Integrable ordinary differential equations on free associative algebras
The authors consider nonlinear ordinary differential equations \(M_t= F(M, C)\) where the unknown \(M= M(t)\) and constant \(C\) belong to a free associative algebra \({\mathcal M}\) and \(F\) is a noncommutative polynomial. They introduce infinitesimal symmetries, the equivalence \(f\sim g\) of elements in \({\mathcal M}\) (\(f\) can be obtained from \(g\) by cyclic permutation), first integrals \(h\in{\mathcal M}\) (defined by \(h_t\sim 0\)), the Fréchet derivative \(a_*\) and the gradient \(\text{grad}_M\) defined by \[ {d\over d\varepsilon} a(M+\varepsilon\delta M, C)\bigl|_{\varepsilon= 0}= a_*(\delta M)\sim \delta M\cdot\text{grad}_M a(M, C) \] of an element \(a(M,C)\in{\mathcal M}\), operator algebra \(\Theta\) generated by the left and the right multiplications, Hamilton operators \({\mathcal O}\in\Theta\) such that the bracket \(\{a,b\}= \text{grad}_Ma\cdot \Theta(\text{grad}_Mb)\) is skew-commutative and satisfies the Jacobi identity, and the Hamilton equations \(M_t= \Theta(\text{grad}_MH(M, C))\). The classical basic results of the formal integrability theory (including the recursion operators) are verified and the classification problem is resolved.
nonlinear ordinary differential equations
scientific article; zbMATH DE number 1535315