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dc.contributor.authorBilge, Ayşe Hümeyra
dc.contributor.authorMizrahi, Eti
dc.date.accessioned2019-06-27T08:01:24Z
dc.date.available2019-06-27T08:01:24Z
dc.date.issued2017
dc.identifier.issn1751-8113
dc.identifier.issn1751-8121
dc.identifier.urihttps://hdl.handle.net/20.500.12469/370
dc.identifier.urihttps://doi.org/10.1088/1751-8121/50/3/035202
dc.description.abstractThe ` 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 derivative F/partial derivative u(m) where um u(m) = partial derivative(m)u/partial derivative x(m). As an integrability test we use the formal symmetry method of Mikhailov-Shabat-Sokolov which is based on the existence of a recursion operator as a formal series. The solvability of its coefficients in the class of local functions gives a sequence of conservation laws called the 'conserved densities' rho((i)) i = -1 1 2 3 ... We apply this method to the classification of scalar evolution equations of orders 3 <= m <= 15 for which rho((-)) = [partial derivative F/partial derivative u(m)](-1/m) and rho((1)) are non-trivial i.e. they are not total derivatives and rho((-1)) is not linear in its highest order derivative. We obtain the 'top level' parts of these equations and their ` top dependencies' with respect to the 'level grading' that we defined in a previous paper as a grading on the algebra of polynomials generated by the derivatives u(b+i) over the ring of C-infinity functions of u u(1) .. u(b). In this setting b and i are called 'base' and 'level' respectively. We solve the conserved density conditions to show that if rho((-)) depends on u u(1) ... u(b) then these equations are level homogeneous polynomials in u(b+i) ... u(m) i >= 1. Furthermore we prove that if rho((3)) is nontrivial then rho((-)) = (alpha mu(2)(b) (3) is trivial then ub 1/3 where b similar to 5 and a .. and mu are functions of u. ub-1. We show that the equations that we obtain form commuting flows and we construct their recursion operators that are respectively of orders 2 and 6 for non-trivial and trivial (3) respectively. Omitting lower order dependencies we show that equations with non-trivial (3) and b = 3 are symmetries of the ` essentially non-linear third order equation'
dc.description.abstractfor trivial rho((3)) the equations with b = 5 are symmetries of a non-quasilinear fifth order equation obtained in previous work while for b = 3 4 they are symmetries of quasilinear fifth order equations.
dc.language.isoEnglish
dc.publisherIOP Publishing Ltd
dc.subjectClassificaiton
dc.subjectDifferential polynomials
dc.subjectEvolution equations
dc.subjectHierarchies
dc.titleOn the classification of scalar evolution equations with non-constant separant
dc.typeArticle
dc.relation.journalJournal of Physics A: Mathematical and Theoretical
dc.identifier.issue3
dc.identifier.volume50
dc.identifier.wosWOS:000390820600002
dc.identifier.doi10.1088/1751-8121/50/3/035202
dc.contributor.khasauthorBilge, Ayşe Hümeyra


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