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

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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

Abstract
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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$.

Keywords: Hankel operators, Lax pair, Szegö equation, integrable systems, soliton resolution, action-angle coordinates..

Mathematics Subject Classification: 35B15, 37K10, 47B3.

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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