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Remarks on the full dispersion Kadomtsev-Petviashvli equation
1. | DMA, Ecole Normale Supérieure et CNRS UMR 8553, 45 rue d'Ulm, 75005 Paris |
2. | Laboratoire de Mathématiques, UMR 8628, Université Paris-Sud et CNRS, 91405 Orsay, France |
References:
[1] |
J. Albert, J. L. Bona and J.-C.Saut, Model equations for waves in stratified fluids,, Proc. Royal Soc. London A, 453 (1997), 1233.
doi: 10.1098/rspa.1997.0068. |
[2] |
D. Alterman and J. Rauch, The linear diffractive pulse equation,, Cathleen Morawetz: A great mathematician, 7 (2000), 263.
|
[3] |
B. Alvarez-Samaniego and D. Lannes, Large time existence for 3d water-waves and asymptotics,, Invent. math., 171 (2008), 485.
doi: 10.1007/s00222-007-0088-4. |
[4] |
W. Ben Youssef and D. Lannes, The long wave limit for a general class of 2D quasilinear hyperbolic problems,, Comm. Partial Differential Equations, 27 (2002), 979.
doi: 10.1081/PDE-120004892. |
[5] |
J. L. Bona, T. Colin and D. Lannes, Long-wave approximation for water waves,, Arch. Ration. Mech. Anal., 178 (2005), 373.
doi: 10.1007/s00205-005-0378-1. |
[6] |
A. de Bouard and J.-C. Saut, Solitary waves of generalized KP equations,, Annales IHP Analyse non Linéaire, 14 (1997), 211.
doi: 10.1016/S0294-1449(97)80145-X. |
[7] |
J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation,, Geom. Funct. Anal., 3 (1993), 315.
doi: 10.1007/BF01896259. |
[8] |
A. Castro, D. Córdoba and F. Gancedo, Singularity formation in a surface wave model,, Nonlinearity, 23 (2010), 2835.
doi: 10.1088/0951-7715/23/11/006. |
[9] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
doi: 10.1007/BF02392586. |
[10] |
T. Colin and D. Lannes, Long-wave short-wave resonance for nonlinear geometric optics,, Duke Math. J., 107 (2001), 351.
doi: 10.1215/S0012-7094-01-10725-4. |
[11] |
M. Ehrnström and H. Kalish, Traveling waves for the Whitham equation,, Diff. Int. Equations, 22 (2009), 1193.
|
[12] |
M. Ehrnström, M. D. Groves and E. Wahlén, On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type,, Nonlinearity, 25 (2012), 2903.
doi: 10.1088/0951-7715/25/10/2903. |
[13] |
R. L. Frank and E. Lenzmann, On the uniqueness and nondegeneracy of ground states of $(-\Delta)^s Q+Q-Q^{\alpha +1}=0$ in $\mathbbR$,, , (2010). Google Scholar |
[14] |
Z. Guo, L. Peng and B. Wang, Decay estimates for a class of wave equations,, J. Funct. Analysis, 254 (2008), 1642.
doi: 10.1016/j.jfa.2007.12.010. |
[15] |
B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media,, Sov. Phys. Dokl., 15 (1970), 539. Google Scholar |
[16] |
C. Klein and J.-C. Saut, Numerical study of blow-up and stability of solutions to generalized Kadomtsev-Petviashvili equations,, J. Nonlinear Science, 22 (2012), 763.
doi: 10.1007/s00332-012-9127-4. |
[17] |
C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of the Burgers equation,, in preparation., (). Google Scholar |
[18] |
C. Klein, C. Sparber and P. Markowich, Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation,, J. Nonl. Sci., 17 (2007), 429.
doi: 10.1007/s00332-007-9001-y. |
[19] |
D. Lannes, The Water Waves Problem: Mathematical Theory and Asymptotics,, Mathematical Surveys and Monographs, (2013).
|
[20] |
D. Lannes, Consistency of the KP approximation, Dynamical systems and differential equations (Wilmington, NC, 2002)., Discrete Cont. Dyn. Syst., (2003), 517.
|
[21] |
D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation,, Nonlinearity, 19 (2006), 2853.
