Painlevé analysis and exact solutions of the nonlinear diffusion equation with a polynomial source (Q2786710)

The authors provide a comprehensive and sequential analysis of the nonlinear reaction-diffusion equation NEWLINE\[NEWLINEu_t=(u^nu_x)_x+\sum_{j=0}^{m}a_ju^j,NEWLINE\]NEWLINE where \(n\) and \(m\) are positive integers.NEWLINENEWLINEThis analysis is evaluated in a comoving frame \(z=kx+\omega t\) by the Painlevé test applied to the obtained ordinary differential equation. As a result, it is shown that the studied PDE is not integrable in general case, but it is possible to do for \(m=n+2\) and \(m=n+3\). Two particular cases (\(n=1\), \(m=3\) and \(n=1\), \(m=4\)) are considered in details. For these cases, the traveling waves are presented not only by series expansions but also as the exact solutions using special functions. The solutions are illustrated graphically.