##
D.E. Pelinovsky and Yu.A. Stepanyants

#
Convergence of Petviashvili's
iteration method for numerical approximation
of stationary solutions of nonlinear wave equations

###
SIAM J. Numer. Anal. 42, 1110-1127 (2004)

**Abstract:**

We analyze an euristic numerical method
[Petviashvili, 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.

**Keywords**:

NONLINEAR EVOLUTION EQUATIONS, SOLITARY WAVES, NUMERICAL APPROXIMATIONS,
ITERATION METHODS, CONVERGENCE AND STABILITY, LINEARIZED OPERATORS