## 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

Zbl 0166.04201

LIE; DIMSYM
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### References:

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