On the classification of scalar evolution equations with non-constant separant
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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 derivative F/partial derivative u(m) where um u(m) = partial derivative(m)u/partial derivative x(m). As an integrability test we use the formal symmetry method of Mikhailov-Shabat-Sokolov which is based on the existence of a recursion operator as a formal series. The solvability of its coefficients in the class of local functions gives a sequence of conservation laws called the 'conserved densities' rho((i)) i = -1 1 2 3 ... We apply this method to the classification of scalar evolution equations of orders 3 <= m <= 15 for which rho((-)) = [partial derivative F/partial derivative u(m)](-1/m) and rho((1)) are non-trivial i.e. they are not total derivatives and rho((-1)) is not linear in its highest order derivative. We obtain the 'top level' parts of these equations and their ` top dependencies' with respect to the 'level grading' that we defined in a previous paper as a grading on the algebra of polynomials generated by the derivatives u(b+i) over the ring of C-infinity functions of u u(1) .. u(b). In this setting b and i are called 'base' and 'level' respectively. We solve the conserved density conditions to show that if rho((-)) depends on u u(1) ... u(b) then these equations are level homogeneous polynomials in u(b+i) ... u(m) i >= 1. Furthermore we prove that if rho((3)) is nontrivial then rho((-)) = (alpha mu(2)(b) (3) is trivial then ub 1/3 where b similar to 5 and a .. and mu are functions of u. ub-1. We show that the equations that we obtain form commuting flows and we construct their recursion operators that are respectively of orders 2 and 6 for non-trivial and trivial (3) respectively. Omitting lower order dependencies we show that equations with non-trivial (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 previous work while for b = 3 4 they are symmetries of quasilinear fifth order equations.