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July  2015, 14(4): 1443-1467. doi: 10.3934/cpaa.2015.14.1443

## A numerical approach to Blow-up issues for Davey-Stewartson II systems

 1 Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078 Dijon Cedex 2 Laboratoire de Mathématiques, UMR 8628, Université Paris-Sud et CNRS, 91405 Orsay

Received  June 2014 Revised  July 2014 Published  April 2015

We provide a numerical study of various issues pertaining to the dynamics of the Davey-Stewartson II systems. In particular we investigate whether or not the properties (blow-up, radiation,...) displayed by the focusing and defocusing Davey-Stewartson II integrable systems persist in the non integrable cases.
Citation: Christian Klein, Jean-Claude Saut. A numerical approach to Blow-up issues for Davey-Stewartson II systems. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1443-1467. doi: 10.3934/cpaa.2015.14.1443
##### References:
 [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering,, London Mathematical Society Lecture Notes series, 149 (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] V. A. Arkadiev, A. K. Pogrebkov and M. C. Polivanov, Inverse scattering transform and soliton solution for Davey-Stewartson II equation,, \emph{Physica D}, 36 (1089), 188. doi: 10.1016/0167-2789(89)90258-3. Google Scholar [4] D. J. Benney and G. J. Roskes, Waves instabilities,, \emph{Stud. Appl. Math.}, 48 (1969), 377. Google Scholar [5] C. Besse and C. H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up,, \emph{Math. Mod. and Meth. in Appl. Sciences}, 8 (1998), 1363. doi: 10.1142/S0218202598000640. Google Scholar [6] C. Besse, N. Mauser and H.-P. Stimming, Numerical study of the Davey-Stewartson system,, \emph{M2AN Math. Model. Numer. Anal.}, 38 (2004), 1035. doi: 10.1051/m2an:2004049. Google Scholar [7] R. Carles, E. Dumas and C. Sparber, Geometric optics and instability for NLS and Davey-Stewartson systems,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1885. doi: 10.4171/JEMS/350. Google Scholar [8] T. Cazenave, and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case,, in \emph{Nonlinear Semigroups, (1987), 18. doi: 10.1007/BFb0086749. Google Scholar [9] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system,, \emph{Comm. Partial Differential Equations}, 17 (1992), 967. doi: 10.1080/03605309208820872. Google Scholar [10] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system,, \emph{Ann.Inst. H. Poincar\' e, 58 (1993), 85. Google Scholar [11] T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems,, \emph{Asymptotic Analysis}, 31 (2002), 69. Google Scholar [12] T. Colin and D. Lannes, Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems,, \emph{Disc. Cont. Dyn. Systems}, 11 (2004), 83. doi: 10.3934/dcds.2004.11.83. Google Scholar [13] A. Davey and K. Stewartson, One three-dimensional packets of water waves,, \emph{Proc. Roy. Soc. Lond. A}, 338 (1974), 101. Google Scholar [14] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves,, \emph{J. Fluid Mech.}, 79 (1977), 703. Google Scholar [15] T. Driscoll, A composite Runge-Kutta Method for the spectral Solution of semilinear PDEs,, \emph{Journal of Computational Physics}, 182 (2002), 357. doi: 10.1006/jcph.2002.7127. Google Scholar [16] A. Fokas, D. Pelinovsky and C. Sulem, Interaction of lumps with a line soliton for the Davey-Stewartson II equation,, \emph{Physica D}, 152-153 (2001), 152. doi: 10.1016/S0167-2789(01)00170-1. Google Scholar [17] J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems,, \emph{Nonlinearity}, 3 (1990), 475. Google Scholar [18] 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 [19] J.