American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Grébert and T. Kappeler, The Defocusing NLS Equation and Its Normal Form, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/131.  Google Scholar

[7]

T. Kappeler, A. Maspero, J.-C. Molnar and P. Topalov, On the convexity of the KdV Hamiltonian,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. Kappeler and J. Pöschel, KdV & KAM, Springer, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. Korotyaev, Estimates for the Hill operator. I, J. Differential Equations, 162 (2000), 1-26. doi: 10.1006/jdeq.1999.3684.  Google Scholar

[15]

E. Korotyaev, Estimates for the Hill operator. II, J. Differential Equations, 223 (2006), 229-260. doi: 10.1016/j.jde.2005.04.017.  Google Scholar

[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.  Google Scholar

[17]

H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math., 30 (1975), 217-274. doi: 10.1007/BF01425567.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J.-C. Molnar, New estimates of the nonlinear Fourier transform for the defocusing NLS equation,, \arXiv{1403.1369}., ().  doi: 10.1093/imrn/rnu208.  Google Scholar

[21]

J.-C. Molnar, On two-sided estimates for the nonlinear fourier transform of KdV,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Grébert and T. Kappeler, The Defocusing NLS Equation and Its Normal Form, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/131.  Google Scholar

[7]

T. Kappeler, A. Maspero, J.-C. Molnar and P. Topalov, On the convexity of the KdV Hamiltonian,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. Kappeler and J. Pöschel, KdV & KAM, Springer, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. Korotyaev, Estimates for the Hill operator. I, J. Differential Equations, 162 (2000), 1-26. doi: 10.1006/jdeq.1999.3684.  Google Scholar

[15]

E. Korotyaev, Estimates for the Hill operator. II, J. Differential Equations, 223 (2006), 229-260. doi: 10.1016/j.jde.2005.04.017.  Google Scholar

[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.  Google Scholar

[17]

H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math., 30 (1975), 217-274. doi: 10.1007/BF01425567.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J.-C. Molnar, New estimates of the nonlinear Fourier transform for the defocusing NLS equation,, \arXiv{1403.1369}., ().  doi: 10.1093/imrn/rnu208.  Google Scholar

[21]

J.-C. Molnar, On two-sided estimates for the nonlinear fourier transform of KdV,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
[2]

M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22
[3]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[4]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[5]

Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[6]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[7]

Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations & Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028
[8]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[9]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[10]

Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[11]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097
[12]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[13]

Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[14]

John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
[15]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[16]

Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri. Non-standard boundary conditions for the linearized Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2625-2654. doi: 10.3934/dcdss.2021066
[17]

Julie Valein. On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021039
[18]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[19]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353
[20]

Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences