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July  2015, 14(4): 1547-1561. doi: 10.3934/cpaa.2015.14.1547

## Remarks on the full dispersion Davey-Stewartson systems

 1 Laboratoire de Mathématiques, Université Paris-Sud, Bât. 430, 91405 Orsay Cedex, France 2 UMR de Mathématiques, Université de Paris-Sud, Bâtiment 425, P.O. Box 91405, Orsay

Received  October 2013 Revised  May 2014 Published  April 2015

We consider the Cauchy problem for the Full Dispersion Davey-Stewartson systems derived in [23] for the modeling of surface water waves in the modulation regime and we investigate some of their mathematical properties, emphasizing in particular the differences with the classical Davey-Stewartson systems.
Citation: 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
##### References:
 [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511623998. Google Scholar [2] M. J. Ablowitz and H. Segur, On the evolution of packets of water waves,, \emph{J. Fluid Mech.}, 92 (1979), 691. doi: 10.1017/S0022112079000835. Google Scholar [3] D. J. Benney and G. J. Roskes, Waves instabilities,, \emph{Stud. Appl. Math.}, 48 (1969), 377. Google Scholar [4] C. Besse and C. H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up,, \emph{Mathematical Models and Methods in Applied Sciences}, 8 (1998), 1363. doi: 10.1142/S0218202598000640. Google Scholar [5] T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson system for quadratic hyperbolic systems,, \emph{Asymptotic Analysis}, 31 (2002), 69. Google Scholar [6] W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system,, \emph{Ann. Inst. H. Poincar\'e, 14 (1997), 615. doi: 10.1016/S0294-1449(97)80128-X. Google Scholar [7] W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach,, \emph{Nonlinearity}, 5 (1992), 497. Google Scholar [8] W. Craig and C. Sulem, Numerical simulation of gravity waves,, \emph{J. Comput. Phys.}, 108 (1993), 73. doi: 10.1006/jcph.1993.1164. Google Scholar [9] A. Davey and K. Stewartson, One three-dimensional packets of water waves,, \emph{Proc. Roy. Soc. Lond. A}, 338 (1974), 101. Google Scholar [10] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves,, \emph{J. Fluid Mech.}, 79 (1977), 703. Google Scholar [11] J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems,, \emph{Nonlinearity}, 3 (1990), 475. Google Scholar [12] J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations,, \emph{J. Nonlinear Sci.}, 6 (1996), 139. doi: 10.1007/s003329900006. Google Scholar [13] J.-M. Ghidaglia and J.-C. Saut, On the Zakharov-Schulman equations,, in \emph{Nonlinear Dispersive Waves} (L. Debnath Ed.), (1992), 83. Google Scholar [14] Z. Guo, L. Peng and B. Wang, Decay estimates for a class of wave equations,, \emph{J. Funct. Analysis}, 254 (2008), 1642. doi: 10.1016/j.jfa.2007.12.010. Google Scholar [15] Z. Guo and Y. Wang 2, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations,, arXiv:1007.4299v3., (). doi: 10.1007/s11854-014-0025-6. Google Scholar [16] N. Hayashi and H. Hirata, Local existence in time of small solutions to the elliptic- hyperbolic Davey-Stewartson system in the usual Sobolev space,, \emph{Proc. Edinburgh Math. Soc.