https://doi.org/10.1140/epjp/s13360-025-06573-w
Regular Article
Delayed complex cubic-quintic Ginzburg–Landau equation in the spring block model of earthquake
1
Department of Physics, Faculty of Science, University of Bamenda, P. O. Box 39, Bambili, Cameroon
2
Unity of Research of Mechanics and Modelling of Physical Systems, Department of Physics, Faculty of Science, University of Dschang, PO Box 69, Dschang, Cameroon
3
Department of Physics, Higher Teacher Training College Bambili, The University of Bamenda, P. O. Box 39, Bambili, Cameroon
Received:
16
March
2025
Accepted:
20
June
2025
Published online:
17
July
2025
In this study, we extend the classical spring-block model for earthquakes by incorporating higher-order nonlinear effects in both the elastic and damping forces. Building on the works of Mofor et al. (Eur Phys J Plus 138:273, 2023), who analyzed the modulation of waves in a system with time delay, cubic nonlinearity in elasticity, and linear damping, we introduce a fifth-order nonlinear elasticity and a nonlinear damping force. The resulting governing equation takes the form of the complex cubic-quintic Ginzburg–Landau equation, which describes the evolution of modulated waves in the system. We investigate the modulational instability criterion, identifying conditions under which supercritical and subcritical waves become unstable or stable. Additionally, we explore the Hopf bifurcation, analyzing its impact on the emergence of complex wave structures in earthquake dynamics. Our results show that the higher-order nonlinear terms significantly alter the stability conditions of propagating waves, affecting energy localization, fault rupture dynamics, and seismic event predictability.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
