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Numerical study of the Davey-Stewartson system

Published online by Cambridge University Press:  15 December 2004

Christophe Besse ,
Norbert J. Mauser  and
Hans Peter Stimming
Christophe Besse
Affiliation:
Laboratoire MIP, UMR 5640, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France. besse@mip.ups-tlse.fr.
Norbert J. Mauser
Affiliation:
Wolfgang Pauli Institute c/o Fakultät f. Math., Universität Wien, Nordbergstr. 15, A 1090 Wien, Austria. mauser@courant.nyu.edu.
Hans Peter Stimming
Affiliation:
Wolfgang Pauli Institute, Wien and ENS Lyon, France. hans.peter.stimming@univie.ac.at.
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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.

Keywords

Nonlinear Schrödinger type equationsurface wavetime-splitting spectral schemefinite time blowup.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 6 , November 2004 , pp. 1035 - 1054
Copyright
© EDP Sciences, SMAI, 2004

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References

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).
M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform. SIAM Stud. Appl. Math., SIAM, Philadelphia 4 (1981).
Arkadiev, V.A., Pogrebkov, A.K. and Polivanov, M.C., Inverse scattering transform method and soliton solutions for the Davey-Stewartson II equation. Physica D 36 (1989) 189–196. CrossRef
Bao, W., Jin, S. and Markowich, P.A., Time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comp. Phys. 175 (2002) 487–524. CrossRef
Bao, W., Mauser, N.J. and Stimming, H.P., Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger-Poisson-Xα model. CMS 1 (2003) 809–831.
Besse, C., 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. CrossRef
Besse, C. and Bruneau, C.H., Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up. Math. Mod. Meth. Appl. Sci. 8 (1998) 1363–1386. CrossRef
Besse, C., Bidégaray, B. and Descombes, S., Order estimates in time of the splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 26–40. CrossRef
Descombes, S., Convergence of a splitting method of high order for reaction-diffusion systems. Math. Comp. 70 (2001) 1481–1501. CrossRef
Djordjević, V.D. and Redekopp, L.G., On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79 (1977) 703–714. CrossRef
Ghidaglia, J.M. and Saut, J.C., On the initial value problem for the Davey-Stewartson systems. Nonlinearity 3 (1990) 475–506. CrossRef
Guzmán-Gomez, M., Asymptotic behaviour of the Davey-Stewartson system. C. R. Math. Rep. Acad. Sci. Canada 16 (1994) 91–96.
Hardin, R.H. and Tappert, F.D., Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations. SIAM Rev. Chronicle 15 (1973) 423.
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.
Hayashi, N. and Hirata, H., Global existence and asymptotic behaviour of small solutions to the elliptic-hyperbolic Davey-Stewartson system. Nonlinearity 9 (1996) 1387–1409. CrossRef
Hayashi, N. and Saut, J.C., Global existence of small solutions to the Davey-Stewartson and Ishimori systems. Diff. Int. Eq. 8 (1995) 1657–1675.
Landman, M.J., Papanicolaou, G.C., Sulem, C. and Sulem, P.-L., Rate of blowup for solutions of the Nonlinear Schrödinger equation at critical dimension. Phys. Rev. A 38 (1988) 3837–3843. CrossRef
Merle, F., Construction of solutions with exactly k blowup points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. 129 (1990) 223–240. CrossRef
Nishinari, K., Abe, K. and Satsuma, J., Multidimensional behaviour of an electrostatic ion wave in a magnetized plasma. Phys. Plasmas 1 (1994) 2559–2565. CrossRef
Ozawa, T., Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems. Proc. R. Soc. A 436 (1992) 345–349. CrossRef
Papanicolaou, G.C., Sulem, C., Sulem, P.-L., Wang, X.P., The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves. Physica D 72 (1994) 61–86. CrossRef
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999)
White, P.W. and Weideman, J.A.C., Numerical simulation of solitons and dromions in the Davey-Stewartson system. Math. Comput. Simul. 37 (1994) 469–479. CrossRef