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 & 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.  Google Scholar

[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.  Google Scholar

[3]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251. doi: 10.1002/cpa.3160320202.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Gérard and S. Grellier, Invariant tori for the cubic Szegö equation,, to appear in Invent. Math., ().   Google Scholar

[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.  Google Scholar

[17]

P. Gérard and S. Grellier, "L'Équation de Szegö Cubique," Séminaire X EDP, École Polytechnique, Palaiseau, 20 octobre 2008.  Google Scholar

[18]

Z. Hani, "Global and Dynamical Aspects of Nonlinear Schrödinger Equations on Compact Manifolds," Ph.D. thesis, UCLA, 2011. Google Scholar

[19]

R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1194. Google Scholar

[20]

S. B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal., 7 (1997), 338-363. doi: 10.1007/PL00001622.  Google Scholar

[21]

P. Lax, Translation invariant spaces, Acta Math., 101 (1959), 163-178. doi: 10.1007/BF02559553.  Google Scholar

[22]

P. Lax, Integral of nonlinear equations of evolution and solitary waves, Comm. Pure and Applied Math., 101 (1968), 467-490.  Google Scholar

[23]

P. Lax, "Linear Algebra," Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1997.  Google Scholar

[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.  Google Scholar

[25]

Y. Martel and F. Merle, Description of two soliton collision for the quartic gKdV equations,, to appear in Annals of Math., ().   Google Scholar

[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.  Google Scholar

[27]

N. K. Nikolskii, "Operators, Functions and Systems: An Easy Reading," Vol.I: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, 92, AMS, 2002.  Google Scholar

[28]

S. P. Novikov, "Theory of Solitons: The Inverse Scattering Method," Moscow: Nauka, 1980.  Google Scholar

[29]

V. V. Peller, "Hankel Operators and Their Applications," Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.  Google Scholar

[30]

O. Pocovnicu, Traveling waves for the cubic Szegö equation on the real line,, to appear in Analysis and PDE, ().   Google Scholar

[31]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics," Vol. I-IV,, Academic Press, (): 1972.   Google Scholar

[32]

T. Tao, Why are solitons stable?, Bulletin (New Series) of the American Mathematical Society, 46 (2009), 1-33.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251. doi: 10.1002/cpa.3160320202.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Gérard and S. Grellier, Invariant tori for the cubic Szegö equation,, to appear in Invent. Math., ().   Google Scholar

[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.  Google Scholar

[17]

P. Gérard and S. Grellier, "L'Équation de Szegö Cubique," Séminaire X EDP, École Polytechnique, Palaiseau, 20 octobre 2008.  Google Scholar

[18]

Z. Hani, "Global and Dynamical Aspects of Nonlinear Schrödinger Equations on Compact Manifolds," Ph.D. thesis, UCLA, 2011. Google Scholar

[19]

R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1194. Google Scholar

[20]

S. B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal., 7 (1997), 338-363. doi: 10.1007/PL00001622.  Google Scholar

[21]

P. Lax, Translation invariant spaces, Acta Math., 101 (1959), 163-178. doi: 10.1007/BF02559553.  Google Scholar

[22]

P. Lax, Integral of nonlinear equations of evolution and solitary waves, Comm. Pure and Applied Math., 101 (1968), 467-490.  Google Scholar

[23]

P. Lax, "Linear Algebra," Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1997.  Google Scholar

[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.  Google Scholar

[25]

Y. Martel and F. Merle, Description of two soliton collision for the quartic gKdV equations,, to appear in Annals of Math., ().   Google Scholar

[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.  Google Scholar

[27]

N. K. Nikolskii, "Operators, Functions and Systems: An Easy Reading," Vol.I: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, 92, AMS, 2002.  Google Scholar

[28]

S. P. Novikov, "Theory of Solitons: The Inverse Scattering Method," Moscow: Nauka, 1980.  Google Scholar

[29]

V. V. Peller, "Hankel Operators and Their Applications," Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.  Google Scholar

[30]

O. Pocovnicu, Traveling waves for the cubic Szegö equation on the real line,, to appear in Analysis and PDE, ().   Google Scholar

[31]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics," Vol. I-IV,, Academic Press, (): 1972.   Google Scholar

[32]

T. Tao, Why are solitons stable?, Bulletin (New Series) of the American Mathematical Society, 46 (2009), 1-33.  Google Scholar

[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.  Google Scholar

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