Exact solutions to some nonlinear PDEs, travelling profiles method

Abstract

We suggest finding exact solutions of equation: \begin{equation*} \frac{\partial u}{\partial t}=(\frac{\partial ^{m}}{\partial x^{m}}u)^{p}, t\geq 0, x\in \mathbb{R}, m, p\in \mathbb{N}, p>1, \end{equation*} by a new method that we call the travelling profiles method. This method allows us to find several forms of exact solutions including the classical forms such as travelling-wave and self-similar solutions.