-M. Ghidaglia and J.-C. Saut, On the Zakharov-Schulman equations,, in \emph{Non-linear Dispersive Waves} (L. Debnath Ed.), (1992), 83. Google Scholar [20] J.-M. Ghidaglia and J.-C. Saut, Nonelliptic Schrödinger evolution equations,, \emph{J. Nonlinear Science}, 3 (1993), 169. doi: 10.1007/BF02429863. Google Scholar [21] N. Hayashi, Local existence in time of solutions to the elliptic-hyperbolic Davey-Stewartson system without smallness condition on the data,, \emph{J. Analyse Math\' ematique}, 73 (1997), 133. doi: 10.1007/BF02788141. Google Scholar [22] N. Hayashi and H. Hirota, 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 [23] N. Hayashi and H. Hirota, Global existence and asymptotic behavior in time 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 [24] P. Kevrekidis, A. R. Nahmod and C. Zeng, Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D,, \emph{Nonlinearity}, 24 (2011), 1523. doi: 10.1088/0951-7715/24/5/007. Google Scholar [25] O.M. Kiselev, Asymptotics of solutions of higher-dimensional integrable equations and their perturbations,, \emph{J. of Mathematical Sciences}, 138 (2006), 6067. doi: 10.1007/s10958-006-0347-8. Google Scholar [26] C. Klein and R. Peter, Numerical study of blow-up in solutions to generalized Korteweg-de Vries equations,, Preprint available at {\tt arXiv:1307.0603}, (). Google Scholar [27] C. Klein and R. Peter, Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations,, \emph{Discr. Cont. Dyn. Syst. B}, 19 (2014). doi: 10.3934/dcdsb.2014.19.1689. Google Scholar [28] C. Klein, C. Sparber and P. Markowich, Numerical study of fractional Nonlinear Schrödinger equations,, Preprint available at {\tt arXiv:1404.6262}, (). doi: 10.1098/rspa.2014.0364. Google Scholar [29] C. Klein and K. Roidot, Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations,, \emph{Phys. D}, 265 (2013), 1. doi: 10.1016/j.physd.2013.09.005. Google Scholar [30] C. Klein and K. Roidot, Numerical study of the semiclassical limit of the Davey-Stewartson II equations,, Prepint available at {\tt arXiv:1401.4745}., (). doi: 10.1088/0951-7715/27/9/2177. Google Scholar [31] C. Klein and K. Roidot, Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations,, \emph{SIAM Journal on Scientific Computing}, 33 (2011). doi: 10.1137/100816663. Google Scholar [32] C. Klein, B. Muite and K. Roidot, Numerical Study of blowup in the Davey-Stewartson system,, \emph{Discr. Cont. Dyn. Syst. B}, 18 (2013), 1361. doi: 10.3934/dcdsb.2013.18.1361. Google Scholar [33] C. Klein, Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation,, \emph{ETNA}, 29 (2008), 116. Google Scholar [34] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions,, \emph{SIAM J. Optimization}, 9 (1998), 112. doi: 10.1137/S1052623496303470. Google Scholar [35] D. Lannes, Water Waves: Mathematical Theory and Asymptotics,, Mathematical Surveys and Monographs, 188 (2013). doi: 10.1090/surv/188. Google Scholar [36] H. Leblond, Electromagnetic waves in ferromagnets,, \emph{J. Phys. A}, 32 (1999), 7907. doi: 10.1088/0305-4470/32/45/308. Google Scholar [37] 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 [38] M. McConnell, A. Fokas, and B. Pelloni, Localised coherent solutions of the DSI and DSII equations a numerical study,, \emph{Mathematics and Computers in Simulation}, 69 (2005), 424. doi: 10.1016/j.matcom.2005.03.007. Google Scholar [39] K. Roidot and N. Mauser, Numerical study of the transverse stability of NLS soliton solutions in several classes of NLS type equations,, preprint, (2014). Google Scholar [40] F. Merle and P. Raphaël, The blow-up dynamic and upper bound rate for critical nonlinear Schrödinger equation,, \emph{Ann. of Math}, 161 (2005), 157. doi: 10.4007/annals.2005.161.157. Google Scholar [41] S. L. Musher, A. M. Rubenchik and V. E. Zakharov, Hamiltonian approach to the description of nonlinear plasma phenomena,, \emph{Phys. Rep.}, 129 (1985), 285. doi: 10.1016/0370-1573(85)90040-7. Google Scholar [42] A. Newell and J. V. Moloney, Nonlinear Optics,, Addison-Wesley, (1992). Google Scholar [43] M. Ohta, Stability and instability of standing waves for the generalized Davey-Stewartson system,, \emph{Diff. Int. Eq.}, 8 (1995), 1775. Google Scholar [44] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system,, \emph{Ann. Inst. H. Poincar\' e, 62 (1995), 69. Google Scholar [45] M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system,, \emph{Ann. Inst. H. Poincar\' e, 63 (1995), 111. Google Scholar [46] 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 [47] G. Papanicolaou, C. Sulem, P.-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary waves,, \emph{Physica D}, 72 (1994), 61. doi: 10.1016/0167-2789(94)90167-8. Google Scholar [48] D. Pelinovsky and C. Sulem, Embedded solitons of the Davey-Stewartson II equation,, in \emph{CRM Proceedings and Lecture Notes} (eds. C. Sulem and I. M. Sigal), 27 (2001), 135. Google Scholar [49] D. E. Pelinovsky, E. A. Rouvinskaya, O. E. Kurkina and B. Deconincks, Short-wave transverse instability of line solitons of the two-dimensional hyperbolic nonlinear Schrödinger equation,, \emph{Theoretical and Mathematical Physics}, 179 (2014), 452. Google Scholar [50] P. A. Perry, Global well-posedness and long time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}(\R^2)$,, preprint, (2012). Google Scholar [51] F. Rousset and N. Tzvetkov, Transverse nonlinear instability for some Hamiltonian PDE's,, \emph{J. Math. Pures Appl.}, 90 (2008), 550. doi: 10.1016/j.matpur.2008.07.004. Google Scholar [52] F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models,, \emph{Ann. IHP, 26 (2009), 477. doi: 10.1016/j.anihpc.2007.09.006. Google Scholar [53] F. Rousset and N. Tzvetkov, A simple criterion of transverse linear instability for solitary waves,, \emph{Math. Res. Lett.}, 17 (2010), 157. doi: 10.4310/MRL.2010.v17.n1.a12. Google Scholar [54] E. I. Schulman, On the integrability of equations of Davey-Stewartson type,, \emph{Theor. Math. Phys.}, 56 (1983), 131. Google Scholar [55] C. Sulem, P.-L. Sulem and H. Frisch, Tracing complex singularities with spectral methods,, \emph{J. Comp. Phys.}, 50 (1983), 138. doi: 10.1016/0021-9991(83)90045-1. Google Scholar [56] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse,, Springer Series in Mathematical Sciences Vol. 139, (1999). Google Scholar [57] L. Y. Sung, An inverse scattering transform for the Davey-Stewartson equations. I,, \emph{J. Math. Anal. Appl.}, 183 (1994), 121. doi: 10.1006/jmaa.1994.1136. Google Scholar [58] L. Y. Sung, An inverse scattering transform for the Davey-Stewartson equations. II,, \emph{J. Math. Anal. Appl.}, 183 (1994), 289. doi: 10.1006/jmaa.1994.1145. Google Scholar [59] L. Y. Sung, An inverse scattering transform for the Davey-Stewartson equations. III,, \emph{J. Math. Anal. Appl.}, 183 (1994), 477. doi: 10.1006/jmaa.1994.1155. Google Scholar [60] L.-Y. Sung, Long-time decay of the solutions of the Davey-Stewartson II equations,, \emph{J. Nonlinear Sci.}, 5 (1995), 433. doi: 10.1007/BF01212909. Google Scholar [61] V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, \emph{J. Appl. Mech. Tech. Phys.}, 2 (1968), 190. Google Scholar [62] 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 [63] V. E. Zakharov and E. I. Schulman, Degenerate dispersion laws, motion invariants and kinetic equations,, \emph{Physica}, 1D (1980), 192. doi: 10.1016/0167-2789(80)90011-1. Google Scholar [64] V. E. Zakharov and E. I. Schulman, Integrability of nonlinear systems and perturbation theory,, in \emph{IWhat is Integrability}? (V. E. Zakharov, (1991), 185. Google Scholar

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##### References:
 [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering,, London Mathematical Society Lecture Notes series, 149 (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] V. A. Arkadiev, A. K. Pogrebkov and M. C. Polivanov, Inverse scattering transform and soliton solution for Davey-Stewartson II equation,, \emph{Physica D}, 36 (1089), 188. doi: 10.1016/0167-2789(89)90258-3. Google Scholar [4] D. J. Benney and G. J. Roskes, Waves instabilities,, \emph{Stud. Appl. Math.}, 48 (1969), 377. Google Scholar [5] C. Besse and C. H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up,, \emph{Math. Mod. and Meth. in Appl. Sciences}, 8 (1998), 1363. doi: 10.1142/S0218202598000640. Google Scholar [6] C. Besse, N. Mauser and H.-P. Stimming, Numerical study of the Davey-Stewartson system,, \emph{M2AN Math. Model. Numer. Anal.}, 38 (2004), 1035. doi: 10.1051/m2an:2004049. Google Scholar [7] R. Carles, E. Dumas and C. Sparber, Geometric optics and instability for NLS and Davey-Stewartson systems,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1885. doi: 10.4171/JEMS/350. Google Scholar [8] T. Cazenave, and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case,, in \emph{Nonlinear Semigroups, (1987), 18. doi: 10.1007/BFb0086749. Google Scholar [9] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system,, \emph{Comm. Partial Differential Equations}, 17 (1992), 967. doi: 10.1080/03605309208820872. Google Scholar [10] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system,, \emph{Ann.Inst. H. Poincar\' e, 58 (1993), 85. Google Scholar [11] T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems,, \emph{Asymptotic Analysis}, 31 (2002), 69. Google Scholar [12] T. Colin and D. Lannes, Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems,, \emph{Disc. Cont. Dyn. Systems}, 11 (2004), 83. doi: 10.3934/dcds.2004.11.83. Google Scholar [13] A. Davey and K. Stewartson, One three-dimensional packets of water waves,, \emph{Proc. Roy. Soc. Lond. A}, 338 (1974), 101. Google Scholar [14] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves,, \emph{J. Fluid Mech.}, 79 (1977), 703. Google Scholar [15] T. Driscoll, A composite Runge-Kutta Method for the spectral Solution of semilinear PDEs,, \emph{Journal of Computational Physics}, 182 (2002), 357. doi: 10.1006/jcph.2002.7127. Google Scholar [16] A. Fokas, D. Pelinovsky and C. Sulem, Interaction of lumps with a line soliton for the Davey-Stewartson II equation,, \emph{Physica D}, 152-153 (2001), 152. doi: 10.1016/S0167-2789(01)00170-1. Google Scholar [17] J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems,, \emph{Nonlinearity}, 3 (1990), 475. Google Scholar [18] 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 [19] J.-M. Ghidaglia and J.-C. Saut, On the Zakharov-Schulman equations,, in \emph{Non-linear Dispersive Waves} (L. Debnath Ed.), (1992), 83. Google Scholar [20] J.-M. Ghidaglia and J.-C. Saut, Nonelliptic Schrödinger evolution equations,, \emph{J. Nonlinear Science}, 3 (1993), 169. doi: 10.1007/BF02429863. Google Scholar [21] N. Hayashi, Local existence in time of solutions to the elliptic-hyperbolic Davey-Stewartson system without smallness condition on the data,, \emph{J. Analyse Math\' ematique}, 73 (1997), 133. doi: 10.1007/BF02788141. Google Scholar [22] N. Hayashi and H. Hirota, 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 [23] N. Hayashi and H. Hirota, Global existence and asymptotic behavior in time 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 [24] P. Kevrekidis, A. R. Nahmod and C. Zeng, Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D,, \emph{Nonlinearity}, 24 (2011), 1523. doi: 10.1088/0951-7715/24/5/007. Google Scholar [25] O.M. Kiselev, Asymptotics of solutions of higher-dimensional integrable equations and their perturbations,, \emph{J. of Mathematical Sciences}, 138 (2006), 6067. doi: 10.1007/s10958-006-0347-8. Google Scholar [26] C. Klein and R. Peter, Numerical study of blow-up in solutions to generalized Korteweg-de Vries equations,, Preprint available at {\tt arXiv:1307.0603}, (). Google Scholar [27] C. Klein and R. Peter, Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations,, \emph{Discr. Cont. Dyn. Syst. B}, 19 (2014). doi: 10.3934/dcdsb.2014.19.1689. Google Scholar [28] C. Klein, C. Sparber and P. Markowich, Numerical study of fractional Nonlinear Schrödinger equations,, Preprint available at {\tt arXiv:1404.6262}, (). doi: 10.1098/rspa.2014.0364. Google Scholar [29] C. Klein and K. Roidot, Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations,, \emph{Phys. D}, 265 (2013), 1. doi: 10.1016/j.physd.2013.09.005. Google Scholar [30] C. Klein and K. Roidot, Numerical study of the semiclassical limit of the Davey-Stewartson II equations,, Prepint available at {\tt arXiv:1401.4745}., (). doi: 10.1088/0951-7715/27/9/2177. Google Scholar [31] C. Klein and K. Roidot, Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations,, \emph{SIAM Journal on Scientific Computing}, 33 (2011). doi: 10.1137/100816663. Google Scholar [32] C. Klein, B. Muite and K. Roidot, Numerical Study of blowup in the Davey-Stewartson system,, \emph{Discr. Cont. Dyn. Syst. B}, 18 (2013), 1361. doi: 10.3934/dcdsb.2013.18.1361. Google Scholar [33] C. Klein, Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation,, \emph{ETNA}, 29 (2008), 116. Google Scholar [34] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions,, \emph{SIAM J. Optimization}, 9 (1998), 112. doi: 10.1137/S1052623496303470. Google Scholar [35] D. Lannes, Water Waves: Mathematical Theory and Asymptotics,, Mathematical Surveys and Monographs, 188 (2013). doi: 10.1090/surv/188. Google Scholar [36] H. Leblond, Electromagnetic waves in ferromagnets,, \emph{J. Phys. A}, 32 (1999), 7907. doi: 10.1088/0305-4470/32/45/308. Google Scholar [37] 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 [38] M. McConnell, A. Fokas, and B. Pelloni, Localised coherent solutions of the DSI and DSII equations a numerical study,, \emph{Mathematics and Computers in Simulation}, 69 (2005), 424. doi: 10.1016/j.matcom.2005.03.007. Google Scholar [39] K. Roidot and N. Mauser, Numerical study of the transverse stability of NLS soliton solutions in several classes of NLS type equations,, preprint, (2014). Google Scholar [40] F. Merle and P. Raphaël, The blow-up dynamic and upper bound rate for critical nonlinear Schrödinger equation,, \emph{Ann. of Math}, 161 (2005), 157. doi: 10.4007/annals.2005.161.157. Google Scholar [41] S. L. Musher, A. M. Rubenchik and V. E. Zakharov, Hamiltonian approach to the description of nonlinear plasma phenomena,, \emph{Phys. Rep.