Analytical solitons, phase-space dynamics, and stability analysis in Katugampola fractional nonlinear telegraph equation

The propagation of waves in complicated media in both space and time in a non-linear fashion is still one of the major problems in mathematics and physics and has numerous important applications including in fluid mechanics, plasma physics and nonlinear optics. Recent models have involved the use of neural symbolic schemes and one-way integration to model these phenomena, but they have not been able to model unified memory and nonlocal effects, or they have limited the available solution space for the analytical solutions. To fill this void, in this study, a generalized nonlinear evolution equation is studied by using the Katugampola fractional derivative approach. A fractional wave transformation is used to transform the governing partial differential equation into an associated ordinary differential equation with success. A powerful hybrid analytical scheme is used to solve the reduced system which consists of a Riccati-Bernoulli sub-ODE method and a Bäcklund transformation. As a result, three different families of exact analytical solutions are built, most of which are kink localized, substantially enriching the solutions which have been recently found in the literature. The analysis of the spatiotemporal profiles is performed using 3D surface plots in the integer-order case, \(\alpha = 1\) ; and 2D plots show that decreasing the Katugampola parameter, \(\alpha <1\) directly affects the shape of the wave pattern and improves its dispersion stability. Moreover, a Hamiltonian analysis of the system reveals the mapping of the energy distribution and conservation laws, phase-portraits, time-series and Maximum Lyapunov Exponent (MLE) analyses indicate a sharp threshold transition from a chaotic regime to a stable periodic regime. The analytical model has a limitation, such as a specific boundary condition and a high computational cost for extreme fractional order, but the analytical results are well verified with numerical simulations. The results provide a predictive and highly general mathematical basis for optimizing the stability and conservation of the wave in complex physical media.