M. E. Elbrolosy

The present work is attentive to studying the qualitative analysis for a nonlinear strain wave equation describing the finite deformation elastic rod taking into account transverse inertia, and shearing strain. The strain wave equation is rewritten as a dynamic system by applying a particular transformation. The bifurcation of the solutions is examined, and the phase portrait is depicted. Based on the bifurcation constraints, the integration of the first integral of the dynamic system along specified intervals leads to real wave solutions. We prove the strain wave equation has periodic, solitary wave solutions and does not possess kink (or anti-kink) solutions. In addition, the set of discovered solutions contains Jacobi-elliptic, trigonometric, and hyperbolic functions. This model contains many kinds of solutions, which are always symmetric or anti-symmetric in space. We study how the change in the physical parameters impacts the solutions that are found. Numerically, the behavior of the strain wave for the elastic rod is examined when particular periodic forces act on it, and moreover, we clarify the existence of quasi-periodic motion. To clarify these solutions, we present a 3D representation of them and the corresponding phase orbit.

The Ivancevic option pricing model comes as an alternative to the Black-Scholes model and depicts a controlled Brownian motion associated with the nonlinear Schrodinger equation. The applicability and practicality of this model have been studied by many researchers, but the analytical approach has been virtually absent from the literature. This study intends to examine some dynamic features of this model. By using the well-known ARS algorithm, it is demonstrated that this model is not integrable in the Painlevé sense. He’s variational method is utilized to create new abundant solutions, which contain the bright soliton, bright-like soliton, kinky-bright soliton, and periodic solution. The bifurcation theory is applied to investigate the phase portrait and to study some dynamical behavior of this model. Furthermore, we introduce a classification of the wave solutions into periodic, super periodic, kink, and solitary solutions according to the type of the phase plane orbits. Some 3D-graphical representations of some of the obtained solutions are displayed. The influence of the model’s parameters on the obtained wave solutions is discussed and clarified graphically.