Numerical study of the Davey-Stewartson system

Abstract

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.Mathematics Subject Classification. 35Q55, 65M12, 65M70, 76B45.

Modulational instability and nonlinear evolution of two-dimensional electrostatic wave packets in ultra-relativistic degenerate dense plasmas

“…Furthermore, the 2D evolution of the nonlocal NLSEs exhibit dromion-like solutions which either decay by the wave dispersion or blowup due to wave nonlinearity in a finite interval of time. The latter are in qualitative agreement with the results 20,21 already found in DS II equations for water waves with finite depth 13 .…”

“…dromion-like solutions (unlike solitons the dromions can have inelastic collisions and can transfer mass or energy) which may either decay due to the dispersion to be enhanced by the static field or exhibit blowup due to nonlinearity, in a finite time 20,21 . However, their applications in physical systems, especially in plasmas are not yet fully understood or less developed till now.…”

Mathematical models for laser-plasma interaction

“…Strictly speaking, we have to consider this wave equation coupled to (11). If ν ei was zero, then we would obtain a simple expression for the right hand side.…”

“…To do this, we assume that ν ei ω 0 and that the field E r is a rapidly oscillating function at the pulsation ω 0 . According to (11), a crude approximation of J r is…”

“…For the introduction of the paraxial approximation with a non zero incident angle see [20]. This paraxial approximation is also used for modelling other physical phenomena such as the propagation of submarine acoustic waves -the corresponding equation is known as parabolic equation-(see [33,36,45]); or the propagation of free surface waves subject to gravity and capillarity (see [11]). …”