Abstract

The dynamics of waves of sign-variable shape has been studied within the framework of the Hopf equation (ut +Fux = 0) with a non-analytic propagation velocity containing the modulus of the function at the zero crossing (F ∼ |u|α). It is shown that Riemann waves exist only for a certain smoothness of the function F[u(x)] at the initial moment of time. Otherwise, the wave immediately overturns (gradient catastrophe). Popular in nonlinear physics the modular Hopf equation, the dispersionless Schamel equation, and the dispersionless logarithmic Korteweg-de Vries equation are considered as examples.

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