Bilge, Ayşe HümeyraÖzkum, GülcanBilge, Ayşe Hümeyra2021-02-072021-02-07201220031-90150031-9015https://hdl.handle.net/20.500.12469/3880https://doi.org/10.1143/JPSJ.81.054001Fifth 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.eninfo:eu-repo/semantics/closedAccessEvolution equationsIntegrabilityClassificationRecursion operatorFormal symmetryArticle581WOS:00030324480000510.1143/JPSJ.81.0540012-s2.0-84860627338N/AN/A