doi: 10.1088/0951-7715/19/12/007. |
[22] |
F. Linares, D. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory,, , (2013). Google Scholar |
[23] |
S. V. Manakov, V. E. Zakharov, L. A. Bordag and V. B. Matveev, Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction,, Phys. Lett. A, 63 (1977), 205.
doi: 10.1016/0375-9601(77)90875-1. |
[24] |
M. Ming, P. Zhang and Z. Zhang, Long-wave approximation to the 3-D capillary-gravity waves,, SIAM J. Math. Anal., 44 (2012), 2920.
doi: 10.1137/11084220X. |
[25] |
L. Molinet, On the asymptotic behavior of solutions to the (generalized) Kadomtsev-Petviashvili-Burgers equations,, J. Diff. Eq., 152 (1999), 30.
doi: 10.1006/jdeq.1998.3522. |
[26] |
L. Molinet, J.-C. Saut and N. Tzvetkov, Remarks on the mass constraint for KP type equations,, SIAM J. Math. Anal., 39 (2007), 627.
doi: 10.1137/060654256. |
[27] |
P. I. Naumkin and I. A. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves,, Translated from the Russian manuscript by Boris Gommerstadt. Translations of Mathematical Monographs, (1994).
|
[28] |
J.-C. Saut, Remarks on the generalized Kadomtsev-Petviashvili equations,, Indiana Univ. Math. J., 42 (1993), 1011.
doi: 10.1512/iumj.1993.42.42047. |
[29] |
H. Takaoka and N. Tzvetkov, On the local regularity of Kadomtsev-Petviashvili-II equation,, IMRN, 8 (2001), 77.
doi: 10.1155/S1073792801000058. |
[30] |
S. Ukaï, Local solutions of the Kadomtsev-Petviashvili equation,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 193.
|
[31] |
M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, Commun. Partial Diff. Equ, 12 (1987), 1133.
doi: 10.1080/03605308708820522. |
[32] |
M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (1983), 567.
|
[33] |
G. B. Whitham, Variational methods and applications to water waves,, Proc. R. Soc. Lond. A, 299 (1967), 6. Google Scholar |
[34] |
G. B. Whitham, Linear and Nonlinear Waves,, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974).
|
show all references
References:
[1] |
J. Albert, J. L. Bona and J.-C.Saut, Model equations for waves in stratified fluids,, Proc. Royal Soc. London A, 453 (1997), 1233.
doi: 10.1098/rspa.1997.0068. |
[2] |
D. Alterman and J. Rauch, The linear diffractive pulse equation,, Cathleen Morawetz: A great mathematician, 7 (2000), 263.
|
[3] |
B. Alvarez-Samaniego and D. Lannes, Large time existence for 3d water-waves and asymptotics,, Invent. math., 171 (2008), 485.
doi: 10.1007/s00222-007-0088-4. |
[4] |
W. Ben Youssef and D. Lannes, The long wave limit for a general class of 2D quasilinear hyperbolic problems,, Comm. Partial Differential Equations, 27 (2002), 979.
doi: 10.1081/PDE-120004892. |
[5] |
J. L. Bona, T. Colin and D. Lannes, Long-wave approximation for water waves,, Arch. Ration. Mech. Anal., 178 (2005), 373.
doi: 10.1007/s00205-005-0378-1. |
[6] |
A. de Bouard and J.-C. Saut, Solitary waves of generalized KP equations,, Annales IHP Analyse non Linéaire, 14 (1997), 211.
doi: 10.1016/S0294-1449(97)80145-X. |
[7] |
J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation,, Geom. Funct. Anal., 3 (1993), 315.
doi: 10.1007/BF01896259. |
[8] |
A. Castro, D. Córdoba and F. Gancedo, Singularity formation in a surface wave model,, Nonlinearity, 23 (2010), 2835.
doi: 10.1088/0951-7715/23/11/006. |
[9] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
doi: 10.1007/BF02392586. |
[10] |
T. Colin and D. Lannes, Long-wave short-wave resonance for nonlinear geometric optics,, Duke Math. J., 107 (2001), 351.
doi: 10.1215/S0012-7094-01-10725-4. |
[11] |
M. Ehrnström and H. Kalish, Traveling waves for the Whitham equation,, Diff. Int. Equations, 22 (2009), 1193.
|
[12] |
M. Ehrnström, M. D. Groves and E. Wahlén, On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type,, Nonlinearity, 25 (2012), 2903.
