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.

[1]

François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems & Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019

[2]

Guillaume Bal, Jiaming Chen, Anthony B. Davis. Reconstruction of cloud geometry from high-resolution multi-angle images. Inverse Problems & Imaging, 2018, 12 (2) : 261-280. doi: 10.3934/ipi.2018011

[3]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067

[4]

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 & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991

[5]

Andrea Braides, Margherita Solci, Enrico Vitali. A derivation of linear elastic energies from pair-interaction atomistic systems. Networks & Heterogeneous Media, 2007, 2 (3) : 551-567. doi: 10.3934/nhm.2007.2.551

[6]

Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 525-545. doi: 10.3934/dcdss.2009.2.525

[7]

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

[8]

Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873

[9]

Bernard Brighi, Tewfik Sari. Blowing-up coordinates for a similarity boundary layer equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 929-948. doi: 10.3934/dcds.2005.12.929

[10]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[11]

Hermano Frid. Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 585-593. doi: 10.3934/dcds.1995.1.585

[12]

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

[13]

Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61

[14]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325

[15]

Boris S. Kruglikov and Vladimir S. Matveev. Vanishing of the entropy pseudonorm for certain integrable systems. Electronic Research Announcements, 2006, 12: 19-28.

[16]

Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889

[17]

Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003

[18]

Hisashi Okamoto, Takashi Sakajo, Marcus Wunsch. Steady-states and traveling-wave solutions of the generalized Constantin--Lax--Majda equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3155-3170. doi: 10.3934/dcds.2014.34.3155

[19]

Carl-Friedrich Kreiner, Johannes Zimmer. Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 915-931. doi: 10.3934/dcds.2009.25.915

[20]

Kaizhi Wang. Action minimizing stochastic invariant measures for a class of Lagrangian systems. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1211-1223. doi: 10.3934/cpaa.2008.7.1211

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]