American Institute of Mathematical Sciences

Advanced
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 & 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

[1]

Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621
[2]

Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397
[3]

Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109
[4]

Shunlian Liu, David M. Ambrose. Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3123-3147. doi: 10.3934/dcds.2019129
[5]

Calin Iulian Martin, Adrián Rodríguez-Sanjurjo. Dispersion relations for steady periodic water waves of fixed mean-depth with two rotational layers. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5149-5169. doi: 10.3934/dcds.2019209
[6]

Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045
[7]

Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331
[8]

Zengji Du, Xiaojie Lin, Yulin Ren. Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021118
[9]

Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic & Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261
[10]

Anwar Ja'afar Mohamad Jawad, Mohammad Mirzazadeh, Anjan Biswas. Dynamics of shallow water waves with Gardner-Kadomtsev-Petviashvili equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1155-1164. doi: 10.3934/dcdss.2015.8.1155
[11]

Caroline Obrecht, J.-C. Saut. Remarks on the full dispersion Davey-Stewartson systems. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1547-1561. doi: 10.3934/cpaa.2015.14.1547
[12]

Fei Guo, Bao-Feng Feng, Hongjun Gao, Yue Liu. On the initial-value problem to the Degasperis-Procesi equation with linear dispersion. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1269-1290. doi: 10.3934/dcds.2010.26.1269
[13]

Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251
[14]

Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021003
[15]

Shengfu Deng. Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1647-1662. doi: 10.3934/dcdss.2016068
[16]

Federico Cacciafesta, Anne-Sophie De Suzzoni. Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4359-4398. doi: 10.3934/dcds.2019177
[17]

Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607
[18]

Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103
[19]

Grégory Faye. Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2473-2496. doi: 10.3934/dcds.2016.36.2473
[20]

Guenbo Hwang, Byungsoo Moon. Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion. Electronic Research Archive, 2020, 28 (1) : 15-25. doi: 10.3934/era.2020002

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences