# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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, 22 (1994), 5. 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. [3] V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Graduate Texts in Mathematics, 60 (1989). [4] J. Bourgain, On the Cauchy problem for periodic KdV-type equations,, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, (1995), 17. [5] J. Bourgain, Aspects of long time behavior of solutions of nonlinear Hamiltonian evolution equations,, Geom. Funct. Anal., 5 (1995), 105. 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. doi: 10.1155/S1073792896000207. [7] J. Bourgain, "Nonlinear Schrödinger Equations,", Hyperbolic equations and frequency interactions (Park City, 5 (1995), 3. [8] J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations,, Ergodic Theory Dynam. Systems, 24 (2004), 1331. 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. 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. 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. 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). 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). [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. 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, 43 (2010), 761. [17] P. Gérard and S. Grellier, "L'Équation de Szegö Cubique,", Séminaire X EDP, (2008). [18] Z. Hani, "Global and Dynamical Aspects of Nonlinear Schrödinger Equations on Compact Manifolds,", Ph.D. thesis, (2011). [19] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,, Phys. Rev. Lett., 27 (1971), 1192. [20] S. B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations,, Geom. Funct. Anal., 7 (1997), 338. doi: 10.1007/PL00001622. [21] P. Lax, Translation invariant spaces,, Acta Math., 101 (1959), 163. doi: 10.1007/BF02559553. [22] P. Lax, Integral of nonlinear equations of evolution and solitary waves,, Comm. Pure and Applied Math., 101 (1968), 467. [23] P. Lax, "Linear Algebra,", Pure and Applied Mathematics, (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, (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. 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 (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, (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] 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.

show all references

##### References:
 [1] E.V. Abakumov, The inverse spectral problem for Hankel operators of finite rank,, (Russian. English, 22 (1994), 5. 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. [3] V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Graduate Texts in Mathematics, 60 (1989). [4] J. Bourgain, On the Cauchy problem for periodic KdV-type equations,, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, (1995), 17. [5] J. Bourgain, Aspects of long time behavior of solutions of nonlinear Hamiltonian evolution equations,, Geom. Funct. Anal., 5 (1995), 105. 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. doi: 10.1155/S1073792896000207. [7] J. Bourgain, "Nonlinear Schrödinger Equations,", Hyperbolic equations and frequency interactions (Park City, 5 (1995), 3. [8] J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations,, Ergodic Theory Dynam. Systems, 24 (2004), 1331. 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. 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. 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. 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). 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). [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. 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, 43 (2010), 761. [17] P. Gérard and S. Grellier, "L'Équation de Szegö Cubique,", Séminaire X EDP, (2008). [18] Z. Hani, "Global and Dynamical Aspects of Nonlinear Schrödinger Equations on Compact Manifolds,", Ph.D. thesis, (2011). [19] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,, Phys. Rev. Lett., 27 (1971), 1192. [20] S. B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations,, Geom. Funct. Anal., 7 (1997), 338. doi: 10.1007/PL00001622. [21] P. Lax, Translation invariant spaces,, Acta Math., 101 (1959), 163. doi: 10.1007/BF02559553. [22] P. Lax, Integral of nonlinear equations of evolution and solitary waves,, Comm. Pure and Applied Math., 101 (1968), 467. [23] P. Lax, "Linear Algebra,", Pure and Applied Mathematics, (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, (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. 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 (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, (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] 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.
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