Abstract
This paper analyzes a nonlinear variational wave equation in which the wave speed is a function of the dependent variable. The wave equation arises is a number of different physical contexts and is the simplest example of an interesting class of nonlinear hyperbolic partial differential equations. We describe a blow-up result for the one-dimensional wave equation which shows that smooth solutions break down in finite time. We illustrate this result with some numerical solutions. We also derive a closed system of equations which describe the interaction between the mean field of a solution and oscillations in its spatial derivative.
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Glassey, R.T., Hunter, J.K., Zheng, Y. (1997). Singularities and Oscillations in a Nonlinear Variational Wave Equation. In: Rauch, J., Taylor, M. (eds) Singularities and Oscillations. The IMA Volumes in Mathematics and its Applications, vol 91. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1972-9_3
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DOI: https://doi.org/10.1007/978-1-4612-1972-9_3
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