Analytical solutions to the (2+1)-dimensional cubic Klein–Gordon equation in the presence of fractional derivatives: A comparative study

Abstract

The study seeks to obtain new analytical solutions for the (2+1)-dimensional cubic Klein–Gordon (cKG) equation using the beta derivative. By applying the unified method to the equation, various types of solitons have been generated, including periodic solitons, periodic solitons with equal and unequal wavelengths, bright solitons, and periodic singular solitons with unequal wavelengths. To demonstrate the fundamental dynamics of the soliton family, three-dimensional and two-dimensional graphs showcasing various novel solutions that satisfy the relevant equations are provided. In relation to fractionality, the bright waveform retains its overall shape, but its smoothness improves as the fractional parameters increase. Conversely, periodic wave solutions show enhanced periodicity as the fractional parameters rise. Additionally, the study provides a comprehensive comparison of solutions derived from models utilizing conformable, M-truncated, and beta derivatives. The investigation explores the effect of the fractional parameter on soliton amplitude, using graphs to illustrate this impact by assigning specific values to the fractional parameter. The properties of the waves can be modified through changes to the model's parameters to produce the appropriate wave profiles. Consequently, the solutions we obtained could be particularly valuable for analyzing physical problems associated with nonlinear complex dynamical systems.

Keywords

• Cubic Klein–Gordon equation
• Beta derivative
• The unified method
• Wave propagation