}, 40 (1997), 563. doi: 10.1017/S0013091500024020. Google Scholar [17] N. Hayashi and H. Hirata, Global existence and scattering of small solutions to the elliptic-hyperbolic Davey-Stewartson system,, \emph{Nonlinearity}, 9 (1996), 1387. doi: 10.1088/0951-7715/9/6/001. Google Scholar [18] A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension,, arXiv:1209.4943v1., (). Google Scholar [19] C. Klein, B. Muite and K. Roidot, Numerical study of the blow-up in the Davey-Stewartson system,, \emph{Discr. Cont. Dyn. Syst. B}, 18 (2013), 1361. doi: 10.3934/dcdsb.2013.18.1361. Google Scholar [20] C. Klein and J.-C. Saut, A numerical approach to blow-up issues for Davey-Stewartson II type systems,, submitted., (). Google Scholar [21] C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations,, submitted., (). Google Scholar [22] Joseph Louis de Lagrange, Mémoire sur la théorie du mouvement des fluides,, Oeuvres compl\etes, (1781), 695. Google Scholar [23] D. Lannes, Water Waves : Mathematical Theory and Asymptotics,, Mathematical Surveys and Monographs, (2013). doi: 10.1090/surv/188. Google Scholar [24] David Lannes, A stability criterion for two-dimensional interfaces and applications,, \emph{Arch. Ration. Mech. Anal.}, 208 (2013), 481. doi: 10.1007/s00205-012-0604-6. Google Scholar [25] D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equation,, \emph{Kinematics and Related Models}, 6 (2013). doi: 10.3934/krm.2013.6.989. Google Scholar [26] F. Linares and G. Ponce, On the Davey-Stewartson systems,, \emph{Ann. Inst. H. Poincar\' e Anal. Non Lin\' eaire}, 10 (1993), 523. Google Scholar [27] C. Obrecht, In preparation., $\$, (). Google Scholar [28] C. Obrecht and K. Roidot, In preparation., $\$, (). Google Scholar [29] T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems,, \emph{Proc. Roy. Soc. London A}, 436 (1992), 345. doi: 10.1098/rspa.1992.0022. Google Scholar [30] G. C. Papanicolaou, C. Sulem, P.-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves,, \emph{Physica D}, 72 (1994), 61. doi: 10.1016/0167-2789(94)90167-8. Google Scholar [31] P. A. Perry, Global well-posedness and long time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}(\R^2)$,, arXiv:1110.5589v2., (). Google Scholar [32] G. Ponce and J.-C. Saut, Well-posedness for the Benney-Roskes-Zakharov- Rubenchik system,, \emph{Discrete Cont. Dynamical Systems}, 13 (2005), 811. doi: 10.3934/dcds.2005.13.811. Google Scholar [33] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation,, Springer-Verlag, 139 (1999). Google Scholar [34] L. Y. Sung, Long time decay of the solutions of the Davey-Stewartson II equations,, \emph{J. Nonlinear Sci.}, 5 (1995), 43. doi: 10.1007/BF01212909. Google Scholar [35] N. Totz, A justification of the modulation approximation to the 3D full water wave problem,, \emph{Comm. Math. Phys.}, 335 (2015), 369. doi: 10.1007/s00220-014-2259-7. Google Scholar [36] N. Totz and S. Wu, A rigorous justification of the modulation approximation to the 2D full water wave problem,, \emph{Comm. Math. Phys.}, 310 (2012), 817. doi: 10.1007/s00220-012-1422-2. Google Scholar [37] V. E. Zakharov, Weakly nonlinear waves on surface of ideal finite depth fluid,, \emph{Amer. Math. Soc. Transl. Ser. 2}, 182 (1998), 167. Google Scholar [38] V. E. Zakharov and A. M. Rubenchik, Nonlinear interaction of high-frequency and low frequency waves,, \emph{Prikl. Mat. Techn. Phys.}, (1972), 84. Google Scholar [39] V. E. Zakharov and E. I Schulman, Degenerate conservation laws, motion invariants and kinetic equations,, \emph{Physica}, 1D (1980), 192. doi: 10.1016/0167-2789(80)90011-1. Google Scholar

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##### References:
 [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511623998. Google Scholar [2] M. J. Ablowitz and H. Segur, On the evolution of packets of water waves,, \emph{J. Fluid Mech.}, 92 (1979), 691. doi: 10.1017/S0022112079000835. Google Scholar [3] D. J. Benney and G. J. Roskes, Waves instabilities,, \emph{Stud. Appl. Math.}, 48 (1969), 377. Google Scholar [4] C. Besse and C. H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up,, \emph{Mathematical Models and Methods in Applied Sciences}, 8 (1998), 1363. doi: 10.1142/S0218202598000640. Google Scholar [5] T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson system for quadratic hyperbolic systems,, \emph{Asymptotic Analysis}, 31 (2002), 69. Google Scholar [6] W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system,, \emph{Ann. Inst. H. Poincar\'e, 14 (1997), 615. doi: 10.1016/S0294-1449(97)80128-X. Google Scholar [7] W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach,, \emph{Nonlinearity}, 5 (1992), 497. Google Scholar [8] W. Craig and C. Sulem, Numerical simulation of gravity waves,, \emph{J. Comput. Phys.}, 108 (1993), 73. doi: 10.1006/jcph.1993.1164. Google Scholar [9] A. Davey and K. Stewartson, One three-dimensional packets of water waves,, \emph{Proc. Roy. Soc. Lond. A}, 338 (1974), 101. Google Scholar [10] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves,, \emph{J. Fluid Mech.}, 79 (1977), 703. Google Scholar [11] J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems,, \emph{Nonlinearity}, 3 (1990), 475. Google Scholar [12] J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations,, \emph{J. Nonlinear Sci.}, 6 (1996), 139. doi: 10.1007/s003329900006. Google Scholar [13] J.-M. Ghidaglia and J.-C. Saut, On the Zakharov-Schulman equations,, in \emph{Nonlinear Dispersive Waves} (L. Debnath Ed.), (1992), 83. Google Scholar [14] Z. Guo, L. Peng and B. Wang, Decay estimates for a class of wave equations,, \emph{J. Funct. Analysis}, 254 (2008), 1642. doi: 10.1016/j.jfa.2007.12.010. Google Scholar [15] Z. Guo and Y. Wang 2, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations,, arXiv:1007.4299v3., (). doi: 10.1007/s11854-014-0025-6. Google Scholar [16] N. Hayashi and H. Hirata, Local existence in time of small solutions to the elliptic- hyperbolic Davey-Stewartson system in the usual Sobolev space,, \emph{Proc. Edinburgh Math. Soc.}, 40 (1997), 563. doi: 10.1017/S0013091500024020. Google Scholar [17] N. Hayashi and H. Hirata, Global existence and scattering of small solutions to the elliptic-hyperbolic Davey-Stewartson system,, \emph{Nonlinearity}, 9 (1996), 1387. doi: 10.1088/0951-7715/9/6/001. Google Scholar [18] A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension,, arXiv:1209.4943v1., (). Google Scholar [19] C. Klein, B. Muite and K. Roidot, Numerical study of the blow-up in the Davey-Stewartson system,, \emph{Discr. Cont. Dyn. Syst. B}, 18 (2013), 1361. doi: 10.3934/dcdsb.2013.18.1361. Google Scholar [20] C. Klein and J.-C. Saut, A numerical approach to blow-up issues for Davey-Stewartson II type systems,, submitted., (). Google Scholar [21] C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations,, submitted., (). Google Scholar [22] Joseph Louis de Lagrange, Mémoire sur la théorie du mouvement des fluides,, Oeuvres compl\etes, (1781), 695. Google Scholar [23] D. Lannes, Water Waves : Mathematical Theory and Asymptotics,, Mathematical Surveys and Monographs, (2013). doi: 10.1090/surv/188. Google Scholar [24] David Lannes, A stability criterion for two-dimensional interfaces and applications,, \emph{Arch. Ration. Mech. Anal.}, 208 (2013), 481. doi: 10.1007/s00205-012-0604-6. Google Scholar [25] D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equation,, \emph{Kinematics and Related Models}, 6 (2013). doi: 10.3934/krm.2013.6.989. Google Scholar [26] F. Linares and G. Ponce, On the Davey-Stewartson systems,, \emph{Ann. Inst. H. Poincar\' e Anal. Non Lin\' eaire}, 10 (1993), 523. Google Scholar [27] C. Obrecht, In preparation., $\$, (). Google Scholar [28] C. Obrecht and K. Roidot, In preparation., $\$, (). Google Scholar [29] T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems,, \emph{Proc. Roy. Soc. London A}, 436 (1992), 345. doi: 10.1098/rspa.1992.0022. Google Scholar [30] G. C. Papanicolaou, C. Sulem, P.-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves,, \emph{Physica D}, 72 (1994), 61. doi: 10.1016/0167-2789(94)90167-8. Google Scholar [31] P. A. Perry, Global well-posedness and long time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}(\R^2)$,, arXiv:1110.5589v2., (). Google Scholar [32] G. Ponce and J.-C. Saut, Well-posedness for the Benney-Roskes-Zakharov- Rubenchik system,, \emph{Discrete Cont. Dynamical Systems}, 13 (2005), 811. doi: 10.3934/dcds.2005.13.811. Google Scholar [33] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation,, Springer-Verlag, 139 (1999). Google Scholar [34] L. Y. Sung, Long time decay of the solutions of the Davey-Stewartson II equations,, \emph{J. Nonlinear Sci.}, 5 (1995), 43. doi: 10.1007/BF01212909. Google Scholar [35] N. Totz, A justification of the modulation approximation to the 3D full water wave problem,, \emph{Comm. Math. Phys.}, 335 (2015), 369. doi: 10.1007/s00220-014-2259-7. Google Scholar [36] N. Totz and S. Wu, A rigorous justification of the modulation approximation to the 2D full water wave problem,, \emph{Comm. Math. Phys.}, 310 (2012), 817. doi: 10.1007/s00220-012-1422-2. Google Scholar [37] V. E. Zakharov, Weakly nonlinear waves on surface of ideal finite depth fluid,, \emph{Amer. Math. Soc. Transl. Ser. 2}, 182 (1998), 167. Google Scholar [38] V. E. Zakharov and A. M. Rubenchik, Nonlinear interaction of high-frequency and low frequency waves,, \emph{Prikl. Mat. Techn. Phys.}, (1972), 84. Google Scholar [39] V. E. Zakharov and E. I Schulman, Degenerate conservation laws, motion invariants and kinetic equations,, \emph{Physica}, 1D (1980), 192. doi: 10.1016/0167-2789(80)90011-1. Google Scholar
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