Seismic pulse propagation: Analytical modeling and exact soliton solutions

Abstract

Seismic wave modeling plays a fundamental role in petroleum and natural gas exploration, earthquake engineering, and environmental sciences. Analytical representations of seismic wave propagation provide insight into subsurface structure and contribute to improve seismic hazard assessment. This paper derives seismic wave equations from the stress-strain relationship and investigates traveling wave solutions of the nonlinear Klein-Gordon equation. The analytical framework is constructed through the Jacobi elliptic function method, which enables the generation of soliton and rogue-wave-type structures within nonlinear dispersive media. In contrast to classical approaches, the proposed formulation allows the wave parameters to be explicitly associated with physically interpretable quantities such as soliton velocity, frequency, and amplitude, revealing their influence on wave evolution and stability. Numerical visualizations are presented to illustrate the transition between bright solitons, dark-bright rogue waves, and hybrid wave patterns, showing how parameter variations modulate the internal wave dynamics. Additionally, the relationship between soliton velocity and characteristic wave velocity has been emphasized, and its impact on the energy distribution, amplitude, and phase dynamics of the generated solutions has been discussed. The results demonstrate that periodic and localized traveling wave modes may emerge for f(u) not equal 0, offering new perspectives on the propagation characteristics of space seismic surface waves in heterogeneous seismic media.