On two-sided estimates for the nonlinear Fourier transform of KdV

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June 2016, 36(6): 3339-3356. doi: 10.3934/dcds.2016.36.3339

On two-sided estimates for the nonlinear Fourier transform of KdV

Jan-Cornelius Molnar ^1,

1.	Winterthrerstrasse 190, 8057 Zurich, Switzerland

Received April 2015 Revised October 2015 Published December 2015

Abstract
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The KdV-equation $u_t = -u_{xxx} + 6uu_x$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we obtain two-sided polynomial estimates of all integer Sobolev norms $||u||_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed, which are linear in the highest order.

Keywords: Birkhoff coordinates., integrable PDEs, Nonlinear Fourier transform, action-angle variables, Korteweg-de Vries equation.

Mathematics Subject Classification: Primary: 37K15; Secondary: 35Q53, 37K1.

Citation: Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339

References:

[1]	J. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. doi: 10.1098/rsta.1975.0035.
[2]	J. Colliander, T. Tao, M. Keel, G. Staffilani and H. Takaoka, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.
[3]	J. Colliander, T. Tao, M. Keel, G. Staffilani and H. Takaoka, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218. doi: 10.1016/S0022-1236(03)00218-0.
[4]	P. Djakov and B. Mityagin, Instability zones of periodic 1-dimensional Schrödinger and Dirac operators, Russian Math. Surveys, 61 (2006), 663-766. doi: 10.1070/RM2006v061n04ABEH004343.
[5]	H. Flaschka and D. W. McLaughlin, Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys., 55 (1976), 438-456. doi: 10.1143/PTP.55.438.
[6]	B. Grébert and T. Kappeler, The Defocusing NLS Equation and Its Normal Form, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/131.
[7]	T. Kappeler, A. Maspero, J.-C. Molnar and P. Topalov, On the convexity of the KdV Hamiltonian,, , ().
[8]	T. Kappeler and B. Mityagin, Gap estimates of the spectrum of Hill's equation and action variables for KdV, Trans. Amer. Math. Soc., 351 (1999), 619-646. doi: 10.1090/S0002-9947-99-02186-8.
[9]	T. Kappeler and B. Mityagin, Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator, SIAM J. Math. Anal., 33 (2001), 113-152. doi: 10.1137/S0036141099365753.
[10]	T. Kappeler, C. Möhr and P. Topalov, Birkhoff coordinates for KdV on phase spaces of distributions, Selecta Math. (N.S.), 11 (2005), 37-98. doi: 10.1007/s00029-005-0009-6.
[11]	T. Kappeler and J. Pöschel, KdV & KAM, Springer, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.
[12]	T. Kappeler and J. Pöschel, On the periodic KdV equation in weighted Sobolev spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 841-853. doi: 10.1016/j.anihpc.2008.03.004.
[13]	T. Kappeler, B. Schaad and P. Topalov, Asymptotics of spectral quantities of Schrödinger operators, in Spectral Geometry, 84, Amer. Math. Soc., Providence, RI, 2012, 243-284. doi: 10.1090/pspum/084/1360.
[14]	E. Korotyaev, Estimates for the Hill operator. I, J. Differential Equations, 162 (2000), 1-26. doi: 10.1006/jdeq.1999.3684.
[15]	E. Korotyaev, Estimates for the Hill operator. II, J. Differential Equations, 223 (2006), 229-260. doi: 10.1016/j.jde.2005.04.017.
[16]	V. A. Marčenko and I. V. Ostrovs ki, A characterization of the spectrum of the Hill operator, Mat. Sb. (N.S.), 97(139) (1975), 540-606, 633-634.
[17]	H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math., 30 (1975), 217-274. doi: 10.1007/BF01425567.
[18]	H. P. McKean and K. L. Vaninsky, Action-angle variables for the cubic Schrödinger equation, Comm. Pure Appl. Math., 50 (1997), 489-562. doi: 10.1002/(SICI)1097-0312(199706)50:6<489::AID-CPA1>3.0.CO;2-4.
[19]	J.-C. Molnar, New estimates of the nonlinear Fourier transform for the defocusing NLS equation, Int. Math. Res. Not., 2015 (2015), 8309-8352. doi: 10.1093/imrn/rnu208.
[20]	J.-C. Molnar, New estimates of the nonlinear Fourier transform for the defocusing NLS equation,, \arXiv{1403.1369}., (). doi: 10.1093/imrn/rnu208.
[21]	J.-C. Molnar, On two-sided estimates for the nonlinear fourier transform of KdV,, , ().
[22]	J. Pöschel, Hill's potentials in weighted Sobolev spaces and their spectral gaps, Math. Ann., 349 (2011), 433-458. doi: 10.1007/s00208-010-0513-7.
[23]	T. Tao, J. Colliander, M. Keel, G. Staffilani and H. Takaoka, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Differential Equations, (2011), 1-7.

