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.
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