}, 129 (1985), 285. doi: 10.1016/0370-1573(85)90040-7. Google Scholar [42] A. Newell and J. V. Moloney, Nonlinear Optics,, Addison-Wesley, (1992). Google Scholar [43] M. Ohta, Stability and instability of standing waves for the generalized Davey-Stewartson system,, \emph{Diff. Int. Eq.}, 8 (1995), 1775. Google Scholar [44] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system,, \emph{Ann. Inst. H. Poincar\' e, 62 (1995), 69. Google Scholar [45] M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system,, \emph{Ann. Inst. H. Poincar\' e, 63 (1995), 111. Google Scholar [46] 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 [47] G. Papanicolaou, C. Sulem, P.-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary waves,, \emph{Physica D}, 72 (1994), 61. doi: 10.1016/0167-2789(94)90167-8. Google Scholar [48] D. Pelinovsky and C. Sulem, Embedded solitons of the Davey-Stewartson II equation,, in \emph{CRM Proceedings and Lecture Notes} (eds. C. Sulem and I. M. Sigal), 27 (2001), 135. Google Scholar [49] D. E. Pelinovsky, E. A. Rouvinskaya, O. E. Kurkina and B. Deconincks, Short-wave transverse instability of line solitons of the two-dimensional hyperbolic nonlinear Schrödinger equation,, \emph{Theoretical and Mathematical Physics}, 179 (2014), 452. Google Scholar [50] P. A. Perry, Global well-posedness and long time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}(\R^2)$,, preprint, (2012). Google Scholar [51] F. Rousset and N. Tzvetkov, Transverse nonlinear instability for some Hamiltonian PDE's,, \emph{J. Math. Pures Appl.}, 90 (2008), 550. doi: 10.1016/j.matpur.2008.07.004. Google Scholar [52] F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models,, \emph{Ann. IHP, 26 (2009), 477. doi: 10.1016/j.anihpc.2007.09.006. Google Scholar [53] F. Rousset and N. Tzvetkov, A simple criterion of transverse linear instability for solitary waves,, \emph{Math. Res. Lett.}, 17 (2010), 157. doi: 10.4310/MRL.2010.v17.n1.a12. Google Scholar [54] E. I. Schulman, On the integrability of equations of Davey-Stewartson type,, \emph{Theor. Math. Phys.}, 56 (1983), 131. Google Scholar [55] C. Sulem, P.-L. Sulem and H. Frisch, Tracing complex singularities with spectral methods,, \emph{J. Comp. Phys.}, 50 (1983), 138. doi: 10.1016/0021-9991(83)90045-1. Google Scholar [56] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse,, Springer Series in Mathematical Sciences Vol. 139, (1999). Google Scholar [57] L. Y. Sung, An inverse scattering transform for the Davey-Stewartson equations. I,, \emph{J. Math. Anal. Appl.}, 183 (1994), 121. doi: 10.1006/jmaa.1994.1136. Google Scholar [58] L. Y. Sung, An inverse scattering transform for the Davey-Stewartson equations. II,, \emph{J. Math. Anal. Appl.}, 183 (1994), 289. doi: 10.1006/jmaa.1994.1145. Google Scholar [59] L. Y. Sung, An inverse scattering transform for the Davey-Stewartson equations. III,, \emph{J. Math. Anal. Appl.}, 183 (1994), 477. doi: 10.1006/jmaa.1994.1155. Google Scholar [60] L.-Y. Sung, Long-time decay of the solutions of the Davey-Stewartson II equations,, \emph{J. Nonlinear Sci.}, 5 (1995), 433. doi: 10.1007/BF01212909. Google Scholar [61] V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, \emph{J. Appl. Mech. Tech. Phys.}, 2 (1968), 190. Google Scholar [62] 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 [63] V. E. Zakharov and E. I. Schulman, Degenerate dispersion laws, motion invariants and kinetic equations,, \emph{Physica}, 1D (1980), 192. doi: 10.1016/0167-2789(80)90011-1. Google Scholar [64] V. E. Zakharov and E. I. Schulman, Integrability of nonlinear systems and perturbation theory,, in \emph{IWhat is Integrability}? (V. E. Zakharov, (1991), 185. Google Scholar
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