The dynamics of waves of sign-variable shape has been studied within the framework of the Hopf equation () with a non-analytic propagation velocity containing the modulus of the function at the zero crossing (). It is shown that Riemann waves exist only for a certain smoothness of the function 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.

中文翻译：

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具有一定模非线性的 Hopf 方程

在 Hopf 方程 () 的框架内研究了符号变量形状波的动力学，其非解析传播速度包含过零处函数的模数 ()。结果表明，黎曼波仅在函数初始时刻具有一定的光滑度时才存在。否则，波浪会立即翻转（梯度灾难）。非线性物理学中流行的模数 Hopf 方程、无色散 Schamel 方程和无色散对数 Korteweg-de Vries 方程被视为示例。