Fractional generalisations of the KP equation employing the conformable or β-fractional derivative have been investigated across a range of generalised, coupled, and higher-dimensional formulations. Danladi, obtained soliton structures for the conformable space-time (n+1)-dimensional generalised KP equation using extended mapping and (G′/G²)-expansion techniques; Demirbilek, extended this line of work to a β-fractional (n+1)-dimensional generalised KP model, combining exact-solution construction with bifurcation and sensitivity analysis of the associated dynamical system [6-7]. Related conformable and beta-fractional KP formulations have been treated with expansion-based methods [9-11], bilinear and lump-solution constructions, and higher-dimensional Boussinesq-coupled extensions, several incorporating bifurcation diagnostics of the reduced planar system as a supplementary analysis rather than a primary object of study [12-14]. Across this strand, the fractional operator is consistently local (conformable or β-type, avoiding the convolution-kernel non-locality of the Riemann–Liouville and Caputo definitions), the equations are typically higher-dimensional or multi-term generalisations of the classical KP equation rather than the equation considered here, and -with the partial exception of dedicated bifurcation studies -the reduced ODE is analysed principally as a vehicle for solution construction rather than as an autonomous dynamical system in its own right.

A broad repertoire of analytical techniques has been applied to fractional KP-type equations, including local-fractional analytical methods, time-fractional expansion techniques for conformable parabolic and Phi-4-type equations, truncated M-fractional approaches, and Hirota bilinear constructions for coupled or M-fractional systems [15-17]. These studies establish that closed-form travelling-wave solutions -typically periodic, kink, or soliton profiles -are obtainable for a wide variety of fractional KP variants, with parallel exact-solution and ansatz-based constructions reported for other conformable time-fractional nonlinear models, including Jimbo–Miwa, Zakharov–Kuznetsov, Cahn–Allen/Cahn–Hilliard, coupled Schrödinger, Kudryashov–Sinelshchikov, Klein–Gordon, and strain-wave equations [18-26], confirming the general tractability of fractional KP-type operators across equation classes. The methodological emphasis throughout this strand is on solution generation: the reduced ODE is manipulated only to the extent required to extract an explicit closed form, and its independent mathematical structure -equilibria, invariants, or orbit classification -is not, in the majority of these studies, treated as a separate object of analysis.

A smaller but mathematically more structural body of work addresses the qualitative dynamics of fractional KP-type travelling-wave equations directly. Spectral and analytical studies of transverse instability in fractional KP-type models have been carried out, and bifurcation and chaos diagnostics have been applied to several fractional KP variants, including shallow-water reductions. These contributions establish that the reduced ODEs associated with fractional KP-type equations do admit rich dynamical behaviour -multiple equilibria, bifurcation with respect to physical or fractional-order parameters, and in some formulations chaotic response under external perturbation. This strand demonstrates that qualitative-dynamics analysis of fractional KP-type reduced equations is an active and recognised line of enquiry; however, the specific planar construction developed in Section: Qualitative Dynamics of the Reduced Travelling-Wave Equation of the present paper -an explicit equivalent first-order system, a conserved energy first integral in closed form, and a proved correspondence between a named exact solution family and a homoclinic orbit of a classified saddle equilibrium -is not present in the surveyed bifurcation-oriented studies for the conformable space-time KP equation considered here.

Lie symmetry reduction and conservation-law construction constitute a further established methodology for fractional KP-type systems. Kumar, performed invariance analysis and derived conservation laws for a (2+1)-dimensional fractional KP system under the Riemann–Liouville derivative; subsequent studies have extended Lie symmetry methods to time-fractional KP variants and examined admitted symmetry algebras for general fractional KP formulations [8]. Symmetry-based conservation laws differ in character from the energy first integral constructed in Section: Conserved Energy First Integral of the present paper: the former are typically derived from the variational or multiplier structure of the full fractional PDE, whereas the first integral obtained here is a direct consequence of the reduced ODE’s autonomous planar structure and is used explicitly to prove the homoclinic-soliton correspondence of Section: Homoclinic Connection and the Bright Soliton, a use not present in the symmetry-based literature reviewed.

Physical interpretations attached to fractional KP-type equations span plasma physics, shallow-water hydrodynamics, cold bosonic atomic systems, fluid mechanics, and neuronal membrane dynamics, reflecting the broad applicability of the underlying mathematical structure. These studies consistently interpret the fractional order as a memory or non-locality index, consistent with the interpretation adopted in Section: Physical Interpretation of the Obtained Results of the present paper, but the physical interpretation offered in each case is typically tied to the specific solution family constructed rather than to a phase-plane classification of the type developed here.

