On the Classification of Fifth Order Quasi-linear Non-constant Separant Scalar Evolution Equations of the KdV-Type
Fifth order, quasi-linear, non-constant separant evolution equations are of the form u(t) = A(partial derivative(5)u/partial derivative x(5)) + (B) over tilde, where A and (B) over tilde are functions of x, t, u and of the derivatives of u with respect to x up to order 4. We use the existence of a "formal symmetry'', hence the existence of "canonical conservation laws'' rho((i)), i = -1, . . . , 5 as an integrability test. We define an evolution equation to be of the KdV-Type, if all odd numbered canonical conserved densities are nontrivial. We prove that fifth order, quasi-linear, non-constant separant evolution equations of KdV type are polynomial in the function a = A(1/5); a = (alpha u(3)(2) + beta u(3) + gamma)(-1/2), where alpha, beta, and gamma are functions of x, t, u and of the derivatives of u with respect to x up to order 2. We determine the u(2) dependency of a in terms of P = 4 alpha gamma - beta(2) > 0 and we give an explicit solution, showing that there are integrable fifth order non-polynomial evolution equations.