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Approximate potential symmetries for partial differential equations. (English) Zbl 0965.35025
This paper deals with a scalar $$k$$-order perturbed partial differential equation (PDE) $$R\{x,u,\varepsilon\}$$, which is written in a conserved form $D_i[f^i(x, u,u_{(1)},\dots, u_{(k-1)})+ \varepsilon g^i(x,u,u_{(1)},\dots, u_{(k-1)})]= 0$ with $$n\geq 2$$ independent variables $$x= (x_1,\dots, x_n)$$, a single dependent variable $$u$$, $$u_{(j)}$$, $$j= 1,\dots, k-1$$ is the collection of $$j$$-order partial derivatives, $$\varepsilon$$ is a small parameter and $$D_i= {\partial\over\partial x_i}+ u_i{\partial\over\partial u}+ u_{ij}{\partial\over\partial u_j}+\cdots$$, $$i= 1,2,\dots, n$$.
The method of approximate potential symmetries for PDEs with a small parameter is introduced. By writing a given perturbed PDE $$R$$ in a conserved form, an associated system $$S$$ with potential variables as additional variables is obtained. Approximate Lie point symmetries admitted by $$S$$ induce approximate potential symmetries of $$R$$. As application of the theory, approximate potential symmetries for a perturbed wave equation and a nonlinear diffusion equation with perturbed convection terms are obtained.

##### MSC:
 35G20 Nonlinear higher-order PDEs 58J70 Invariance and symmetry properties for PDEs on manifolds 35A30 Geometric theory, characteristics, transformations in context of PDEs
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