Table 1: Comparative Summary of Representative Literature on Fractional KP-Type Equations

Note: Table entries in rows 1-18 are drawn from the literature database compiled for this study; full bibliographic detail is provided for the three entries independently verified and cited in the text ([6,7,8]).

Table 4: Classification of Contributions Established within this Manuscript

The conformable fractional derivative, introduced by Khalil, Al Horani, Yousef, and Sababheh [27] and further developed by Abdeljawad [28], provides a local definition of fractional differentiation that avoids the convolution-kernel structure of the Riemann–Liouville and Caputo operators.

Definition 1. Let f : (0, ∞) → ℝ and 0 < α ≤ 1. The conformable fractional derivative of order α of f at t > 0 is defined by

Tαf(t)=limε→0(f(t+εt1−α)−f(t)ε),t>0(1)

Provided the limit exists. When f is differentiable in the classical sense, Definition 1 reduces to Tᵅ(f)(t) = t¹⁻ᵅ (df/dt)(t), so that the conformable derivative of order α is the classical derivative rescaled by a power-law weight in the independent variable. This local representation is the mechanism by which the conformable operator retains the classical differentiation calculus while still carrying an explicit fractional-order parameter.

Let f and g be α-differentiable functions for some α and let λ, μ ∈ ℝ. The following properties, established in [27,28], are used throughout the derivations of Sections Methodology, Formulation and Exact Analytical Solutions and Qualitative Dynamics of the Reduced Travelling-Wave Equation.

Property (v) is the property exploited in the travelling-wave transformation of Section Methodology: it permits the fractional derivatives in the governing equation to be converted directly into classical derivatives of a single-variable profile function u(ξ), without any auxiliary integral representation.

The sine–cosine method, introduced by Wazwaz [29], is a direct algebraic technique for constructing exact travelling-wave solutions of nonlinear evolution equations, requiring no auxiliary linear equation, perturbation parameter, or truncated series. It is applied here to a nonlinear fractional partial differential equation of the general form.

P(u,Dtαu,Dxβu,Dyγu,DtαDtαu,DtαDxβu,DxβuDxβu,DyγuDyγu,…)=0(2)

where u = u(x,y,t) and Dₜᵅ, Dₓᵝ, Dᵧᵞ denote conformable fractional derivatives of orders α, β, γ with respect to t, x, and y respectively.

The wave variable

ξ=(aβ)xβ+(bγ)yγ−(cα)tα(3)

is introduced, with a, b, c non-zero real constants. By property (v) of Section: Calculus Properties of the Conformable Derivative, setting u(x, y, t) = u(ξ) converts each conformable partial derivative in Equation (2) into a classical ordinary derivative of u with respect to ξ, reducing the fractional PDE to the ODE

R(u′,u″,u‴,…)=0(4)

which is subsequently integrated as many times as the equation structure permits, with integration constants set to zero, to obtain the lowest-order equivalent ODE.

Following Wazwaz, the solution is sought in one of two complementary trial forms [29]:

u(x,y,t)=ψsinρ(μξ),|ξ|≤πμ(5)

u(x,y,t)=ψcosρ(μξ),|ξ|≤π2μ(6)

where ψ, μ, and ρ are real parameters determined by the method.

Differentiating the sine ansatz twice with respect to ξ gives

u′(ξ)=ψρμsinρ−1(μξ)cos(μξ)(7)

u″(ξ)=ψρ(ρ−1)μ2sinρ−2(μξ)−ψρ2μ2sinρ(μξ)(8)

and analogously for the cosine ansatz,

u′(ξ)=−ψρμcosρ−1(μξ)sin(μξ)(9)

u″(ξ)=ψρ(ρ−1)μ2cosρ−2(μξ)−ψρ2μ2cosρ(μξ)(10)

Substituting the chosen ansatz and its derivatives into the reduced ODE (4), the exponent ρ is fixed by balancing the highest-degree nonlinear term against the highest-order derivative term. The remaining coefficients are then obtained by collecting terms of like powers of sin(μξ) or cos(μξ) and setting each coefficient to zero, yielding a finite algebraic system in ψ, μ, and ρ that is solved explicitly in Section Formulation and Exact Analytical Solutions.

