# American Institute of Mathematical Sciences

September  2011, 31(3): 607-649. doi: 10.3934/dcds.2011.31.607

## Explicit formula for the solution of the Szegö equation on the real line and applications

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

Received  January 2010 Revised  April 2011 Published  August 2011

We consider the cubic Szegö equation

$i\partial$$t$$u=$$\Pi$$(|u|^{2}u)$

in the Hardy space $L^2_+$$(\mathbb{R}) on the upper half-plane, where \Pi is the Szegö projector. It is a model for totally non-dispersive evolution equations and is completely integrable in the sense that it admits a Lax pair. We find an explicit formula for solutions of the Szegö equation. As an application, we prove soliton resolution in H^s for all s\geq 0, for generic rational function data. As for non-generic data, we construct an example for which soliton resolution holds only in H^s, 0\leq s<1/2, while the high Sobolev norms grow to infinity over time, i.e. \lim_{t\to\pm\infty}\|u(t)\|_{H^s}=\infty, s>1/2. As a second application, we construct explicit generalized action-angle coordinates by solving the inverse problem for the Hankel operator H_u appearing in the Lax pair. In particular, we show that the trajectories of the Szegö equation with generic rational function data are spirals around Lagrangian toroidal cylinders \mathbb{T}^N$$\times$$\mathbb{R}^N$.
Citation: Oana Pocovnicu. Explicit formula for the solution of the Szegö equation on the real line and applications. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 607-649. doi: 10.3934/dcds.2011.31.607
##### References:
 [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. doi: 10.1007/BF02355284. [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. doi: 10.1007/BF01895664. [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. doi: 10.1155/S1073792896000207. [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. doi: 10.1017/S0143385703000750. [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. doi: 10.1007/s00222-010-0242-2. [10] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251. doi: 10.1002/cpa.3160320202. [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. doi: 10.1002/mma.1670050108. [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. doi: 10.1088/0305-4470/36/7/102. [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. doi: 10.1103/PhysRevLett.19.1095. [15] P. Gérard and S. Grellier, Invariant tori for the cubic Szegö equation,, to appear in Invent. Math., (). [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. doi: 10.1007/PL00001622. [21] P. Lax, Translation invariant spaces, Acta Math., 101 (1959), 163-178. doi: 10.1007/BF02559553. [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., (). [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. doi: 10.1007/BF02392468. [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, (). [31] M. Reed and B. Simon, "Methods of Modern Mathematical Physics," Vol. I-IV,, Academic Press, (): 1972. [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.

show all references

##### References:
 [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. doi: 10.1007/BF02355284. [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. doi: 10.1007/BF01895664. [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. doi: 10.1155/S1073792896000207. [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. doi: 10.1017/S0143385703000750. [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. doi: 10.1007/s00222-010-0242-2. [10] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251. doi: 10.1002/cpa.3160320202. [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. doi: 10.1002/mma.1670050108. [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. doi: 10.1088/0305-4470/36/7/102. [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. doi: 10.1103/PhysRevLett.19.1095. [15] P. Gérard and S. Grellier, Invariant tori for the cubic Szegö equation,, to appear in Invent. Math., (). [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. doi: 10.1007/PL00001622. [21] P. Lax, Translation invariant spaces, Acta Math., 101 (1959), 163-178. doi: 10.1007/BF02559553. [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., (). [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. doi: 10.1007/BF02392468. [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, (). [31] M. Reed and B. Simon, "Methods of Modern Mathematical Physics," Vol. I-IV,, Academic Press, (): 1972. [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.
 [1] Yuan Li, Shou-Fu Tian. Inverse scattering transform and soliton solutions of an integrable nonlocal Hirota equation. Communications on Pure and Applied Analysis, 2022, 21 (1) : 293-313. doi: 10.3934/cpaa.2021178 [2] François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems and Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019 [3] Thomas Kappeler, Riccardo Montalto. Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022048 [4] Joseph Thirouin. Classification of traveling waves for a quadratic Szegő equation. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3099-3122. doi: 10.3934/dcds.2019128 [5] Guillaume Bal, Jiaming Chen, Anthony B. Davis. Reconstruction of cloud geometry from high-resolution multi-angle images. Inverse Problems and Imaging, 2018, 12 (2) : 261-280. doi: 10.3934/ipi.2018011 [6] Xiaojuan Deng, Xing Zhao, Mengfei Li, Hongwei Li. Limited-angle CT reconstruction with generalized shrinkage operators as regularizers. Inverse Problems and Imaging, 2021, 15 (6) : 1287-1306. doi: 10.3934/ipi.2021019 [7] Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067 [8] Aristophanes Dimakis, Folkert Müller-Hoissen. Bidifferential graded algebras and integrable systems. Conference Publications, 2009, 2009 (Special) : 208-219. doi: 10.3934/proc.2009.2009.208 [9] Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873 [10] Andrea Braides, Margherita Solci, Enrico Vitali. A derivation of linear elastic energies from pair-interaction atomistic systems. Networks and Heterogeneous Media, 2007, 2 (3) : 551-567. doi: 10.3934/nhm.2007.2.551 [11] Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 525-545. doi: 10.3934/dcdss.2009.2.525 [12] Jon Johnsen. Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators. Electronic Research Archive, 2019, 27: 20-36. doi: 10.3934/era.2019008 [13] Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Hassan D. Sidibé. Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991 [14] Bernard Brighi, Tewfik Sari. Blowing-up coordinates for a similarity boundary layer equation. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 929-948. doi: 10.3934/dcds.2005.12.929 [15] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [16] Benjamin Dodson, Cristian Gavrus. Instability of the soliton for the focusing, mass-critical generalized KdV equation. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1767-1799. doi: 10.3934/dcds.2021171 [17] Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889 [18] Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109 [19] Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61 [20] Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

2020 Impact Factor: 1.392