  Three new iteration methods, namely the squared-operator method, the modified
squared-operator method, and the power-conserving squared-operator method, for
solitary waves in general scalar and vector nonlinear wave equations are
proposed. These methods are based on iterating new differential equations whose
linearization operators are squares of those for the original equations,
together with acceleration techniques. The first two methods keep the
propagation constants fixed, while the third method keeps the powers (or other
arbitrary functionals) of the solution fixed. It is proved that all these
methods are guaranteed to converge to any solitary wave (either ground state or
not) as long as the initial condition is sufficiently close to the
corresponding exact solution, and the time step in the iteration schemes is
below a certain threshold value. Furthermore, these schemes are
fast-converging, highly accurate, and easy to implement. If the solitary wave
exists only at isolated propagation constant values, the corresponding
squared-operator methods are developed as well. These methods are applied to
various solitary wave problems of physical interest, such as higher-gap vortex
solitons in the two-dimensional nonlinear Schr\"odinger equations with periodic
potentials, and isolated solitons in Ginzburg-Landau equations, and some new
types of solitary wave solutions are obtained. It is also demonstrated that the
modified squared-operator method delivers the best performance among the
methods proposed in this article.
