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Teoreticheskaya i Matematicheskaya Fizika, 2003, Volume 137, Number 1, Pages 74–86
DOI: https://doi.org/10.4213/tmf246
(Mi tmf246)
 

Extended Rotation and Scaling Groups for Nonlinear Evolution Equations

P. G. Esteveza, C. Quab

a University of Salamanca
b Northwest University
References:
Abstract: A $(1+1)$-dimensional nonlinear evolution equation is invariant under the rotation group if it is invariant under the infinitesimal generator $V=x\partial_u-u\partial_x$. Then the solution satisfies the condition $u_x=-x/u$. For equations that do not admit the rotation group, we provide an extension of the rotation group. The corresponding exact solution can be constructed via the invariant set $R_0=\{u:u_x=x F(u)\}$ of a contact first-order differential structure, where $F$ is a smooth function to be determined. The time evolution on $R_0$ is shown to be governed by a first-order dynamical system. We introduce an extension of the scaling groups characterized by an invariant set that depends on two constants $\epsilon$ and $n\ne1$. When $\epsilon=0$, it reduces to the invariant set $S_0$ introduced by Galaktionov. We also introduce a generalization of both the scaling and rotation groups, which is described by an invariant set $E_0$ with parameters $a$ and $b$. When $a=0$ or $b=0$, it respectively reduces to $R_0$ or $S_0$. These approaches are used to obtain exact solutions and reductions of dynamical systems of nonlinear evolution equations.
Keywords: differential evolution equations, rotation group, scaling group.
English version:
Theoretical and Mathematical Physics, 2003, Volume 137, Issue 1, Pages 1419–1429
DOI: https://doi.org/10.1023/A:1026052622703
Bibliographic databases:
Language: Russian
Citation: P. G. Estevez, C. Qu, “Extended Rotation and Scaling Groups for Nonlinear Evolution Equations”, TMF, 137:1 (2003), 74–86; Theoret. and Math. Phys., 137:1 (2003), 1419–1429
Citation in format AMSBIB
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\paper Extended Rotation and Scaling Groups for Nonlinear Evolution Equations
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\vol 137
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\pages 74--86
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\transl
\jour Theoret. and Math. Phys.
\yr 2003
\vol 137
\issue 1
\pages 1419--1429
\crossref{https://doi.org/10.1023/A:1026052622703}
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  • https://www.mathnet.ru/eng/tmf/v137/i1/p74
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