• Previous Article
    Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms
  • CPAA Home
  • This Issue
  • Next Article
    On the pointwise decay estimate for the wave equation with compactly supported forcing term
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

show all references

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

[1]

Christian Klein, Benson Muite, Kristelle Roidot. Numerical study of blow-up in the Davey-Stewartson system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1361-1387. doi: 10.3934/dcdsb.2013.18.1361

[2]

Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077

[3]

T. Colin, D. Lannes. Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 83-100. doi: 10.3934/dcds.2004.11.83

[4]

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

[5]

Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11

[6]

Jing Lu, Yifei Wu. Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1641-1670. doi: 10.3934/cpaa.2015.14.1641

[7]

Olivier Goubet, Manal Hussein. Global attractor for the Davey-Stewartson system on $\mathbb R^2$. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1555-1575. doi: 10.3934/cpaa.2009.8.1555

[8]

María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic & Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024

[9]

Christian Klein, Ralf Peter. Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1689-1717. doi: 10.3934/dcdsb.2014.19.1689

[10]

Uchida Hidetake. Analytic smoothing effect and global existence of small solutions for the elliptic-hyperbolic Davey-Stewartson system. Conference Publications, 2001, 2001 (Special) : 182-190. doi: 10.3934/proc.2001.2001.182

[11]

Zaihui Gan, Boling Guo, Jian Zhang. Sharp threshold of global existence for the generalized Davey-Stewartson system in $R^2$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 913-922. doi: 10.3934/cpaa.2009.8.913

[12]

Frank Merle, Hatem Zaag. O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 435-450. doi: 10.3934/dcds.2002.8.435

[13]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399

[14]

José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43

[15]

Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure & Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435

[16]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[17]

Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025

[18]

Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315

[19]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

[20]

Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]