On the Classifications of Scalar Evolution Equations with Non-constant Separant
The "separant" of the evolution equation u_t=F, where F is some differentiable function of the derivatives of u up to order m is the partial derivative \partial F}/{\partial u_m} where u_m={\partial^m u}/{\partial x}^m. We apply the formal symmetry method proposed in [MSS (1991)] to the classification of scalar evolution equations of orders m\le 15, with non-trivial ρ^{(-1)}=\left[\partial F/\partial u_m\right]^{-1/m} and rho^{(1). We obtain the "top level" parts of these equations and their "top dependencies" with respect to the "level grading" defined in [Mizrahi, Bilge (2013)]. We show that if rho^{(-1)} depends on u,u_1,\dots,u_b, where b is the base level, then, these equations are level homogeneous polynomials in u_{b+i},\dots ,u_m, i\ge 1 and the coefficient functions are determined up to their dependencies on u,u_1,\dots,u_{b-1}. We prove that if ρ^{(3)} is non-trivial, then ρ^{(-1)}=(αu_b^2+βu_b+γ)^{1/2}, with b\le 3 while if ρ^{(3)} is trivial, then rho^{(-1)}=(λu_b+μ)^{1/3}, where b\le 5 and alpha, beta, gamma, lambda and mu are functions of u,\dots,u_{b-1}. We show that these equations form commuting flows and we construct their recursion
