Soliton dynamics of the Landau–Ginzburg–Higgs and generalized Kadomtsev–Petviashvili modified equal width Burgers equations for the truncated M-fractional derivative

Fractional partial differential equations are essential for a more accurate description of complex physical phenomena than their integer-order versions. The equations offer better insight into different dynamic processes in physics and engineering with the inclusion of the memory effects and non-local interactions associated with fractional derivatives. In the present research, we are interested in the soliton dynamics of two important fractional partial differential equations: the Landau–Ginzburg–Higgs equation and the generalized Kadomtsev–Petviashvili modified equal width Burgers equation for the truncated M-fractional derivative. The Landau–Ginzburg–Higgs equation is central to field theory and condensed matter physics, used to model phase transitions, super-conductivity, and other critical phenomena. The generalized Kadomtsev–Petviashvili modified equal width Burgers equation is an important model for the generation and analysis of long-wave structures in nonlinear dispersive and dissipative media. We use the extended hyperbolic function method, a robust analytical method for finding exact solutions to nonlinear fractional differential equations, to study these equations. In the present analysis, we build a range of solution types such as bright solitons, dark solitons, kink-type waves, dark-singular solitons and periodic solutions, which reveal the qualitative richness of the models under the effect of the truncated M-fractional derivative. The investigation also examines how various fractional-order parameters influence the derived solutions and how the soliton structures change as the fractional order changes from non-integer to integer. To enable a thorough comprehension of these effects, we depict two-dimensional and three-dimensional graphical plots of solutions for different fractional orders. These plots emphasize the progressive evolution of soliton profiles and the effects of fractional differentiation on wave propagation.