Hybrid Solutions and Qualitative Analysis of (2+1)-Dimensional Nonlinear Wave Equations

Abstract

This thesis investigates the (2+1)-dimensional integrable Kadomtsev-Petviashvili equation, employing Hirota’s bilinear form to explore various exact solutions, including breather waves, periodic waves, and two-wave interactions, as well as kink-cross, coscosh, and sin-cosh type solutions. Additionally, soliton solutions such as bright and dark envelope solitons, periodic solutions, and solutions involving the ratio of exponential functions have been thoroughly examined. These solutions are visualized through interactive 3D and density graphs generated using MATLAB, offering a comprehensive understanding of the dynamic behaviour of the KP equation. A significant contribution of this work is the qualitative analysis of the system’s chaotic behaviour, focusing on bifurcation and sensitivity. The resonant higher-order nonlinear Schrödinger equation is transformed into ordinary differential equations using traveling wave transformations and further converted into a planar dynamical system using the Galilean transformation. The qualitative dynamics of this time-dependent system are then investigated through chaos theory, with phase portraits, time series, and Poincaré maps employed to identify chaotic behavior. The sensitivity of the models is examined under four distinct initial conditions, highlighting the system’s responsiveness to small changes. VIII