On the Integrability of Homogeneous Scalar Evolution Equations

We determine the existence of (infinitely many) symmetries for equations of the formut=uk+f(u, …, uk−1)when they areλ-homogeneous (with respect to the scalinguk λ+k) withλ>0. Algorithms are given to determine whether a system has a symmetry (also independent oftandx). If it has one generalized symmetry, we prove it has infinitely many and these can be found using recursion operators or master symmetries. The method of proof uses the symbolic method and results from diophantine approximation theory. We list the 10 integrable hierarchies. The methods can in principle be applied to theλ 0 case, as we illustrate for one example withλ=0, which seems to be new. In principle they can also be used for systems of evolution equations, but so far this has only been demonstrated for one class of examples.