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ABSTRACT
Problem
The space-time fractional Kadomtsev–Petviashvili (KP) equation, formulated with the conformable fractional derivative, models weakly nonlinear, weakly dispersive wave propagation in media exhibiting memory and non-local transport, yet the mathematical structure underlying its travelling-wave solutions -as opposed to the solutions themselves has not been systematically characterised.
Gap
Existing studies on conformable and β-fractional KP-type equations construct explicit solution families through expansion, bilinear, or symmetry-based methods, but do not establish the equivalent planar dynamical system of the reduced travelling-wave equation, classify its equilibria, or identify which exact solutions correspond to which phase-plane orbits.
Objective and Methods
A fractional travelling-wave transformation reduces the governing equation to a nonlinear ordinary differential equation, to which the sine–cosine method is applied, yielding six exact closed-form solutions. The reduced equation is then analysed independently as a planar autonomous system. Main results. The equilibria of the planar system are classified by linear stability as centre or saddle according to the sign of D = 3b² - ac; a conserved energy first integral is derived; and the bright soliton solution is proved to coincide exactly with the homoclinic orbit of the saddle equilibrium, giving a structural explanation for its existence that is independent of the algebraic derivation and Verification. Every derivation is verified by direct substitution, symbolic differentiation, and recovery of the classical limit α,β,γ → 1; an inconsistency in the amplitude and width constants present in an earlier derivation is identified and corrected, with the corrected values validated through three independent routes, including agreement with the classical Korteweg-de Vries amplitude-to-equilibrium ratio. Physical significance. The fractional parameters α,β,γ are shown to reparametrise the wave variable without altering the qualitative (centre/saddle) classification of the reduced system, separating the roles of memory-driven propagation and intrinsic solution-family structure.
Contribution
This work integrates exact-solution construction with rigorous planar qualitative dynamics for the conformable fractional KP equation, extending the largely solution-generating scope of prior literature toward a structural, theorem-driven mathematical treatment.
Keywords: Fractional Kadomtsev-Petviashvili Equation; Conformable Fractional Derivative; Travelling-Wave Analysis; Qualitative Dynamics; Dynamical Systems; Phase-Plane Analysis; Homoclinic Orbit; Analytical Framework; Sine-Cosine Method; First Integral