérard, Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem, in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, 157-182. | Zbl

[11] P. Gérard & S. Grellier, L'équation de Szegő cubique, Séminaire X-EDP, École polytechnique, 2008. | Zbl

[12] M. Grillakis, J. Shatah & W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), 308-348. | Zbl

[13] T. Kappeler & J. Pöschel, KdV & KAM, Ergebnisse Math. Grenzg. 45, Springer, 2003.

[14] L. Kronecker, Zur Theorie der Elimination einer Variablen aus zwei algebraischen Gleichungen, Monatsber. königl. preuss. Akad. Wiss. (1881), 535-600, reprinted in Mathematische Werke, vol. 2, 113-192, Chelsea, 1968. | JFM

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

[16] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490. | MR | Zbl

[17] Z. Nehari, On bounded bilinear forms, Ann. of Math. 65 (1957), 153-162. | MR | Zbl

[18] F. Nier, Bose-Einstein condensates in the lowest Landau level: Hamiltonian dynamics, Rev. Math. Phys. 19 (2007), 101-130. | MR | Zbl

[19] N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs 92, Amer. Math. Soc., 2002. | MR | Zbl

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

[21] V. V. Peller, Hankel operators of class ${𝔖}_p$ and their applications (rational approximation, Gaussian processes, the problem of majorization of operators), Math. USSR Sb. 41 (1982), 443-479. | Zbl

[22] V. V. Peller, Hankel operators and their applications, Springer Monographs in Math., Springer, 2003. | MR | Zbl

[23] W. Rudin, Real and complex analysis, third éd., McGraw-Hill Book Co., 1987, Analyse réelle et complexe, Masson, 1980. | MR | Zbl

[24] N. Tzvetkov, À la frontière entre EDP semi- et quasi-linéaires, HDR, Université Paris-Sud Orsay, 2003.

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

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

[27] V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, USSR Comput. Math. Math. Phys. 3 (1963), 1407-1456 (english), Zh. Vuch. Mat. 3 (1963), 1032-1066 (russian). | Zbl

[28] V. E. Zakharov & A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34 (1972), 62-69. | MR