An extended Petviashvili method for the numerical generation of traveling and localized waves

•We analyze a family of fixed-point iterations to compute traveling waves.•The methods extend the Petviashvili iteration.•They can be used for nonlinear terms with homogeneities of different degree.•Isolated solitary waves and waves in systems with symmetries are generated.

A family of fixed-point iterations is proposed for the numerical computation of traveling waves and localized ground states. The methods are extended versions of Petviashvili type, and they are applicable when the nonlinear term of the system contains homogeneous functions of different degree. The methods are described and applied to several examples of interest, that calibrate their efficiency.