Journal of Computational Physics
A mode elimination technique to improve convergence of iteration methods for finding solitary waves
Journal of Computational Physics
Newton-conjugate-gradient methods for solitary wave computations
Journal of Computational Physics
On the uniform convergence of the Chebyshev interpolants for solitons
Mathematics and Computers in Simulation
Simulation of coherent structures in nonlinear Schrödinger-type equations
Journal of Computational Physics
A numerical scheme for periodic travelling-wave simulations in some nonlinear dispersive wave models
Journal of Computational and Applied Mathematics
Solitary Wave Benchmarks in Magma Dynamics
Journal of Scientific Computing
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We analyze a heuristic numerical method suggested by V. I. Petviashvili in 1976 for approximation of stationary solutions of nonlinear wave equations. The method is used to construct numerically the solitary wave solutions, such as solitons, lumps, and vortices, in a space of one and higher dimensions. Assuming that the stationary solution exists, we find conditions when the iteration method converges to the stationary solution and when the rate of convergence is the fastest. The theory is illustrated with examples of physical interest such as generalized Korteweg--de Vries, Benjamin--Ono, Zakharov--Kuznetsov, Kadomtsev--Petviashvili, and Klein--Gordon equations.