We consider the one-dimensional dynamics of nonlinear non-dispersive waves. The problem can be mapped onto a linear one by means of the hodograph transform. We propose an approximate scheme for solving the corresponding Euler-Poisson equation which is valid for any kind of nonlinearity. The approach is exact for monoatomic classical gas and agrees very well with exact results and numerical simulations for other systems. We also provide a simple and accurate determination of the wave breaking time for typical initial conditions.
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