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Symmetry and singularity properties of a system of ordinary differential equations arising in the analysis of the nonlinear telegraph equations. (English) Zbl 1129.34030

The author presents a symmetry analysis and determines the conservation laws for the system of differential equations arising in the theory of nonlinear telegraph equations
\[ u{^2}u'''-4uu'u''+3u'{^3}=0,\quad u{^2}v''-3uu'v'+(3u'{^2}-uu'')v=0 . \]
In terms of Lie point symmetries according to the classification scheme of [G. M. Mubarakzyanov, Izv. Vyssh. Uchebn. Zaved. Mat. 34, 99–106 (1963; Zbl 0166.04201)] this system has five Lie point symmetries \(\Gamma_1=\partial_x\), \(\Gamma_2=x\partial_x\), \(\Gamma_3=u\partial_u\), \(\Gamma_4=u\partial_v\), \(\Gamma_5=v\partial_v\) with the Lie algebra \(\{ 2A_1\oplus A_1 \}+A_2\). The symmetry \(\Gamma_4\) indicates that the dependent variable \(v\) can be replaced by the other dependent variable \(u\) and allows to construct the solutions of the system. By means of the substitutions \(u=\alpha\chi^p\), \(v=\beta\chi^q\), where \(\chi=x-x_0\) and \(x_0\) is the location of the putative movable singularity, the author performs the singularity analysis of the system with its interpretation in terms of the possibility of the existence of a subsidiary solution.

MSC:

34C14 Symmetries, invariants of ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
35L70 Second-order nonlinear hyperbolic equations

Citations:

Zbl 0166.04201

Software:

DIMSYM; LIE
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References:

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