Numerical simulation of solitons and dromions in the Davey-Stewartson system
Mathematics and Computers in Simulation - Special issue: solitons, nonlinear wave equations and computation
A fast spectral algorithm for nonlinear wave equations with linear dispersion
Journal of Computational Physics
Exponential time differencing for stiff systems
Journal of Computational Physics
A composite Runge-Kutta method for the spectral solution of semilinear PDEs
Journal of Computational Physics
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
Generalized integrating factor methods for stiff PDEs
Journal of Computational Physics
Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems
SIAM Journal on Numerical Analysis
Journal of Computational Physics
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Purely dispersive partial differential equations such as the Korteweg-de Vries equation, the nonlinear Schrödinger equation, and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow-up. To numerically study such phenomena, fourth order time-stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Kadomtsev-Petviashvili and the Davey-Stewartson equations, two integrable equations in $2+1$ dimensions: these methods are exponential time-differencing, integrating factors, time-splitting, implicit Runge-Kutta, and Driscoll's composite Runge-Kutta method. The accuracy in the numerical conservation of integrals of motion is discussed.