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<p>This study highlights the impact of stress nonlinearity on seismic-wave dynamics. When deformations are no longer elastic, linear elasticity is no longer valid, and the medium response becomes nonlinear. In this work, we use the Hardin–Drnevich law for nonlinear shear stresses and a nonlinear Hooke-type law for compression. By incorporating these constitutive laws, we derive a displacement-field equation analogous to a nonlinear Klein–Gordon model with anharmonic, cubic, and quartic interactions embedded in a substrate potential. The resulting amplitude equation contains singular-curve terms that provide additional insight into seismic-wave behavior. Wave amplitudes are obtained using a bifurcation approach based on phase portraits. Under different parameter regimes, distinct classes of solutions are found, and the corresponding constants related to nonlinear coefficients are extracted. We further show how singular curves affect stress evolution in the propagation medium.</p>

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Phase plane dynamics of seismic wave subjected to nonlinear stresses at upper gap condition

  • 期刊论文

  • 发表日期：

    2026年7月13日

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  • Guy Merlin Nkeumaleu,
  • Kagho Ymele Laurent,
  • Desire Ndjanfang

摘要

This study highlights the impact of stress nonlinearity on seismic-wave dynamics. When deformations are no longer elastic, linear elasticity is no longer valid, and the medium response becomes nonlinear. In this work, we use the Hardin–Drnevich law for nonlinear shear stresses and a nonlinear Hooke-type law for compression. By incorporating these constitutive laws, we derive a displacement-field equation analogous to a nonlinear Klein–Gordon model with anharmonic, cubic, and quartic interactions embedded in a substrate potential. The resulting amplitude equation contains singular-curve terms that provide additional insight into seismic-wave behavior. Wave amplitudes are obtained using a bifurcation approach based on phase portraits. Under different parameter regimes, distinct classes of solutions are found, and the corresponding constants related to nonlinear coefficients are extracted. We further show how singular curves affect stress evolution in the propagation medium.