## Symmetry-based algorithms for invertible mappings of polynomially nonlinear PDE to linear PDE.(English)Zbl 1486.12006

The paper under review is a sequel to the previous work of the authors [Z. Mohammadi et al., in: Proceedings of the 44th international symposium on symbolic and algebraic computation, ISSAC ’19, Beijing, China, July 15–18, 2019. New York, NY: Association for Computing Machinery (ACM). 331–338 (2019; Zbl 1467.35005)] where they introduced the MapDE algorithm to determine the existence of analytic invertible mappings of an input (source) differential polynomial system (DPS) to a specific target DPS, and sometimes by heuristic integration an explicit form of the mapping. In this paper the authors extend MapDE to determine if a source nonlinear DPS can be mapped to a linear differential system.
The main contribution of the paper is the presentation of an algorithmic method for determining the mapping of a nonlinear system to a linear system when it exists. Using a technique of [S. Kumei and G. W. Bluman, SIAM J. Appl. Math. 42, 1157–1173 (1982; Zbl 0506.35003)], the authors exploit the fact that the target system $$\hat{R}$$, which corresponds to the source system of differential equations $$R$$, must admit a sub-pseudo group corresponding to the superposition property that linear systems by definition must satisfy. Once existence is established, a second stage can determine features of the map and sometimes by integration, explicit forms of the mapping. For an algorithmic treatment using differential elimination (differential algebra), the authors limit their consideration to systems of differential polynomials, with coefficients from $$\mathbb{Q}$$ or some computable extension of $$\mathbb{Q}$$ in $$\mathbb{C}$$. (Thus the input system $$R$$ should be a system of DPS.) The paper contains examples that illustrate the approach; the authors also illustrate the powerful maximal symmetry groups facility as a natural tool to be used in conjunction with MapDE.

### MSC:

 12H05 Differential algebra 68W30 Symbolic computation and algebraic computation 35A30 Geometric theory, characteristics, transformations in context of PDEs

### Citations:

Zbl 1467.35005; Zbl 0506.35003

### Software:

CRACK; SADE; GeM; MapDE; LieAlgebrasOfVectorFields; DifferentialGeometry
Full Text:

### References:

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