EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6624 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generalized Riccati Equation Mapping Method: 27 Solutions for Josephson Junctions and Nonlinear Optics Models Sirasrete Phoosree1, Jiraphat Phookwantong2, Marisa Senmoh2, Jiraporn Sanjun3, Weerachai Thadee2,∗ 1 Education Program in Mathematics, Faculty of Education, Suratthani Rajabhat University, Mueang, Suratthani, 84100, Thailand 2 Department of General Education, Faculty of Liberal Arts, Rajamangala University of Technology Srivijaya, Mueang, Songkhla, 90000, Thailand 3 Mathematics Program, Faculty of Science and Technology, Suratthani Rajabhat University, Mueang, Suratthani, 84100, Thailand Abstract. The generalized Riccati equation mapping method, combined with Jumarie’s Rie- mann–Liouville fractional derivative, is employed to obtain 27 distinct travelling wave solutions for two nonlinear space-time fractional (2+1)-dimensional models: the cubic Klein-Gordon equa- tion, which is crucial for describing the propagation of fluxions in Josephson junctions and the Ablowitz-Kaup-Newell-Segur equation, a model with key applications in the field of nonlinear op- tics. Among the diverse solutions obtained, kink and periodic wave behaviors were identified and graphically represented through three-dimensional, two-dimensional, and contour plots to illustrate their dynamic characteristics. In comparison to other alternative methods, the generalized Riccati equation mapping method provides a broader array of solutions to these equations. This technique illustrates the solution’s behavior as a wave in various shapes, as evidenced by the findings of this research. 2020 Mathematics Subject Classifications: 35C07, 35G20, 35R11 Key Words and Phrases: Fractional nonlinear partial differential equations, generalized Ric- cati equation mapping method, travelling wave solutions, cubic Klein-Gordon equation, Ablowitz- Kaup-Newell-Segur equation 1. Introduction Nonlinear evolution equations (NLEEs) emerge in various scientific and technical do- mains, such as particle physics, biophysics, beam propagation, ecology, nonlinear optics, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6624 Email addresses: sirasrete.pho@sru.ac.th (S. Phoosree), jirapat.p@rmutsv.ac.th (J. Phookwantong), marisa.s@rmutsv.ac.th (M. Senmoh), jiraporn.san@sru.ac.th (J. Sanjun), weerachai.t@rmutsv.ac.th (W. Thadee) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 2 of 24 fluid mechanics, marine engineering, elasticity theory, solid-state physics, and quantum field theory. They offer essential models for elucidating intricate real-world events. Exact solutions to nonlinear evolution equations are essential, as they provide crucial insights into the qualitative behavior of nonlinear systems and act as benchmarks for validating numerical approaches and simulations. Numerous strong and successful approaches have been developed to address the NLEEs, including the Kudyashov method [1], generalized Kudryashov method [2], first integral method [3], the Riccati-Bernoulli sub-ode method [4, 5], generalized Riccati equation mapping method [6], modified Khater method [7], gen- eralized Khater method [8], Sardar sub-equation method [9], modified Sardar sub-equation method [10, 11], exponential rational function method [12, 13], G′/G-expansion method [14, 15], G′/G2-expansion method [16, 17], functional variable method [18, 19], and sim- ple equation method [20, 21]. In the process of finding an exact solution of the NLLEs, the researchers convert the nPDEs to ordinary differential equations (ODEs) using many approaches, including the conformable fractional derivative [12], Caputo [22, 23], Riemann- Liouville [24], modified Riemann-Liouville [25], and Jumarie’s modified Riemann-Liouville [26], etc. One of the definitions that is used the most often in the field of study is the Jumarie’s Riemann-Liouville fractional derivative [6, 21]. Travelling wave solutions are fundamental in the analysis of nonlinear partial differen- tial equations (PDEs), as they reduce complex spatiotemporal dynamics into ordinary dif- ferential equations and simultaneously provide physically meaningful descriptions of wave phenomena. In solid-state physics, kink-type travelling waves model fluxon propagation in Josephson junctions and crystal dislocations [27], while in nonlinear optics, AKNS-type equations capture optical solitons and related interactions [28]. The importance of such solutions is amplified in the fractional-order setting, where memory and hereditary effects emerge and yield more realistic models of anomalous transport and dispersive processes. By adjusting the fractional order α, one can tune wave velocity and profile beyond what is possible in the classical case, thereby enriching the diversity of exact solutions. Although several analytical methods, including the (1/G′)-expansion and functional variable tech- niques [28], have been employed to construct travelling waves, these approaches typically yield only limited classes of hyperbolic or trigonometric forms. In contrast, the generalized Riccati equation mapping method adopted in this work generates a broader spectrum of hyperbolic, trigonometric, and rational travelling wave solutions, underscoring both its mathematical versatility and physical relevance. The Riemann-Liouville derivative of Jumarie was presented in the following manner in the year 2006 [29]: Definition 1. Riemann-Liouville’s fractional derivative of Jumarie can be expressed in the following