The equation under investigation is the space-time fractional KP equation in the conformable sense,

DtαDxβu+Dx4βu−6Dxβ(uDxβu)+3Dy2γu=0,0α,β,γ≤1(11)

Equation (11) provides a conformable fractional extension of the KP-type model used in this study. Because the conformable derivative employed here is local for differentiable functions, the fractional orders α, β, and γ should be interpreted mathematically through the coordinate weighting in the travelling-wave transformation rather than as direct measures of memory or non-local transport. Any stronger physical interpretation would require an additional constitutive or empirical model.

Applying transformation (3) to Equation (11), integrating once with respect to ξ and discarding the constant of integration, gives

a4u‴−6a2uu′+(3b2−ac)u′=0(12)

A second integration, again with the integration constant set to zero, yields the reduced travelling-wave equation

a4u″−3a2u2+(3b2−ac)u=0(13)

Equation (13) is the central mathematical object of this paper. Sections: Application of the Cosine Ansatz, Parameter Determination and Exact Analytical Solutions construct its exact solutions via the sine–cosine ansatz; Section: Qualitative Dynamics of the Reduced Travelling-Wave Equation analyses Equation (13) directly as a planar dynamical system, independently of any particular ansatz.

Substituting u(ξ) = ψ cosᵖ(μξ) and Equation (10) into Equation (13) gives

a4ψμ2ρ(ρ−1)cosρ−2(μξ)−a4ψρ2μ2cosρ(μξ)−3a2ψ2cos2ρ(μξ)+(3b2−ac)ψcosρ(μξ)=0(14)

Matching the exponents ρ−2 and 2ρ of the two terms carrying the highest-degree power of cos(μξ) forces

ρ−2=2ρ⇒ρ=−2(15)

With ρ = −2 fixed, Equation (14) contains only two distinct powers of cos(μξ), namely cos⁻⁴(μξ) and cos⁻²(μξ); collecting their coefficients separately and equating each to zero produces the algebraic system

a4ψμ2ρ(ρ−1)=3a2ψ2(16)

−a4ψρ2μ2=−(3b2−ac)ψ(17)

This section treats Equation (13) as an autonomous second-order ordinary differential equation in its own right, independently of the sine–cosine ansatz of Section: Formulation and Exact Analytical Solutions, and characterizes its solution structure through the associated planar dynamical system. This is the principal theoretical contribution of the present paper: whereas Section: Formulation and Exact Analytical Solutions constructs six explicit closed-form solutions, this section explains, at the structural level, why solutions of exactly two morphological types -bounded solitary waves and unbounded periodic or singular waves -are the ones admitted by Equation (13), and identifies the precise phase-plane object occupied by each solution family.

Lemma 1: (Equivalent planar system). Let v(ξ) = u′(ξ). Equation (13) is equivalent to the autonomous planar system

u′=v(27)

v′=3a2u2−[3b2−aca4]u(28)

in the sense that a pair (u(ξ), v(ξ)) solves system (27) – (28) if and only if u(ξ) solves Equation (13).

Differentiating the first equation of the system with respect to ξ and substituting the second gives u″ = v′ = (3/a²)u² − [(3b² − ac)/a⁴]u, which rearranges directly to a⁴u″ − 3a²u² + (3b² − ac)u = 0, identical to Equation (13). Conversely, given any solution u(ξ) of Equation (13), the pair (u(ξ), u′(ξ)) satisfies (27) by the definition of v and satisfies (28) because u″ = (3/a²)u² − [(3b² − ac)/a⁴]u is precisely Equation (13) rearranged. This establishes the stated equivalence in both directions.

Equilibria of (27)-(28) satisfy v = 0 and (3/a²)u² − (D/a⁴)u = 0, hence u[(3/a²)u − D/a⁴] = 0. Therefore u = 0 or u = D/(3a²). For D ≠ 0 these give the two distinct equilibria E₀ and E₁ stated above; for D = 0 they coincide at the origin. The Jacobian at (u, 0) is

J ( u ) = 0 , 1 6 a 2 u − D a 4 , 0

Its characteristic equation is λ 2 = 6 a 2 u * − D a 4 at an equilibrium u * . At E 0 , λ 2 = − D a 4 . At E 1 , with u * = D 3 a 2 , λ 2 = D a 4 . Thus, for D > 0 , E 0 has purely imaginary eigenvalues and is a centre while E 1 has real eigenvalues of opposite sign and is a saddle. For D < 0 , the classification is exchanged. For D = 0 , the Jacobian at the single equilibrium is 0 1 0 0 and both eigenvalues are zero, so the equilibrium is non-hyperbolic and requires separate nonlinear analysis.

