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Nonlinear superposition principles obtained by Lie symmetry methods. (English) Zbl 0893.35020
Given a partial or ordinary differential equation (1) f(x,u,u x i ,u x i x j ,)=0, a nonlinear superposition principle is a binary operation F defined on the set of all its solutions. Thus, w=F(u,v) solves (1) whenever u,v do. This concept is attributed to S. E. Jones and W. F. Ames [J. Math. Anal. Appl. 17, 484-487 (1967; Zbl 0145.13201)]. Out of the scope of the present paper are nonlinear superposition principles with restricted domain, such as those resulting from permutability of Bäcklund transformations. Exploiting close similarity to symmetries, applicable techniques to find superposition principles embeddable in a 1-parametric family are suggested that also reveal a linearizing transformation for the equation (1). Presented are classification results for second-order PDEs in two independent variables. The authors also discuss the existence of isolated superposition principles and suggest a method to find them.
Reviewer: M.Marvan (Opava)
35G20General theory of nonlinear higher-order PDE
35A30Geometric theory for PDE, characteristics, transformations
20N02Sets with a single binary operation (groupoids)