The symmetry algebra of the differential-difference equation \documentclass[12pt]minimal\begindocument\$\dotu_n = N/D\$\enddocumentu?n=N/D with D = un+1 - un?1 and N = P(un)un+1un?1 + Q(un)(un+1 + un?1) + R(un), where P, Q, and R are arbitrary analytic functions is shown to have the dimension 1 ≤ dimL ≤ 5. When P, Q, and R are specific second order polynomials in un (depending on 6 constants) this is the integrable discretization of the Krichever-Novikov equation. We find 3 cases when the arbitrary functions are not polynomials and the symmetry algebra satisfies dimL = 2. These cases are shown not to be integrable. The symmetry algebras are used to reduce the equations to purely difference ones. The symmetry group is also used to impose periodicity un+N = un and thus to reduce the differential-difference equation to a system of N coupled ordinary three points difference equations.

Levi, D., Ricca, E., Thomova, Z., Winternitz, P. (2014). Lie group analysis of a generalized Krichever-Novikov differential-difference equation. JOURNAL OF MATHEMATICAL PHYSICS, 55(10), 103503 [10.1063/1.4896989].

### Lie group analysis of a generalized Krichever-Novikov differential-difference equation

#### Abstract

The symmetry algebra of the differential-difference equation \documentclass[12pt]minimal\begindocument\$\dotu_n = N/D\$\enddocumentu?n=N/D with D = un+1 - un?1 and N = P(un)un+1un?1 + Q(un)(un+1 + un?1) + R(un), where P, Q, and R are arbitrary analytic functions is shown to have the dimension 1 ≤ dimL ≤ 5. When P, Q, and R are specific second order polynomials in un (depending on 6 constants) this is the integrable discretization of the Krichever-Novikov equation. We find 3 cases when the arbitrary functions are not polynomials and the symmetry algebra satisfies dimL = 2. These cases are shown not to be integrable. The symmetry algebras are used to reduce the equations to purely difference ones. The symmetry group is also used to impose periodicity un+N = un and thus to reduce the differential-difference equation to a system of N coupled ordinary three points difference equations.
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2014
Levi, D., Ricca, E., Thomova, Z., Winternitz, P. (2014). Lie group analysis of a generalized Krichever-Novikov differential-difference equation. JOURNAL OF MATHEMATICAL PHYSICS, 55(10), 103503 [10.1063/1.4896989].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11590/301931`
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