Conditional and Lie symmetry of nonlinear wave equation. (English) Zbl 0949.35092
In the paper the author considers the nonlinear wave equation
\[
u_{00} + m \frac{u_0}{x_0} - p \frac{F(u)u_1}{x_1} - (F(u)u_1)_1 = 0,
\]
where \( u=u(x_0,x_1)\), \( u_\mu = \dfrac{\partial u}{\partial x_\mu}\), \( u_{\mu\mu}=\dfrac{\partial^2 u}{\partial x_\mu^2}\), \( \mu = 0,1\), \( F(u)\) is an arbitrary differentiable function, \( F'(u)\neq 0\), \( F(u)>0\), \( m\) and \(p\) are arbitrary constants. The conditional invariance of the wave equation with the nonlinearity \( F(u)=u \) is found.
Reviewer: V. I. Guljaev (Kyïv)
MSC:
| 35L70 | Second-order nonlinear hyperbolic equations |
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
| 35C05 | Solutions to PDEs in closed form |
| 58J70 | Invariance and symmetry properties for PDEs on manifolds |
References:
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| [2] | Ovsyannikov L.V., Group Analysis of Differential Equations, Academic Press, New York, 1982, 400p. · Zbl 0485.58002 |
| [3] | Fushchych W., Shtelen W. and Serov N., Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Dordrecht, Kluwer Academic Publishers, 1993, 436p. · Zbl 0838.58043 |
| [4] | Fushchych W.I., Serov M.I., Repeta V.K., Conditional symmetry, reduction and exact solutions of nonlinear wave equation, Dopovidi Akademii Nauk Ukrainy, (Proceedings of the Academy of Sciences of Ukraina), 1991, N 5, 29-36. |
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