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December  2013, 6(4): 989-1009. doi: 10.3934/krm.2013.6.989

Remarks on the full dispersion Kadomtsev-Petviashvli equation

David Lannes 1, and  Jean-Claude Saut 2,

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

Received  September 2013 Revised  September 2013 Published  November 2013

We consider in this paper the Full Dispersion Kadomtsev-Petviashvili Equation (FDKP) introduced in [19] in order to overcome some shortcomings of the classical KP equation. We investigate its mathematical properties, emphasizing the differences with the Kadomtsev-Petviashvili equation and their relevance to the approximation of water waves. We also present some numerical simulations.
Keywords: full dispersion, KP equation, weakly transverse waves, nonlocal dispersion., water waves, Whitham equation.
Mathematics Subject Classification: 35Q53, 35R11, 76B15, 76B4.
Citation: David Lannes, Jean-Claude Saut. Remarks on the full dispersion Kadomtsev-Petviashvli equation. Kinetic and Related Models, 2013, 6 (4) : 989-1009. doi: 10.3934/krm.2013.6.989
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), 1233-1260. doi: 10.1098/rspa.1997.0068.

[2]

D. Alterman and J. Rauch, The linear diffractive pulse equation, Cathleen Morawetz: A great mathematician, Methods Appl. Anal., 7 (2000), 263-274.

[3]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3d water-waves and asymptotics, Invent. math., 171 (2008), 485-541. 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), 979-1020. 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), 373-410. 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), 211-236. 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), 315-341. doi: 10.1007/BF01896259.

[8]

A. Castro, D. Córdoba and F. Gancedo, Singularity formation in a surface wave model, Nonlinearity, 23 (2010), 2835-2847. 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), 229-243. doi: 10.1007/BF02392586.

[10]

T. Colin and D. Lannes, Long-wave short-wave resonance for nonlinear geometric optics, Duke Math. J., 107 (2001), 351-419. 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), 1193-1210.

[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), 2903-2936. 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 $\mathbbR$, arXiv:1009.4042, 2010.

[14]

Z. Guo, L. Peng and B. Wang, Decay estimates for a class of wave equations, J. Funct. Analysis, 254 (2008), 1642-1660. 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), 539-541.

[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), 763-811. 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., (). 

[18]

C. Klein, C. Sparber and P. Markowich, Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation, J. Nonl. Sci., 17 (2007), 429-470. doi: 10.1007/s00332-007-9001-y.

[19]

D. Lannes, The Water Waves Problem: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, Volume 188, American Mathematical Society, Providence Rhode Island, 2013.

[20]

D. Lannes, Consistency of the KP approximation, Dynamical systems and differential equations (Wilmington, NC, 2002). Discrete Cont. Dyn. Syst., (2003) Suppl. 517-525.

[21]

D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation, Nonlinearity, 19 (2006), 2853-2875. 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, arXiv:1302.7146v1 [math.AP] 28 Feb 2013.

[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), 205-206. 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), 2920-2948. 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), 30-74. 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), 627-641. 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, 133. American Mathematical Society, Providence, RI, 1994.

[28]

J.-C. Saut, Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42 (1993), 1011-1026. 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), 77-114. doi: 10.1155/S1073792801000058.

[30]

S. Ukaï, Local solutions of the Kadomtsev-Petviashvili equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 193-209.

[31]

M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Commun. Partial Diff. Equ,. 12 (1987), 1133-1173. doi: 10.1080/03605308708820522.

[32]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576.

[33]

G. B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. Lond. A, 299 (1967), 6-25.

[34]

G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 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), 1233-1260. doi: 10.1098/rspa.1997.0068.

[2]

D. Alterman and J. Rauch, The linear diffractive pulse equation, Cathleen Morawetz: A great mathematician, Methods Appl. Anal., 7 (2000), 263-274.

[3]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3d water-waves and asymptotics, Invent. math., 171 (2008), 485-541. 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), 979-1020. 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), 373-410. 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), 211-236. 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), 315-341. doi: 10.1007/BF01896259.

[8]

A. Castro, D. Córdoba and F. Gancedo, Singularity formation in a surface wave model, Nonlinearity, 23 (2010), 2835-2847. 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), 229-243. doi: 10.1007/BF02392586.

[10]

T. Colin and D. Lannes, Long-wave short-wave resonance for nonlinear geometric optics, Duke Math. J., 107 (2001), 351-419. 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), 1193-1210.

[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), 2903-2936. 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 $\mathbbR$, arXiv:1009.4042, 2010.

[14]

Z. Guo, L. Peng and B. Wang, Decay estimates for a class of wave equations, J. Funct. Analysis, 254 (2008), 1642-1660. 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), 539-541.

[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), 763-811. 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., (). 

[18]

C. Klein, C. Sparber and P. Markowich, Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation, J. Nonl. Sci., 17 (2007), 429-470. doi: 10.1007/s00332-007-9001-y.

[19]

D. Lannes, The Water Waves Problem: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, Volume 188, American Mathematical Society, Providence Rhode Island, 2013.

[20]

D. Lannes, Consistency of the KP approximation, Dynamical systems and differential equations (Wilmington, NC, 2002). Discrete Cont. Dyn. Syst., (2003) Suppl. 517-525.

[21]

D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation, Nonlinearity, 19 (2006), 2853-2875. 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, arXiv:1302.7146v1 [math.AP] 28 Feb 2013.

[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), 205-206. 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), 2920-2948. 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), 30-74. 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), 627-641. 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, 133. American Mathematical Society, Providence, RI, 1994.

[28]

J.-C. Saut, Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42 (1993), 1011-1026. 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), 77-114. doi: 10.1155/S1073792801000058.

[30]

S. Ukaï, Local solutions of the Kadomtsev-Petviashvili equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 193-209.

[31]

M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Commun. Partial Diff. Equ,. 12 (1987), 1133-1173. doi: 10.1080/03605308708820522.

[32]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576.

[33]

G. B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. Lond. A, 299 (1967), 6-25.

[34]

G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

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