Proposition 2: (First integral). The scalar function

E(u,v)=12v2−u3a2+[D2a4]u2(30)

is conserved along every trajectory of system (27) – (28), i.e. dE/dξ ≡ 0.

Differentiating (30) along a trajectory, dE/dξ = v · v′ − (3/a²)u² · u′ + (D/a⁴)u · u′ = v · [v′ − (3/a²)u² + (D/a⁴)u]. By Equation (28), the bracketed term equals v′ minus itself, hence zero, so dE/dξ = 0 identically.

Proposition 2 shows that Equation (13) is completely integrable at the planar level: every trajectory of (27) – (28) lies on a level curve E(u, v) = E₀ of the conserved quantity (30), and the global qualitative behaviour of solutions of Equation (13) is determined entirely by the topology of these level curves relative to the equilibria of Theorem 3.

Corollary 1 (Soliton–homoclinic correspondence). In the soliton regime D < 0, the trajectory (u₅(ξ), u₅′(ξ)) generated by solution (25) satisfies E(u₅,u₅′) ≡ 0 and u₅(ξ) → 0, u₅′(ξ) → 0 as ξ → ±∞. It is therefore precisely the homoclinic orbit of the saddle equilibrium E₀ = (0,0) identified in Theorem 3.

Proof / verification. Since E₀ = (0,0) gives E(0,0) = 0 by direct evaluation of (30), the level set E(u,v) = 0 is the natural candidate for a homoclinic loop based at the saddle. Symbolic substitution of solution (25) and its derivative into Equation (30) gives E(u₅(ξ),u₅′(ξ)) = 0 identically for all ξ verified independently, for representative parameter values, to machine precision. The asymptotic decay sech²(·) → 0 as ξ → ±∞ gives u₅ → E₀ in both directions of ξ, the defining property of a homoclinic orbit.

This correspondence gives a structural, phase-plane explanation -rather than a purely algebraic one -for the existence of the bright soliton: it exists precisely because D < 0 forces the origin into a saddle configuration, and a homoclinic loop can form only at a hyperbolic (saddle-type) equilibrium, never at a centre. The remaining solution families occupy the complementary regions of the same phase portraits. The singular soliton u₆, obtained from the same parameter values via the μ → iμ substitution combined with the csch² rather than sech² branch, corresponds to the unbounded trajectory on the far side of the same separatrix, escaping to infinity in finite ξ, consistent with its divergence at ξ = 0. In the complementary regime D > 0, the origin is a centre, so no homoclinic loop exists there; the periodic solutions u₁–u₄, which diverge at the zeros of cos(·) or sin(·) rather than remaining bounded, correspond to unbounded trajectories lying outside the closed orbits encircling the centre at E₀. The bounded periodic orbits that do encircle this centre would instead be represented by Jacobi elliptic function solutions, which lie outside the sine–cosine ansatz and are noted here only to delineate precisely the boundary of the present method's solution space.

As a further, literature-independent verification (the third route summarized in Table 5), the ratio between the soliton amplitude and the nontrivial equilibrium value obtained here, ψ/u* = [D/(2a²)]/[D/(3a²)] = 3/2, reproduces the corresponding ratio for the classical Korteweg–de Vries solitary wave, whose amplitude is likewise three-halves of the nonzero fixed point of its travelling-wave equation [34]. Agreement with this independent classical benchmark supports the corrected parameter values of Equation (18) and the internal consistency of the qualitative-dynamics construction.

Corollary 2 (Recovery of the classical KP equation and fractional-order invariance of the phase portrait). As α, β, γ → 1, Equation (11) reduces identically to the classical KP equation

∂t(ux)+uxxxx−6∂x(uux)+3uyy=0

with ξ → ax + by − ct. The reduced ODE (13), the parameter relations (18), the six solutions (21)-(26), the planar system (27)–(28), its equilibria, the first integral of Proposition 2, and the homoclinic correspondence of Corollary 1 retain their displayed form in this limit because the fractional orders enter through the travelling-wave coordinate rather than the reduced algebraic coefficients.

By Definition 1 and property (v) of Section 3.2, Tᵅ(f)(t) → df/dt as α → 1 for differentiable f, and analogously for the spatial operators as β, γ → 1. The corrected Equation (11) therefore reduces term-by-term to ∂ₜ(uₓ) + uₓₓₓₓ − 6∂ₓ(uuₓ) + 3uᵧᵧ = 0. The wave variable (3) reduces to ξ = ax + by − ct. Equations (12)-(13) were derived from Equation (11) and the transformation (3) using algebraic and classical differential manipulations that do not contain α, β, or γ explicitly. Hence the reduced ODE, the parameter relations (18), the six solutions (21)-(26), the planar system (27)-(28), the equilibrium classification, the first integral, and the homoclinic correspondence retain the same displayed form in the classical limit α = β = γ = 1.

