Download PDF
ABSTRACT
The time-fractional Kadomtsev–Petviashvili (KP) equation and describes weakly nonlinear, weakly dispersive waves with slowly varying transverse profile, and it occurs in the shallow-water hydrodynamics, in nonlinear optical fibre transmission and in ion-acoustic plasma waves. The equation is investigated here in the framework of conformable fractional derivative where the classical product, quotient and chain rules of ordinary calculus are preserved, and thus a fractional travelling-wave transformation is applied to get the governing PDE directly transformed to a nonlinear ODE. This equation is then solved in turn with the sine-cosine method and the tanh-coth method, yielding 5 periodic and bright-soliton solutions, and 7 solutions of the kink, singular and further soliton type, respectively. All the solutions are verified by direct substitution and the errors in the residuals are less than 10−8 in the range of parameters tested. The paper then goes beyond these exact solutions to give a complete dynamical-systems treatment of the reduced equation. The ODE is written as a planar Hamiltonian system, the equilibrium points are classified, a stability theorem is established for both branches, and it is demonstrated that the value k = −1 is a true bifurcation point where the periodic-wave family (k > −1) and bright soliton family (k < −1) switch stability. A spectral analysis is also performed: The bright soliton is spectrally stable, as is verified by linearising around the bright soliton and then calculating the discrete spectrum of the Pöschl–Teller potential, and the periodic branch is shown to be Floquet stable for the appropriate parameter range. All of these conclusions are confirmed by independent Lyapunov exponent calculations in the planar system. It is found that the width and smoothness of the wave profile depends on the parameter α, the bifurcation structure is found to remain the same, and the classical KP soliton solutions are recovered as α approaches 1.
Keywords: Time-Fractional Kadomtsevpetviashvili Equation; Conformable Fractional Derivative; Sine-Cosine Method; Tanh-Coth Method; Soliton Solutions; Stability Analysis; Bifurcation; Spectral Analysis; Lyapunov Exponents