Abstract. 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 ...

Deterministic and stochastic nonlinear partial differential equations with mixed features (e.g. partial hyperbolicity and anisotropicity) have played an important role in describing the models in physics, chemistry, finance and other real-world phenomena. Moreover, sitting at the interface between probability theory, mathematical analysis and theory of parabolic and hyperbolic partial different...

In this paper, a singularity structure analysis of the (2+1)-dimensional Zakharov equation is carried out and it is shown that it admits the Painleve property. The bilinear form of this equation is derived from the Painleve analysis in a straightforward manner. Using this bilinear form, we have constructed the simply one soliton solution by the Hirota method. We have then presented the localize...