# American Institute of Mathematical Sciences

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.  doi: 10.1098/rspa.1997.0068.  Google Scholar [2] D. Alterman and J. Rauch, The linear diffractive pulse equation,, Cathleen Morawetz: A great mathematician, 7 (2000), 263.   Google Scholar [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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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 $\mathbbR$,, , (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.  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.   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.  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.  doi: 10.1007/s00332-007-9001-y.  Google Scholar [19] D. Lannes, The Water Waves Problem: Mathematical Theory and Asymptotics,, Mathematical Surveys and Monographs, (2013).   Google Scholar [20] D. Lannes, Consistency of the KP approximation, Dynamical systems and differential equations (Wilmington, NC, 2002)., Discrete Cont. Dyn. Syst., (2003), 517.   Google Scholar [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.  Google Scholar [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.  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.  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.  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.  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, (1994).   Google Scholar [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.  Google Scholar [29] H. Takaoka and N. Tzvetkov, On the local regularity of Kadomtsev-Petviashvili-II equation,, IMRN, 8 (2001), 77.  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.   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.  doi: 10.1080/03605308708820522.  Google Scholar [32] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (1983), 567.   Google Scholar [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).   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.  doi: 10.1098/rspa.1997.0068.  Google Scholar [2] D. Alterman and J. Rauch, The linear diffractive pulse equation,, Cathleen Morawetz: A great mathematician, 7 (2000), 263.   Google Scholar [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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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 $\mathbbR$,, , (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.  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.   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.  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.  doi: 10.1007/s00332-007-9001-y.  Google Scholar [19] D. Lannes, The Water Waves Problem: Mathematical Theory and Asymptotics,, Mathematical Surveys and Monographs, (2013).   Google Scholar [20] D. Lannes, Consistency of the KP approximation, Dynamical systems and differential equations (Wilmington, NC, 2002)., Discrete Cont. Dyn. Syst., (2003), 517.   Google Scholar [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.  Google Scholar [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.  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.  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.  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.  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, (1994).   Google Scholar [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.  Google Scholar [29] H. Takaoka and N. Tzvetkov, On the local regularity of Kadomtsev-Petviashvili-II equation,, IMRN, 8 (2001), 77.  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.   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.  doi: 10.1080/03605308708820522.  Google Scholar [32] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (1983), 567.   Google Scholar [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).   Google Scholar
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