14 research outputs found

    Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited

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    We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x + (4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no recursion operator or master symmetry was known so far, and prove that the system (*) admits infinitely many local generalized symmetries that are constructed using a nonlocal {\em two-term} recursion relation rather than from a recursion operator.Comment: 10 pages, LaTeX; minor changes in terminology; some references and definitions adde

    A unified approach to computation of integrable structures

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    We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. We adopt a coordinate based approach and aim to provide a tutorial to the computations.Comment: 19 pages, based on a talk on the SPT 2011 conference, http://www.sptspt.it/spt2011/ ; v2, v3: minor correction

    Symmetry classification of third-order nonlinear evolution equations. Part I: Semi-simple algebras

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    We give a complete point-symmetry classification of all third-order evolution equations of the form ut=F(t,x,u,ux,uxx)uxxx+G(t,x,u,ux,uxx)u_t=F(t,x,u,u_x, u_{xx})u_{xxx}+G(t,x,u,u_x, u_{xx}) which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.Comment: 53 page
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