An Implicit Finite Difference Scheme for Focusing Solutions of the Generalized Davey-stewartson System

where u and φ1, φ2 are, respectively, the complexand the real-valued functions of spatial coordinates x, y and the time t. The parameters σ, κ, γ,m1,m2, λ, n are real constants and σ is normalized as | σ |= 1. The GDS system has been derived to model 2 + 1 dimensional wave propagation in a bulk medium composed of an elastic material with couple stresses [1]. The parametric relation (λ − 1)(m2 −m1) = n follows from the structure of the physical constants and plays a key role in the analysis of these equations. The GDS system is classified according to the signs of parameters (σ,m1,m2, λ). In this study, we consider (+,+,+,+) elliptic-elliptic-elliptic (EEE) and (–,+,+,+) hyperbolic-elliptic-elliptic (HEE) cases. In this study, the generalized Davey-Stewartson (GDS) system is solved by a numerical method which is based on an extension of the relaxation method introduced in [2]. In the hyperbolicelliptic-elliptic case, we numerically test the relaxation method by using the analytical blow-up profile. In the elliptic-elliptic-elliptic case, we compare the numerical results with the analytical global existence and blow-up results [3] for certain ranges of parameters. Numerical tests show that the relaxation method does not miss the blow-up phenomena and provides accurate results for the GDS system. REFERENCES [1] Babaoglu, C., Erbay, S., 2004. Two dimensional wave packets in an elastic solid with couple stresses, Int. J. Nonlinear Mech. 39, 941-949. [2] Besse, C., Bruneau, C. H., 1998. Numerical study of elliptic-hyperbolic Davey-Stewartson system: Dromions simulation and blow-up, Math. Model Meth. Appl. Sci. 8, 1363-1386. [3] Eden, A., Erbay, H. A., Muslu, G. M., 2008. Closing the gap in the purely elliptic generalized Davey-Stewartson system, Non. Anal. TMA 69, 2575-2581.