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<p>We consider the tsunami wave equation with singular coefficients and prove that it has a very weak solution. We show the uniqueness and consistency of the very weak solution with the classical one in an appropriate sense. In one space dimension, we analyze the behavior of the waves in singular topographies. We observe the appearance of a substantial reflected wave, travelling in the opposite direction from the point of singularity. Its structure and strength are analyzed numerically. In particular, we illustrate the limiting behavior of the solution to the regularized problems when the regularizing parameter tends to zero. A surprising conclusion is that while, in general, the solution of the equation may not exist in the “classical” sense, the limit of the net of solutions of the regularized problems may exist, calling it the limiting very weak solution.</p>

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Singular Hyperbolic Type Equations and Tsunami Propagation for Irregular Topographies

  • 期刊论文

  • 发表日期：

    2025年6月21日

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  • Arshyn Altybay,
  • Michael Ruzhansky,
  • Mohammed Elamine Sebih,
  • Niyaz Tokmagambetov

摘要

We consider the tsunami wave equation with singular coefficients and prove that it has a very weak solution. We show the uniqueness and consistency of the very weak solution with the classical one in an appropriate sense. In one space dimension, we analyze the behavior of the waves in singular topographies. We observe the appearance of a substantial reflected wave, travelling in the opposite direction from the point of singularity. Its structure and strength are analyzed numerically. In particular, we illustrate the limiting behavior of the solution to the regularized problems when the regularizing parameter tends to zero. A surprising conclusion is that while, in general, the solution of the equation may not exist in the “classical” sense, the limit of the net of solutions of the regularized problems may exist, calling it the limiting very weak solution.