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

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

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.
Citation: David Lannes, Jean-Claude Saut. Remarks on the full dispersion Kadomtsev-Petviashvli equation. Kinetic & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341. doi: 10.1007/BF01896259.  Google Scholar

[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.  Google Scholar

[9]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[11]

M. Ehrnström and H. Kalish, Traveling waves for the Whitham equation, Diff. Int. Equations, 22 (2009), 1193-1210.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. 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), 763-811. doi: 10.1007/s00332-012-9127-4.  Google Scholar

[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), 429-470. doi: 10.1007/s00332-007-9001-y.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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. 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), 205-206. doi: 10.1016/0375-9601(77)90875-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

H. Takaoka and N. Tzvetkov, On the local regularity of Kadomtsev-Petviashvili-II equation, IMRN, 8 (2001), 77-114. doi: 10.1155/S1073792801000058.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[33]

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

[34]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341. doi: 10.1007/BF01896259.  Google Scholar

[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.  Google Scholar

[9]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[11]

M. Ehrnström and H. Kalish, Traveling waves for the Whitham equation, Diff. Int. Equations, 22 (2009), 1193-1210.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. 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), 763-811. doi: 10.1007/s00332-012-9127-4.  Google Scholar

[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), 429-470. doi: 10.1007/s00332-007-9001-y.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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. 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), 205-206. doi: 10.1016/0375-9601(77)90875-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

H. Takaoka and N. Tzvetkov, On the local regularity of Kadomtsev-Petviashvili-II equation, IMRN, 8 (2001), 77-114. doi: 10.1155/S1073792801000058.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[33]

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

[34]

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

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