Corollary 2 establishes that the conformable fractional KP equation is a genuine generalization of the classical model at the level of the governing PDE and the wave variable, while the algebraic and dynamical structure of the reduced travelling-wave problem is fractional-order-invariant. This distinction is used directly in the physical interpretation of Section Physical Interpretation of the Obtained Results: the fractional parameters modulate how physical space and time map onto ξ, not the intrinsic solution-family structure available at fixed ξ.

The exact solutions of Section: Formulation and Exact Analytical Solutions, together with the phase-plane structure of Section: Qualitative Dynamics of the Reduced Travelling-Wave Equation, admit a physical reading that extends beyond their mathematical form. Each subsection below interprets one aspect of the mathematics established previously, following the discipline that every physical statement made here is traceable to a specific theorem, proposition, or corollary rather than to independent speculation.

The expressions for travelling waves u₁,u₄ are inverse-square trig functions. The phase plane for regime D = 3b² − ac > 0 has the origin as a centre, and the nontrivial equilibrium as a saddle. These four sine-cosine solutions in the form of equations (21), (22), (23), (24) are singular at the zeros of the involved trigonometric functions and thus, they are not bounded periodic closed orbits around the centre. The closed periodic orbits found for the planar system are contained in a larger solution class, including the Jacobi-elliptic representations, whose ansatz is not in the current form of sine-cosine. For α, β, γ < 1, these fractional indices are not a direct measure of memory or sub-diffusion, but rather a report of the coordinate transformation between (x,y,t) and ξ for the chosen parameter set that should be reported as such.

(aβ)xβ+(bγ)yγ−(cα)tα introduces power-law stretching in both the spatial and temporal

The solution which is localised hyperbolic is u₅ for the regime D < 0. Corollary 1 shows that the trajectory (u₅(ξ), u₅′(ξ)) is a homoclinic orbit that's a solution of the conserved first integral and goes to the saddle equilibrium E₀ = (0, 0), as ξ → ±∞. It provides a phase-plane analysis to account for the existence of the bright solitary-wave profile that is modelled above by the sine-cosine algebraic construction, but remains independent of that construction. When using the conformable formulation as in this work, and assuming the symmetry interpretations below, there are no changes in the equilibrium classification nor homoclinic mechanism up to the ξ-plane level. Their effect, on the other hand, is represented by the mapping from physical coordinates (x, y, t) to ξ.

The principal physical consequence of the conformable fractional derivative is the introduction of memory effects into the wave dynamics. The operator Tᵅ(f)(t) = t¹⁻ᵅ(df/dt) carries an implicit dependence on the accumulated evolution of the coordinate through the power-law weight t¹⁻ᵅ, which decays slowly as t increases when α < 1. Effects of this character are documented in geophysical fluid dynamics, biological fluid mechanics, and electrical signal propagation through media with fractal geometry. The fractional order α is naturally read as an index of hereditary strength, with α = 1 recovering the memoryless limit (Corollary 2) and α → 0 corresponding to maximal long-range memory.

The three fractional order parameters control, respectively, the scaling factor for the temporal coordinate (α), the scaling factor for the x-directional coordinate (β), and the scaling factor in the y-directional coordinate (γ) in the travelling-wave representation. If the underlying function is differentiable, the conformable derivative is a local one, and is given in the form of a coordinate-weighted classical derivative. Treating the fractional orders one could therefore say the mapping from the physical variables (x, y, t) to the travelling-wave coordinate ξ is modified. If α = β = γ = 1, then the classical limit is regained where these are integer coordinates. Thus values of α, β, and γ that are less than 1 affect the coordinate parametrisation of the travelling wave; on the other hand, Corollary 2 provides an indication that they do not affect the centre/saddle classification of the reduced planar system. The interpretation of memory or history from the conformable derivative should not be assumed, as it involves a true non-local fractional operation or another constitutive model.

Two representative parameter sets are used for the surface plots Set A (α = β = γ = 0.5) and Set B (α = 0.8, β = γ = 0.5), both with a = 1, b = 2, c = 1, i.e. D = 3b² − ac = 11 > 0 (periodic regime) together with a soliton-regime parameter set a = 1, b = 2, c = 13, for which D = 3(2²) − (1)(13) = −1 < 0, used for Figures 2-6.