The cubic Szegő equation
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 5, p. 761-810

We consider the following Hamiltonian equation on the ${L}^{2}$ Hardy space on the circle, $i{\partial }_{t}u={\Pi \left(|u|}^{2}u\right)\phantom{\rule{4pt}{0ex}},$ 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.

On considère l’équation hamiltonienne suivante sur l’espace de Hardy du cercle $i{\partial }_{t}u={\Pi \left(|u|}^{2}u\right)\phantom{\rule{4pt}{0ex}},$$\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.

DOI : https://doi.org/10.24033/asens.2133
Classification:  35B15,  37K10,  47B35
Keywords: nonlinear schrödinger equations, integrable hamiltonian systems, Lax pairs, Hankel operators
@article{ASENS_2010_4_43_5_761_0,
author = {G\'erard, Patrick and Grellier, Sandrine},
title = {The cubic Szeg\H o equation},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {Ser. 4, 43},
number = {5},
year = {2010},
pages = {761-810},
doi = {10.24033/asens.2133},
zbl = {1228.35225},
mrnumber = {2721876},
language = {en},
url = {http://www.numdam.org/item/ASENS_2010_4_43_5_761_0}
}

Gérard, Patrick; Grellier, Sandrine. The cubic Szegő equation. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 5, pp. 761-810. doi : 10.24033/asens.2133. http://www.numdam.org/item/ASENS_2010_4_43_5_761_0/

[1] V. I. Arnold, Mathematical methods of classical mechanics, Springer, 1978. | MR 690288 | Zbl 0386.70001

[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 1396718 | Zbl 0855.35112

[3] H. Brezis & T. Gallouët, Nonlinear Schrödinger evolution equations, Nonlinear Anal. 4 (1980), 677-681. | MR 582536 | Zbl 0451.35023

[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 1003.35113

[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 1067.58027

[6] N. Burq, P. Gérard & N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), 187-223. | Zbl 1092.35099

[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 1116.35109

[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 1187.35232

[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 1106.35096

[11] P. Gérard & S. Grellier, L'équation de Szegő cubique, Séminaire X-EDP, École polytechnique, 2008. | Zbl 1213.35397

[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 0711.58013

[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 13.0114.02

[15] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications 19, Oxford Univ. Press, 2000. | MR 1857574 | Zbl 0960.35001

[16] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490. | MR 235310 | Zbl 0162.41103

[17] Z. Nehari, On bounded bilinear forms, Ann. of Math. 65 (1957), 153-162. | MR 82945 | Zbl 0077.10605

[18] F. Nier, Bose-Einstein condensates in the lowest Landau level: Hamiltonian dynamics, Rev. Math. Phys. 19 (2007), 101-130. | MR 2293086 | Zbl 1129.82023

[19] N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs 92, Amer. Math. Soc., 2002. | MR 1864396 | Zbl 1007.47001

[20] T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal. 14 (1990), 765-769. | MR 1049119 | Zbl 0715.35073

[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 0478.47015

[22] V. V. Peller, Hankel operators and their applications, Springer Monographs in Math., Springer, 2003. | MR 1949210 | Zbl 1030.47002

[23] W. Rudin, Real and complex analysis, third éd., McGraw-Hill Book Co., 1987, Analyse réelle et complexe, Masson, 1980. | MR 662565 | Zbl 0278.26001

[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 745511 | Zbl 0585.35019

[26] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), 567-576. | MR 691044 | Zbl 0527.35023

[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 0147.44303

[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 406174