# Numerical study of the Davey-Stewartson system

Christophe Besse; Norbert J. Mauser; Hans Peter Stimming

- Volume: 38, Issue: 6, page 1035-1054
- ISSN: 0764-583X

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topBesse, Christophe, Mauser, Norbert J., and Stimming, Hans Peter. "Numerical study of the Davey-Stewartson system." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.6 (2004): 1035-1054. <http://eudml.org/doc/246034>.

@article{Besse2004,

abstract = {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.},

author = {Besse, Christophe, Mauser, Norbert J., Stimming, Hans Peter},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {nonlinear Schrödinger type equation; surface wave; time-splitting spectral scheme; finite time blowup; surface waves; time splitting spectral method; blow up},

language = {eng},

number = {6},

pages = {1035-1054},

publisher = {EDP-Sciences},

title = {Numerical study of the Davey-Stewartson system},

url = {http://eudml.org/doc/246034},

volume = {38},

year = {2004},

}

TY - JOUR

AU - Besse, Christophe

AU - Mauser, Norbert J.

AU - Stimming, Hans Peter

TI - Numerical study of the Davey-Stewartson system

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2004

PB - EDP-Sciences

VL - 38

IS - 6

SP - 1035

EP - 1054

AB - 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.

LA - eng

KW - nonlinear Schrödinger type equation; surface wave; time-splitting spectral scheme; finite time blowup; surface waves; time splitting spectral method; blow up

UR - http://eudml.org/doc/246034

ER -

## References

top- [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). Zbl0762.35001MR1149378
- [2] M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform. SIAM Stud. Appl. Math., SIAM, Philadelphia 4 (1981). Zbl0472.35002MR642018
- [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. Zbl0698.35150
- [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. Zbl1006.65112
- [5] W. Bao, N.J. Mauser and H.P. Stimming, Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger-Poisson-X$\alpha $ model. CMS 1 (2003) 809–831. Zbl1160.81497
- [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. Zbl0911.65072
- [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. Zbl0940.76044
- [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. Zbl1026.65073
- [9] S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems. Math. Comp. 70 (2001) 1481–1501. Zbl0981.65107
- [10] V.D. Djordjević and L.G. Redekopp, On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79 (1977) 703–714. Zbl0351.76016
- [11] J.M. Ghidaglia and J.C. Saut, On the initial value problem for the Davey-Stewartson systems. Nonlinearity 3 (1990) 475–506. Zbl0727.35111
- [12] M. Guzmán-Gomez, Asymptotic behaviour of the Davey-Stewartson system. C. R. Math. Rep. Acad. Sci. Canada 16 (1994) 91–96. Zbl0808.35139
- [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. Zbl0907.35120
- [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. Zbl0906.35096
- [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. Zbl0827.35120
- [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. Zbl0707.35021
- [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. Zbl0754.35114
- [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. Zbl0815.35104
- [22] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999) Zbl0928.35157MR1696311
- [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. Zbl0812.65119

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