Numerical study of the Davey-Stewartson system
Besse, Christophe ; Mauser, Norbert J. ; Stimming, Hans Peter
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004), p. 1035-1054 / Harvested from Numdam
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We deal with numerical analysis and simulations of the Davey-Stewartson equations which model, for example, the evolution of water surface waves. This time dependent PDE system is particularly interesting as a generalization of the 1-d integrable NLS to 2 space dimensions. We use a time splitting spectral method where we give a convergence analysis for the semi-discrete version of the scheme. Numerical results are presented for various blow-up phenomena of the equation, including blowup of defocusing, elliptic-elliptic Davey-Stewartson systems and simultaneous blowup at multiple locations in the focusing elliptic-elliptic system. Also the modeling of exact soliton type solutions for the hyperbolic-elliptic (DS2) system is studied.

Publié le : 2004-01-01
DOI : https://doi.org/10.1051/m2an:2004049
Classification:  35Q55,  65M12,  65M70,  76B45 
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     author = {Besse, Christophe and Mauser, Norbert J. and Stimming, Hans Peter},
     title = {Numerical study of the Davey-Stewartson system},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {38},
     year = {2004},
     pages = {1035-1054},
     doi = {10.1051/m2an:2004049},
     mrnumber = {2108943},
     zbl = {1080.65095},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2004__38_6_1035_0}
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Besse, Christophe; Mauser, Norbert J.; Stimming, Hans Peter. Numerical study of the Davey-Stewartson system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 1035-1054. doi : 10.1051/m2an:2004049. http://gdmltest.u-ga.fr/item/M2AN_2004__38_6_1035_0/

[1] M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, London Math. Soc. Lect. Note Series 149 (1991). | MR 1149378 | Zbl 0762.35001

[2] M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform. SIAM Stud. Appl. Math., SIAM, Philadelphia 4 (1981). | MR 642018 | Zbl 0472.35002

[3] V.A. Arkadiev, A.K. Pogrebkov and M.C. Polivanov, Inverse scattering transform method and soliton solutions for the Davey-Stewartson II equation. Physica D 36 (1989) 189-196. | Zbl 0698.35150

[4] W. Bao, S. Jin and P.A. Markowich, Time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comp. Phys. 175 (2002) 487-524. | Zbl 1006.65112

[5] W. Bao, N.J. Mauser and H.P. Stimming, Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger-Poisson-Xα model. CMS 1 (2003) 809-831. | Zbl pre02247546

[6] C. Besse, Schéma de relaxation pour l'équation de Schrödinger non linéaire et les systèmes de Davey et Stewartson. C. R. Acad. Sci. Paris I 326 (1998) 1427-1432. | Zbl 0911.65072

[7] C. Besse and C.H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up. Math. Mod. Meth. Appl. Sci. 8 (1998) 1363-1386. | Zbl 0940.76044

[8] C. Besse, B. Bidégaray and S. Descombes, Order estimates in time of the splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 26-40. | Zbl 1026.65073

[9] S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems. Math. Comp. 70 (2001) 1481-1501. | Zbl 0981.65107

[10] V.D. Djordjević and L.G. Redekopp, On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79 (1977) 703-714. | Zbl 0351.76016

[11] J.M. Ghidaglia and J.C. Saut, On the initial value problem for the Davey-Stewartson systems. Nonlinearity 3 (1990) 475-506. | Zbl 0727.35111

[12] M. Guzmán-Gomez, Asymptotic behaviour of the Davey-Stewartson system. C. R. Math. Rep. Acad. Sci. Canada 16 (1994) 91-96. | Zbl 0808.35139

[13] R.H. Hardin and F.D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations. SIAM Rev. Chronicle 15 (1973) 423.

[14] N. Hayashi, Local existence in time solutions to the elliptic-hyperbolic Davey-Stewartson system without smallness condition on the data. J. Anal. Math. LXXIII (1997) 133-164. | Zbl 0907.35120

[15] N. Hayashi and H. Hirata, Global existence and asymptotic behaviour of small solutions to the elliptic-hyperbolic Davey-Stewartson system. Nonlinearity 9 (1996) 1387-1409. | Zbl 0906.35096

[16] N. Hayashi and J.C. Saut, Global existence of small solutions to the Davey-Stewartson and Ishimori systems. Diff. Int. Eq. 8 (1995) 1657-1675. | Zbl 0827.35120

[17] M.J. Landman, G.C. Papanicolaou, C. Sulem and P.-L. Sulem, Rate of blowup for solutions of the Nonlinear Schrödinger equation at critical dimension. Phys. Rev. A 38 (1988) 3837-3843.

[18] F. Merle, Construction of solutions with exactly k blowup points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. 129 (1990) 223-240. | Zbl 0707.35021

[19] K. Nishinari, K. Abe and J. Satsuma, Multidimensional behaviour of an electrostatic ion wave in a magnetized plasma. Phys. Plasmas 1 (1994) 2559-2565.

[20] T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems. Proc. R. Soc. A 436 (1992) 345-349. | Zbl 0754.35114

[21] G.C. Papanicolaou, C. Sulem, P.-L. Sulem, X.P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves. Physica D 72 (1994) 61-86. | Zbl 0815.35104

[22] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999) | MR 1696311 | Zbl 0928.35157

[23] P.W. White and J.A.C. Weideman, Numerical simulation of solitons and dromions in the Davey-Stewartson system. Math. Comput. Simul. 37 (1994) 469-479. | Zbl 0812.65119