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References:

[1]	J. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. doi: 10.1098/rsta.1975.0035.
[2]	J. Colliander, T. Tao, M. Keel, G. Staffilani and H. Takaoka, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.
[3]	J. Colliander, T. Tao, M. Keel, G. Staffilani and H. Takaoka, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218. doi: 10.1016/S0022-1236(03)00218-0.
[4]	P. Djakov and B. Mityagin, Instability zones of periodic 1-dimensional Schrödinger and Dirac operators, Russian Math. Surveys, 61 (2006), 663-766. doi: 10.1070/RM2006v061n04ABEH004343.
[5]	H. Flaschka and D. W. McLaughlin, Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys., 55 (1976), 438-456. doi: 10.1143/PTP.55.438.
[6]	B. Grébert and T. Kappeler, The Defocusing NLS Equation and Its Normal Form, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/131.
[7]	T. Kappeler, A. Maspero, J.-C. Molnar and P. Topalov, On the convexity of the KdV Hamiltonian,, , ().
[8]	T. Kappeler and B. Mityagin, Gap estimates of the spectrum of Hill's equation and action variables for KdV, Trans. Amer. Math. Soc., 351 (1999), 619-646. doi: 10.1090/S0002-9947-99-02186-8.
[9]	T. Kappeler and B. Mityagin, Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator, SIAM J. Math. Anal., 33 (2001), 113-152. doi: 10.1137/S0036141099365753.
[10]	T. Kappeler, C. Möhr and P. Topalov, Birkhoff coordinates for KdV on phase spaces of distributions, Selecta Math. (N.S.), 11 (2005), 37-98. doi: 10.1007/s00029-005-0009-6.
[11]	T. Kappeler and J. Pöschel, KdV & KAM, Springer, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.
[12]	T. Kappeler and J. Pöschel, On the periodic KdV equation in weighted Sobolev spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 841-853. doi: 10.1016/j.anihpc.2008.03.004.
[13]	T. Kappeler, B. Schaad and P. Topalov, Asymptotics of spectral quantities of Schrödinger operators, in Spectral Geometry, 84, Amer. Math. Soc., Providence, RI, 2012, 243-284. doi: 10.1090/pspum/084/1360.
[14]	E. Korotyaev, Estimates for the Hill operator. I, J. Differential Equations, 162 (2000), 1-26. doi: 10.1006/jdeq.1999.3684.
[15]	E. Korotyaev, Estimates for the Hill operator. II, J. Differential Equations, 223 (2006), 229-260. doi: 10.1016/j.jde.2005.04.017.
[16]	V. A. Marčenko and I. V. Ostrovs ki, A characterization of the spectrum of the Hill operator, Mat. Sb. (N.S.), 97(139) (1975), 540-606, 633-634.
[17]	H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math., 30 (1975), 217-274. doi: 10.1007/BF01425567.
[18]	H. P. McKean and K. L. Vaninsky, Action-angle variables for the cubic Schrödinger equation, Comm. Pure Appl. Math., 50 (1997), 489-562. doi: 10.1002/(SICI)1097-0312(199706)50:6<489::AID-CPA1>3.0.CO;2-4.
[19]	J.-C. Molnar, New estimates of the nonlinear Fourier transform for the defocusing NLS equation, Int. Math. Res. Not., 2015 (2015), 8309-8352. doi: 10.1093/imrn/rnu208.
[20]	J.-C. Molnar, New estimates of the nonlinear Fourier transform for the defocusing NLS equation,, \arXiv{1403.1369}., (). doi: 10.1093/imrn/rnu208.
[21]	J.-C. Molnar, On two-sided estimates for the nonlinear fourier transform of KdV,, , ().
[22]	J. Pöschel, Hill's potentials in weighted Sobolev spaces and their spectral gaps, Math. Ann., 349 (2011), 433-458. doi: 10.1007/s00208-010-0513-7.
[23]	T. Tao, J. Colliander, M. Keel, G. Staffilani and H. Takaoka, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Differential Equations, (2011), 1-7.

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