Noncommutative and nonassociative pseudo-analysis and its applications on nonlinear partial differential equations
Considered is a pair of functions \(\oplus\) and \(\odot\) from \(C^2 \mathbb R^2\) satisfying the conditions: (1)~the equation \(x \odot x = a\) is uniquely solvable for every \(a\); (2)~\(\odot\) is right distributive over \(\oplus\). By solving (1) and (2) as a system of functional equations, the authors prove theorems characterizing any such pair essentially in terms of two arbitrary functions from \(C^2\mathbb R\). (One degree of freedom is missed in Theorem 3, namely \(x \oplus y = c x + y\) and \(x \odot y = x y\) is a solution; this does not influence the rest of the paper.) The research is interesting by its relations to superposition principles in PDE theory. Considering a nonlinear PDE, a superposition principle is a function \(F\) of two variables such that \(F(u,v)\) is a solution whenever \(u,v\) are so. The authors give several examples of superposition principles satisfying the conditions~(1) and~(2) above.
scientific article; zbMATH DE number 1496300
scientific article