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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, 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, J. Fluid Mech., 92 (1979), 691-715. 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, Physica D, 36 (1089), 188-197. doi: 10.1016/0167-2789(89)90258-3.  Google Scholar

[4]

D. J. Benney and G. J. Roskes, Waves instabilities, Stud. Appl. Math., 48 (1969), 377-385. Google Scholar

[5]

C. Besse and C. H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up, Math. Mod. and Meth. in Appl. Sciences, 8 (1998), 1363-1386. doi: 10.1142/S0218202598000640.  Google Scholar

[6]

C. Besse, N. Mauser and H.-P. Stimming, Numerical study of the Davey-Stewartson system, M2AN Math. Model. Numer. Anal., 38 (2004), 1035-1054. 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, J. Eur. Math. Soc., 14 (2012), 1885-1921. 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 Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, DC, 1987), 18-29, Lecture Notes in Math., 1394, Springer, Berlin, 1989. doi: 10.1007/BFb0086749.  Google Scholar

[9]

R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Partial Differential Equations, 17 (1992), 967-988. doi: 10.1080/03605309208820872.  Google Scholar

[10]

R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann.Inst. H. Poincaré, Phys. Théor., 58 (1993), 85-104.  Google Scholar

[11]

T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, Asymptotic Analysis, 31 (2002), 69-91.  Google Scholar

[12]

T. Colin and D. Lannes, Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems, Disc. Cont. Dyn. Systems, 11 (2004), 83-100. doi: 10.3934/dcds.2004.11.83.  Google Scholar

[13]

A. Davey and K. Stewartson, One three-dimensional packets of water waves, Proc. Roy. Soc. Lond. A, 338 (1974), 101-110.  Google Scholar

[14]

V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714.  Google Scholar

[15]

T. Driscoll, A composite Runge-Kutta Method for the spectral Solution of semilinear PDEs, Journal of Computational Physics, 182 (2002), 357-367. 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, Physica D, 152-153 (2001), 189-198. 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, Nonlinearity, 3 (1990), 475-506.  Google Scholar

[18]

J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci., 6 (1996), 139-145. doi: 10.1007/s003329900006.  Google Scholar

[19]

J.-M. Ghidaglia and J.-C. Saut, On the Zakharov-Schulman equations, in Non-linear Dispersive Waves (L. Debnath Ed.), World Scientific, (1992), 83-97.  Google Scholar

[20]

J.-M. Ghidaglia and J.-C. Saut, Nonelliptic Schrödinger evolution equations, J. Nonlinear Science, 3 (1993), 169-195. 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, J. Analyse Mathématique, 73 (1997), 133-164. 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, Proc. Edinburgh Math. Soc., 40 (1997), 563-581. 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, Nonlinearity, 9 (1996), 1387-1409. 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, Nonlinearity, 24 (2011), 1523-1538. 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, J. of Mathematical Sciences, 138 (2006), 6067-6230. 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, Discr. Cont. Dyn. Syst. B, 19 (2014), doi:10.3934/dcdsb.2014.19.1689 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, Phys. D, 265 (2013), 1-25. 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, 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, Discr. Cont. Dyn. Syst. B, 18 (2013), 1361-1387. 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, ETNA, 29 (2008), 116-135.  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, SIAM J. Optimization, 9 (1998), 112-147. doi: 10.1137/S1052623496303470.  Google Scholar

[35]

D. Lannes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, volume 188, 2013, AMS, Providence. doi: 10.1090/surv/188.  Google Scholar

[36]

H. Leblond, Electromagnetic waves in ferromagnets, J. Phys. A, 32 (1999), 7907-7932. doi: 10.1088/0305-4470/32/45/308.  Google Scholar

[37]

F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 523-548.  Google Scholar

[38]

M. McConnell, A. Fokas, and B. Pelloni, Localised coherent solutions of the DSI and DSII equations a numerical study, Mathematics and Computers in Simulation, 69 (2005), 424-438. 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, arXiv:1401.5349v1 (2014). Google Scholar

[40]

F. Merle and P. Raphaël, The blow-up dynamic and upper bound rate for critical nonlinear Schrödinger equation, Ann. of Math, 161 (2005), 157-222. 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, Phys. Rep., 129 (1985), 285-366. 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, Diff. Int. Eq., 8 (1995), 1775-1788.  Google Scholar

[44]

M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. H. Poincaré, Phys. Théor., 62 (1995), 69-80.  Google Scholar

[45]

M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. H. Poincaré, Phys. Théor., 63 (1995), 111-117.  Google Scholar

[46]

T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems, Proc.Roy. Soc. London A, 436 (1992), 345-349. 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, Physica D, 72 (1994), 61-86. doi: 10.1016/0167-2789(94)90167-8.  Google Scholar

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D. Pelinovsky and C. Sulem, Embedded solitons of the Davey-Stewartson II equation, in CRM Proceedings and Lecture Notes (eds. C. Sulem and I. M. Sigal), Volume 27 (2001), 135-145.  Google Scholar

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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, Theoretical and Mathematical Physics, 179 (2014), 452-461. Google Scholar

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P. A. Perry, Global well-posedness and long time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}(\R^2)$, preprint, arXiv:1110.5589v2, (2012). Google Scholar

