On the Classification of Fifth Order Quasi-linear Non-constant Separant Scalar Evolution Equations of the KdV-Type

Abstract

Fifth order, quasi-linear, non-constant separant evolution equations are of the form \(u_{t} = A(\partial^{5}u/\partial x^{5}) + \tilde{B}\), where A and \(\tilde{B}\) 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” ρ ( 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 = (α u 3 2 + β u 3 + γ) -1/2 , where α, β, and γ 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 = 4αγ- β 2 > 0 and we give an explicit solution, showing that there are integrable fifth order non-polynomial evolution equations.