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• W3105976651 abstract "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 F}/{partial u_m} where u_m={partial^m u}/{partial x}^m. We apply the formal symmetry method proposed in [MSS (1991)] to the classification of scalar evolution equations of orders mle 15, with non-trivial rho^{(-1)}=left[partial F/partial u_mright]^{-1/m} and rho^{(1). We obtain the top level parts of these equations and their top dependencies with respect to the level grading defined in [Mizrahi, Bilge (2013)]. We show that if rho^{(-1)} depends on u,u_1,dots,u_b, where b is the base level, then, these equations are level homogeneous polynomials in u_{b+i},dots ,u_m, ige 1 and the coefficient functions are determined up to their dependencies on u,u_1,dots,u_{b-1}. We prove that if rho^{(3)} is non-trivial, then rho^{(-1)}=(alpha u_b^2+beta u_b+gamma)^{1/2}, with ble 3 while if rho^{(3)} is trivial, then rho^{(-1)}=(lambda u_b+mu)^{1/3}, where ble 5 and alpha, beta, gamma, lambda and mu are functions of u,dots,u_{b-1}. We show that these equations form commuting flows and we construct their recursion operators that are respectively of orders 2 and 6 for non-trivial and trivial rho^(3) respectively. Omitting lower order dependencies, we show that equations with non-trivial rho^(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 [Bilge,(2005)] while for b=3,4 they are symmetries of quasilinear fifth order equations and we outline the transformations to polynomial equations where $u$ has zero scaling weight, suggesting that the hierarchies that we obtain could be transformable to known equations possibly by introducing non-locality." @default.
• W3105976651 created "2020-11-23" @default.
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• W3105976651 date "2016-12-09" @default.
• W3105976651 modified "2023-10-18" @default.
• W3105976651 title "On the classification of scalar evolution equations with non-constant separant" @default.
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• W3105976651 doi "https://doi.org/10.1088/1751-8121/50/3/035202" @default.
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