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Extinction of multiple shocks in the modular Burgers’ equation

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Abstract

We consider multiple shock waves in the Burgers’ equation with a modular advection term. It was previously shown that the modular Burgers’ equation admits a traveling viscous shock with a single interface, which is stable against smooth and exponentially localized perturbations. In contrast, we suggest in the present work with the help of energy estimates and numerical simulations that the evolution of shock waves with multiple interfaces leads to finite-time coalescence of two consecutive interfaces. We formulate a precise scaling law of the finite-time extinction supported by the interface equations and by numerical simulations.

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The data of numerical computations is available upon request to the corresponding author.

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Acknowledgements

An early stage of this work was completed during the undergraduate research project of Y. Ackermann and E. Redfearn. The later stage of this work was completed during the visit of D. E. Pelinovsky to KIT as a part of Humboldt Reseach Award from Alexander von Humboldt Foundation. The project is supported by the RSF grant 19-12-00253.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Dmitry E. Pelinovsky

  2. Institute of Applied Physics RAS, Nizhny Novgorod, 603950, Russia

    Dmitry E. Pelinovsky

  3. Karlsruhe Institute of Technology, Englerstraße 2, 76131, Karlsruhe, Germany

    Björn de Rijk

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  1. Dmitry E. Pelinovsky
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  2. Björn de Rijk
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Correspondence to Dmitry E. Pelinovsky.

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Pelinovsky, D.E., de Rijk, B. Extinction of multiple shocks in the modular Burgers’ equation. Nonlinear Dyn 111, 3679–3687 (2023). https://doi.org/10.1007/s11071-022-07873-x

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