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Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy
On two-sided estimates for the nonlinear Fourier transform of KdV
1. | Winterthrerstrasse 190, 8057 Zurich, Switzerland |
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. |
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. |
[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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