[51]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability for some Hamiltonian PDE's, J. Math. Pures Appl., 90 (2008), 550-590. doi: 10.1016/j.matpur.2008.07.004.  Google Scholar

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F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. IHP, Analyse Non Linéaire, 26 (2009), 477-496. doi: 10.1016/j.anihpc.2007.09.006.  Google Scholar

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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, 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, J. Fluid Mech., 92 (1979), 691-715. 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, Physica D, 36 (1089), 188-197. doi: 10.1016/0167-2789(89)90258-3.  Google Scholar

[4]

D. J. Benney and G. J. Roskes, Waves instabilities, Stud. Appl. Math., 48 (1969), 377-385. Google Scholar

[5]

C. Besse and C. H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up, Math. Mod. and Meth. in Appl. Sciences, 8 (1998), 1363-1386. doi: 10.1142/S0218202598000640.  Google Scholar

[6]

C. Besse, N. Mauser and H.-P. Stimming, Numerical study of the Davey-Stewartson system, M2AN Math. Model. Numer. Anal., 38 (2004), 1035-1054. 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, J. Eur. Math. Soc., 14 (2012), 1885-1921. 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 Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, DC, 1987), 18-29, Lecture Notes in Math., 1394, Springer, Berlin, 1989. doi: 10.1007/BFb0086749.  Google Scholar

[9]

R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Partial Differential Equations, 17 (1992), 967-988. doi: 10.1080/03605309208820872.  Google Scholar

[10]

R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann.Inst. H. Poincaré, Phys. Théor., 58 (1993), 85-104.  Google Scholar

[11]

T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, Asymptotic Analysis, 31 (2002), 69-91.  Google Scholar

[12]

T. Colin and D. Lannes, Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems, Disc. Cont. Dyn. Systems, 11 (2004), 83-100. doi: 10.3934/dcds.2004.11.83.  Google Scholar

[13]

A. Davey and K. Stewartson, One three-dimensional packets of water waves, Proc. Roy. Soc. Lond. A, 338 (1974), 101-110.  Google Scholar

[14]

V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714.  Google Scholar

[15]

T. Driscoll, A composite Runge-Kutta Method for the spectral Solution of semilinear PDEs, Journal of Computational Physics, 182 (2002), 357-367. 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, Physica D, 152-153 (2001), 189-198. 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, Nonlinearity, 3 (1990), 475-506.  Google Scholar

[18]

J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci., 6 (1996), 139-145. doi: 10.1007/s003329900006.  Google Scholar

[19]

J.-M. Ghidaglia and J.-C. Saut, On the Zakharov-Schulman equations, in Non-linear Dispersive Waves (L. Debnath Ed.), World Scientific, (1992), 83-97.  Google Scholar

[20]

J.-M. Ghidaglia and J.-C. Saut, Nonelliptic Schrödinger evolution equations, J. Nonlinear Science, 3 (1993), 169-195. 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, J. Analyse Mathématique, 73 (1997), 133-164. 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, Proc. Edinburgh Math. Soc., 40 (1997), 563-581. 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, Nonlinearity, 9 (1996), 1387-1409. 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, Nonlinearity, 24 (2011), 1523-1538. 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, J. of Mathematical Sciences, 138 (2006), 6067-6230. 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, Discr. Cont. Dyn. Syst. B, 19 (2014), doi:10.3934/dcdsb.2014.19.1689 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, Phys. D, 265 (2013), 1-25. 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, 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, Discr. Cont. Dyn. Syst. B, 18 (2013), 1361-1387. 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, ETNA, 29 (2008), 116-135.  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, SIAM J. Optimization, 9 (1998), 112-147. doi: 10.1137/S1052623496303470.  Google Scholar

[35]

D. Lannes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, volume 188, 2013, AMS, Providence. doi: 10.1090/surv/188.  Google Scholar

[36]

H. Leblond, Electromagnetic waves in ferromagnets, J. Phys. A, 32 (1999), 7907-7932. doi: 10.1088/0305-4470/32/45/308.  Google Scholar

[37]

F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 523-548.  Google Scholar

[38]

M. McConnell, A. Fokas, and B. Pelloni, Localised coherent solutions of the DSI and DSII equations a numerical study, Mathematics and Computers in Simulation, 69 (2005), 424-438. 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, arXiv:1401.5349v1 (2014). Google Scholar

[40]

F. Merle and P. Raphaël, The blow-up dynamic and upper bound rate for critical nonlinear Schrödinger equation, Ann. of Math, 161 (2005), 157-222. 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, Phys. Rep., 129 (1985), 285-366. 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, Diff. Int. Eq., 8 (1995), 1775-1788.  Google Scholar

[44]

M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. H. Poincaré, Phys. Théor., 62 (1995), 69-80.  Google Scholar

[45]

M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. H. Poincaré, Phys. Théor., 63 (1995), 111-117.  Google Scholar

[46]

T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems, Proc.Roy. Soc. London A, 436 (1992), 345-349. 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, Physica D, 72 (1994), 61-86. 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 CRM Proceedings and Lecture Notes (eds. C. Sulem and I. M. Sigal), Volume 27 (2001), 135-145.  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, Theoretical and Mathematical Physics, 179 (2014), 452-461. 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, arXiv:1110.5589v2, (2012). Google Scholar

[51]

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