Local Analysis of Nonlinear Equations

Abstract

Many nonlinear mappings, connected with equations of mathematical physics have some special properties which allow to obtain explicit results by the traditional method of normal form. For example, the mappings from Thomas and KdV equations

$$\matrix{ {F:u \to {u_{xy}} + \alpha {u_x} + \beta {u_y} + \gamma {u_x}{u_y},\left( {\alpha ,\beta ,\gamma = const} \right)} \cr {G:u \to {u_{xxx}} + 6u{u_x} + {u_y},} \cr } $$

belong to this class. It’s easy to see that: The origin is a fixed point of F and G; F and G are analytic in a neighborhood of the origin and also are translational invariant. In the following, we consider only such mappings.