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<p>The dynamics of the Burridge–Knopoff (BK) model have been modified by taking into account simultaneously the fractional order and the long-range interactions (LRI) of each block. It is shown that the dynamics of the model becomes a nonlinear fractional Schrödinger equation, where the dispersion and non-linearity parameters depend strongly on the order of fractional derivative and long-range coefficients. We found that, for low values of the LRI parameter, the system exhibits an unusual behaviour, indicating a probable earthquake warning. A high value of LRI leads to crucial phenomena. An earthquake can occur at a high value of LRI. At a high value of LRI, the wave’s amplitude increases with time, showing that our system has high energy, which justifies the catastrophic processes observed in earthquakes. The results show that when the fractional order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> increases, the velocity and amplitude of the waves decrease, and when the exponent <i>s</i> increases, the velocity decreases and amplitude of the seismic waves increases. Consequently, for low values of <i>s</i>, the system exhibits highly localised waves with high amplitude and slow velocity. The numerical results are in perfect agreement with the theory and show that the nonlinear dynamics of seismic soliton wave can be well understood when we introduce LRI and fractional derivative order.</p>

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Extended Burridge–Knopoff model associated with fractional order and long-range interactions

  • 期刊论文

  • 发表日期：

    2026年2月7日

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  • Clement Parfait Bounoung,
  • Françoise Martine Enyegue A Nyam,
  • Henock Ngoubi,
  • Sarskolin Fosso Kegne,
  • Tabod Charles Tabod

摘要

The dynamics of the Burridge–Knopoff (BK) model have been modified by taking into account simultaneously the fractional order and the long-range interactions (LRI) of each block. It is shown that the dynamics of the model becomes a nonlinear fractional Schrödinger equation, where the dispersion and non-linearity parameters depend strongly on the order of fractional derivative and long-range coefficients. We found that, for low values of the LRI parameter, the system exhibits an unusual behaviour, indicating a probable earthquake warning. A high value of LRI leads to crucial phenomena. An earthquake can occur at a high value of LRI. At a high value of LRI, the wave’s amplitude increases with time, showing that our system has high energy, which justifies the catastrophic processes observed in earthquakes. The results show that when the fractional order \(\gamma \) γ increases, the velocity and amplitude of the waves decrease, and when the exponent s increases, the velocity decreases and amplitude of the seismic waves increases. Consequently, for low values of s, the system exhibits highly localised waves with high amplitude and slow velocity. The numerical results are in perfect agreement with the theory and show that the nonlinear dynamics of seismic soliton wave can be well understood when we introduce LRI and fractional derivative order.