F-expansion method analysis of Navier-Bernoulli hypothesis and Love’s hypothesis driven nonlinear dispersive waves in Murnaghan’s cylindrical elastic rod

This study explores the nonlinear dispersive waves within an elastic rod, which can expand in size during the propagation of long waves. Utilizing the Navier-Bernoulli and Love’s hypotheses, we derive these waves within the framework of compressible Murnaghan materials. Unlike previous studies that primarily focus on classical elastic models, we investigate nonlinear wave dynamics in compressible Murnaghan materials, which better describe the behavior of highly deformable elastic rods. A key novelty of our work lies in applying the F-expansion method to obtain exact solutions in terms of Jacobi elliptic functions, an approach not previously utilized for this class of materials. We systematically classify these solutions into six distinct groups, providing the necessary conditions for their existence and expanding the known solution space for nonlinear dispersive waves. Furthermore, we perform graphical simulations to illustrate the behavior of the obtained solutions, offering new insights into their physical characteristics. Finally, we analyze the role of frequency dispersion in these nonlinear waves, highlighting key observations that contribute to a deeper understanding of wave propagation in elastic media.