Discussion on Soliton Solution via Complete Discrimination System Approach with Bifurcation Analysis

Abstract

Nonlinear partial differential equations (NLPDEs) are significant because they are applied in almost every field of study, including biology, chemistry, physics, fiber optics, mechan- ics, atmospheric, and electronics science. The nonlinear Schr¨odinger equation (NLSE) is a unique class of NLPDEs. The NLSE have applications in physical, biological, and engi- neering research. Soliton is an important description in the NLSE analysis, and it has been a popular focus of research in the field of nonlinear models during the past two decades. The establishment of optical solitons, in particular, has provided a theoretical basis for the growth of nonlinear optics. Bifurcation analysis is an important tool in this process since it assists in identifying critical points, categorizing bifurcation types, and visualizing the sys- tem’s behavior using bifurcation diagrams, providing substantial insights into the system’s overall dynamics and behavior. A bifurcation can also create or destroy soliton solutions, or it can affect the stability features of existing solitons. In this thesis, our objective is to illustrate the bifurcation, wave structure and topological properties of the Chiral NLSE (CNLSE) with Bohm potential (CNLSE-BP) and cubic- quintic NLSE (CQNLSE) with an additional anti-cubic nonlinear term (CQNLSE-AC) by the complete discriminant system (CDS) of polynomial method (CDSPM). We study the bifurcation analysis of our governing model; bifurcation analysis is helpful in finding how systems change and exhibit various behaviors in response to changes in parameters or initial conditions. Furthermore, we also get optical solitons and wave structure like as Jacobian elliptic function (JEF), hyperbolic function, trigonometric function solutions, and rational function solution as well as also convert the JEF into solitary wave (SW) solutions. A sensi- tivity study is also performed under different initial conditions. Moreover, the results offer a way to investigate optical solitons and exact solutions of pulse propagation in optical fibers.