doi: 10.1088/0951-7715/25/10/2903. |
[13] |
R. L. Frank and E. Lenzmann, On the uniqueness and nondegeneracy of ground states of $(-\Delta)^s Q+Q-Q^{\alpha +1}=0$ in $\mathbbR$,, , (2010). Google Scholar |
[14] |
Z. Guo, L. Peng and B. Wang, Decay estimates for a class of wave equations,, J. Funct. Analysis, 254 (2008), 1642.
doi: 10.1016/j.jfa.2007.12.010. |
[15] |
B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media,, Sov. Phys. Dokl., 15 (1970), 539. Google Scholar |
[16] |
C. Klein and J.-C. Saut, Numerical study of blow-up and stability of solutions to generalized Kadomtsev-Petviashvili equations,, J. Nonlinear Science, 22 (2012), 763.
doi: 10.1007/s00332-012-9127-4. |
[17] |
C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of the Burgers equation,, in preparation., (). Google Scholar |
[18] |
C. Klein, C. Sparber and P. Markowich, Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation,, J. Nonl. Sci., 17 (2007), 429.
doi: 10.1007/s00332-007-9001-y. |
[19] |
D. Lannes, The Water Waves Problem: Mathematical Theory and Asymptotics,, Mathematical Surveys and Monographs, (2013).
|
[20] |
D. Lannes, Consistency of the KP approximation, Dynamical systems and differential equations (Wilmington, NC, 2002)., Discrete Cont. Dyn. Syst., (2003), 517.
|
[21] |
D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation,, Nonlinearity, 19 (2006), 2853.
doi: 10.1088/0951-7715/19/12/007. |
[22] |
F. Linares, D. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory,, , (2013). Google Scholar |
[23] |
S. V. Manakov, V. E. Zakharov, L. A. Bordag and V. B. Matveev, Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction,, Phys. Lett. A, 63 (1977), 205.
doi: 10.1016/0375-9601(77)90875-1. |
[24] |
M. Ming, P. Zhang and Z. Zhang, Long-wave approximation to the 3-D capillary-gravity waves,, SIAM J. Math. Anal., 44 (2012), 2920.
doi: 10.1137/11084220X. |
[25] |
L. Molinet, On the asymptotic behavior of solutions to the (generalized) Kadomtsev-Petviashvili-Burgers equations,, J. Diff. Eq., 152 (1999), 30.
doi: 10.1006/jdeq.1998.3522. |
[26] |
L. Molinet, J.-C. Saut and N. Tzvetkov, Remarks on the mass constraint for KP type equations,, SIAM J. Math. Anal., 39 (2007), 627.
doi: 10.1137/060654256. |
[27] |
P. I. Naumkin and I. A. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves,, Translated from the Russian manuscript by Boris Gommerstadt. Translations of Mathematical Monographs, (1994).
|
[28] |
J.-C. Saut, Remarks on the generalized Kadomtsev-Petviashvili equations,, Indiana Univ. Math. J., 42 (1993), 1011.
doi: 10.1512/iumj.1993.42.42047. |
[29] |
H. Takaoka and N. Tzvetkov, On the local regularity of Kadomtsev-Petviashvili-II equation,, IMRN, 8 (2001), 77.
doi: 10.1155/S1073792801000058. |
[30] |
S. Ukaï, Local solutions of the Kadomtsev-Petviashvili equation,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 193.
|
[31] |
M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, Commun. Partial Diff. Equ, 12 (1987), 1133.
doi: 10.1080/03605308708820522. |
[32] |
M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (1983), 567.
|
[33] |
G. B. Whitham, Variational methods and applications to water waves,, Proc. R. Soc. Lond. A, 299 (1967), 6. Google Scholar |
[34] |
G. B. Whitham, Linear and Nonlinear Waves,, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974).
|
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