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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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