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 ∂ 5 u ∂x 5 +B, where A andB 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 = (αu 2 3 + βu 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.

‘Level grading’ a new graded algebra structure on differential polynomials: application to the classification of scalar evolution equations

“…We give explicit formulas for order m = 7, 9 and give the formulas for order m = 11, 13, 15 in a closed form with the explicit form of the coefficients B 0 . We observed that, at all orders, a satisfies the same equations given in (10) The occurrence of the same form for the separant a suggests strongly that these equations belong to a hierarchy. The same form of a has occurred in the classification of fifth order equation [10], where it has been noted that these equations would be intrinsically related to the class of fully nonlinear third order equations [1],…”

“…The classification of 5th order, constant separant evolution equations is given in [3]. The non-constant separant case is studied by the MSS method in [10] where "KdV-like" equations are defined to be the ones that admit an unbroken sequence of conserved densities at all orders. Although the classification is not complete, it has been shown that the non-constant separant KdV-type equations are of the form u t = a 5 u 5 + Bu 2 4 + Cu 4 + G, where B, C and G are polynomial in a and u 3 and a new exact solution is given [10].…”

On the classification of scalar evolution equations with non-constant separant

“…This form can also be obtained from the triviality of ρ (3) , whose explicit expression is given in [11]. This expression involves…”

“…We aimed to obtain a similar uniqueness result for general, non-polynomial integrable equations and we applied the method of formal symmetries, based on the existence of "canonical densities" [7], to the classification of integrable evolution equations in 1 + 1 dimensions, u t = F [u] [1,8,11,9].…”

“…. , 5 and obtained a preliminary classification of quasilinear 5th order equation with non-trivial ρ (i) , i = 1, 2, 3 [11]. In subsequent work on the classification problem, we noticed that the triviality or non-triviality of the canonical density ρ (3) is the key element that determines the form of integrable equations.…”