Direct method to construct integrals for \(N\)th-order autonomous ordinary difference equations. (English) Zbl 1141.37025
Summary: A direct method to construct integrals for an \(N\)th-order autonomous ordinary difference equation (ODE): \(w_{n+N}=F(w_n,\dots ,w_{n+N - 1})\) is presented. As an illustration we first consider the third-order autonomous ODE \(w_{n+3}=F(w_n,w_{n+1},w_{n+2})\) and identify the forms of \(F\) for which two independent integrals exist. The effectiveness of the method to construct two or more integrals for fourth- and fifth-order ODEs is also demonstrated. The question of integrability of each of the identified difference equations with more than one integral is also discussed.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 39A12 | Discrete version of topics in analysis |
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