Lie symmetry analysis and explicit formulas for solutions of some third-order difference equations

Abstract

Lie group theory is applied to rational difference equations of the form xn+1 = xn−2xn xn−1(an + bnxn−2xn) , where (an)n∈N0 , (bn)n∈N0 are non-zero real sequences. Consequently, new symmetries are derived and exact solutions, in unified manner, are constructed. Based on some constraints in the expression of the symmetry generators, we split these solutions into different categories. This work generalises a recent result by Ibrahim

Key words: Difference equation, symmetry, group invariant solutions, periodicity.