The Hopf equation with certain modular nonlinearities

•The Hopf equation with a non-analytic propagation velocity is considered.•Riemann waves exist only for a certain smoothness of the velocity function.•Within the modular Hopf equation, a sine wave is transformed into a Riemann wave and exists for a finite time.•Initial sine wave breaks immediately w...

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Bibliographic Details
Published in:Physics letters. A 2024-05, Vol.507, p.129489, Article 129489
Main Authors: Pelinovsky, Efim, Talipova, Tatiana, Didenkulova, Ekaterina
Format: Article
Language:English
Subjects:
Hopf equation
Korteweg-de Vries-type equation
Nonlinear wave process in the physics
Riemann waves
Schamel equation
Wave breaking
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Summary:•The Hopf equation with a non-analytic propagation velocity is considered.•Riemann waves exist only for a certain smoothness of the velocity function.•Within the modular Hopf equation, a sine wave is transformed into a Riemann wave and exists for a finite time.•Initial sine wave breaks immediately within the Schamel equation.•Sign-variable disturbance of any shape breaks immediately within the log-KdV equation. 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.
ISSN:0375-9601
1873-2429
DOI:10.1016/j.physleta.2024.129489