On the Classifications of Scalar Evolution Equations with Non-constant Separant

Abstract

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 \rho^{(-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 \rho^{(3)} is non-trivial, then \rho^{(-1)}=(\alpha u_b^2+\beta u_b+\gamma)^{1/2}, with b\le 3 while if \rho^{(3)} is trivial, then rho^{(-1)}=(\lambda u_b+\mu)^{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 operators that are respectively of orders 2 and 6 for non-trivial and trivial rho^(3) respectively. Omitting lower order dependencies, we show that equations with non-trivial rho^(3) and b=3 are symmetries of the "essentially non-linear third order equation". For trivial rho^(3), the equations with b=5 are symmetries of a non-quasilinear fifth order equation obtained in [Bilge,(2005)] while for b=3,4 they are symmetries of quasilinear fifth order equations and we outline the transformations to polynomial equations where $u$ has zero scaling weight, suggesting that the hierarchies that we obtain could be transformable to known equations possibly by introducing non-locality.