Algebraic traveling wave solutions, Darboux polynomials and polynomial solutions (Q1617225)

A traveling wave solution \(u= U(x-ct)\) of a partial differential equation \(u_{xx}= F(u,u_x,u_t)\) is called an algebraic traveling wave solution if there exists a polynomial \(p\) such that \(p(U,U')= 0\). The author completely characterizes the existence of algebraic traveling wave solutions of the partial differential equation \[ u_t= du_{xx}- a(u-u_1)(u_2-u)(u-u_3). \] For doing that, necessary and sufficient conditions are derived for a linear differential equation to have a polynomial solution.