Exact and Interaction solutions by using Hirota Bilinear forms and Qualitative analysis of an NLEE

Abstract

This thesis explores two models of the integrable nonlinear partial differential equations namely, the newly proposed (3+1)-dimensional generalized Korteweg-de Vries (gKdV) equation and the (2+1)-dimensional extended Kadomtsev–Petviashvili (eKP) equation. Hirota’s bilinear form has been employed to extract some exact solutions for both equations. The gKdV equation yielded the following solutions: bright and dark envelope solitons, periodic solutions, a singular solution, cos − cosh Type, sin − cosh Type, and a ratio of exponential functions solution. Other interaction solutions included the Solitary Wave Solution, Lump Solution, Kink-Cross Rational Solution, and Cross-Kink Periodic Solution. Using Matlab software, these solutions were also illustrated graphically in 3-D and density plots, which enhanced the understanding of the physical characteristics of the gKdV equation. Furthermore, the eKp equation is investigated to yield multiple exact solutions, which were displayed in a similar way using density and 3-D plots. Apart from obtaining solutions, the qualitative behavior of the eKP equation has been studied through bifurcation, chaos, and sensitivity analyses. Chaotic behavior was identified and examined using methods like phase portraits, Poincaré maps, time series analysis, and Lyapunov exponents. The visual representation of the analysis results provided further insights into the chaotic dynamics and stability of the system.