Studies on mathematical models of seismic wave

Seismic wave studies are essential for academic research and practical applications in fields such as petroleum and natural gas exploration, earthquake engineering, and ecological science. Mathematical models of seismic waves play a crucial role in understanding their propagation, aiding earthquake engineering practices, subsurface exploration, and revealing insights into Earth's deep structure and subsurface layers. Accurate modeling of seismic wave propagation is critical for understanding the behavior of the model and making educated drilling and production choices. This thesis proposed seismic wave equations for elastic wave propagation in a uniform space by using stress and strain theory, the most important motivation of this thesis is to analyze the behavior of the space P-wave in time for f(u) ̸= 0 using analytical approaches. We first propose different types of evolution equations both linear and nonlinear to obtain rational function solutions utilizing the bilinear form with the help of the Hirota bilinear operator for the first time. The homoclinic technique, which is based on the basic principles of the Hirota bilinear method and produces solutions in different forms, was developed within the scope of this thesis and is the original part of the thesis. Using the homoclinic technique proposed within the scope of the thesis, papers were published in prestigious international journals. The φ^6-model expansion method and Jacobi elliptic function method are also used to derive soliton solutions such as dark, bright, kink, lump, and so on. The graphs given for these solutions are important tools to explain the physical understanding of the dynamics affecting the soliton. We investigate the stability analysis of both wave equations and solutions separately and then the energy represented by the wave model is analyzed to gain a solid understanding of wave interactions. Stable wave solutions guarantee that energy estimates are precise and legitimate, which is important for applications such as seismic risk estimation, and optimizing energy transmission in communication networks. By guaranteeing that wave equations have stable solutions, we may securely study and use the energy transmitted by these waves in a variety of practical and scientific situations. The concept of energy storage in seismic waves is crucial for understanding their behavior and impact. Kinetic energy which is associated with wave particle motion, and potential energy which is stored in the elastic deformation of rocks were examined. When stress accumulates along a fault, potential energy builds up until it's released as an earthquake. Energy flux and the total energy are also considered.