Phase Space Analysis for the Direct Algebraic Method for Nonlinear Evolution and Wave Equations

The basis of the direct algebraic method employing a series in real exponential functions for solving nonlinear evolution and wave equations is examined in terms of the general theory of autonomous ordinary differential equations (ODEs) and phase space analysis. The closed form solutions obtained by the direct algebraic method are identified with special trajectories in phase space that connect a critical point to itself or other critical points. The phase portrait is investigated for ODEs resulting from a combined KdV and mKdV equation, a KdV-like equation with fifth-degree nonlinearity, a special case of the Calogero–Degasperis–Fokas modified mKdV equation, and a generalized Fisher equation. The phase space analysis serves to clarify and partially justify the direct algebraic method.