way: Dα t Θ(t) =  Θ(t) , α = 0, 1 Γ(1− α) d dt ∫ t 0 (t− β)−α(Θ(β)−Θ(0))dβ , 0 < α < 1, dn dtn Dα−n t Θ(t) , n ≤ α < n+ 1, n ≥ 1, (1) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 3 of 24 where represents an order of the fractional derivative. The following is a list of features that were raised in 2009 regarding various attributes of the fractional derivative order of Jumarie’s Riemann-Liouville [30]: Dα xx Ω = Γ(Ω + 1) Γ(Ω− θ + 1) xΩ−α, Ω ≥ 0, (2) Dα x [Θ(x)Υ(x)] = Θ(x)Dα xΥ(x) + Υ(x)Dα xΘ(x), (3) Dα xΘ[Υ(x)] = Θ′ Ω[Υ(x)][Dα xΥ(x)] (4) = Dα ΥΘ[Υ(x)][Υ′(x)]α. (5) These represent direct results that result from the equality dαx(t) = Γ(1 + α)dx(t). This is true for functions that are not susceptible of being differentiated. In Equations (3) and (4), the function is not differentiable, however in Equation (5), it is differentiable. It is possible to modify a variety of nonlinear phenomena by using the nonlinear (2+1)- dimensional cubic Klein-Gordon (cKG) equation. These phenomena [27], include the propagation of fluxions in Josephson junctions, the propagation of dislocation in crystals, and the behavior of elementary particles. The equation that represents the nonlinear (2+1)-dimensional cKG equation is as follows [20, 31]: uxx + uyy − utt + δu+ λu3 = 0, (6) where u = u(x, y, t) , x and y are considered to represent the distance of propagation and t represents the time, δ and λ are non-zero constants. Sanjun and Chankaew [20] have developed four distinct exact solutions to the cKG equation. These solutions were realized by the use of the simple equation method. The solutions have the form of hyperbolic and trigonometric form. Ablowitz, Kaup, Newell, and Segur came up with the nonlinear (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equations in 1970. These equations were inspired by the applications to nonlinear optics, which were the primary motivation for their cre- ation. It is possible to convert the AKNS equation to a few nonlinear evolution equations, including the nonlinear Schrödinger equation, the sine-Gordon equations, the KdV equa- tion, and a few more. The nonlinear (2+1)-dimensional AKNS equation of the fourth order, which is nonlinear, with the parameter ξ in this form [1, 28], 4uxt + uxxxt + 8uxuxy + 4uxxuy − ξuxx = 0, (7) where u = u(x, y, t), x and y are considered to represent the distance of propagation and t represents the time, ξ is non-zero constants. The AKNS problem has two solutions, which were created by Durur and Yokus [28] in 2021 by using the (1/G′)-expansion method. In terms of form, the solutions are hyperbolic type. Several analytical methods have been applied to fractional nonlinear equations, yet they typically generate only a narrow class of hyperbolic or trigonometric solutions. Their S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 4 of 24 effectiveness also decreases when extended to higher-dimensional or fractional-order prob- lems. This limitation reveals a clear research gap: the need for a systematic approach that can provide a wider variety of exact solutions. In this work, we address this is- sue by employing the generalized Riccati equation mapping method with Jumarie’s Rie- mann–Liouville fractional derivative to derive 27 analytic solutions of the cKG and AKNS equations, including kink-type and periodic wave behaviors. The following is the outline of the paper: within Section 2, a comprehensive description of the algorithm for the generalized Riccati equation mapping method may be found. In Section 3, the method is used to solve the nonlinear evolution equations that were explored in the previous section. The discussion and results are presented in Section 4. To summarize, the conclusion is presented in Section 5. 2. Algorithm of Generalized Riccati Equation Mapping Method For the purpose of solving fractional partial differential equations, we will be discussing the generalized Riccati equation mapping method in this section. An illustration of the general form of fractional PDEs is as follows: Ψ(u, ux, uy, ut, uxx, uyx, utx, . . .) = 0, (8) where Ψ is a polynomial of u(x, y, t) and its partial derivatives, devoting particular at- tention to the highest order derivatives and nonlinear terms in relation to the functions. Within the framework of the generalized Riccati equation mapping method, the following is an overview of the first step involved: Step 1. Wave transforming The traveling wave solution to fractional partial differential equations is a solution that meets the following conditions: u(x, y, t) = U(β), β = axα Γ(α+ 1) + byα Γ(α+ 1) − ctα Γ(α+ 1) , (9) where β is a broad word for a wave that is moving, and c is a constant that represents the wave’s velocity. We refer to a wave as stationary when c equals zero. When the value of c is more than zero, the wave goes in a positive direction, whereas when c is less than zero, the wave moves in a negative direction. Taking Equation (8) and decreasing it to an ODE Φ ( U, dU dβ , d2U dβ2 , d3U dβ3 , . . . ) = 0, (10) where Φ is a polynomial in U(β) and its derivatives. Step 2. Assumption of solution When written out in the form of a finite series, the solution to Equation (10) is expressed as U(β) = N∑ i=0 ρiH i(β), (11) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 5 of 24 where ρi are real constants and ρN ̸= 0, and H(β) is dependent on the generalized Riccati equation mapping method [6], the following is what it says. H ′(β) = r + cH(β) + kH2(β), (12) where r, s and k are the constants that are not always zero. For Equation (12), there are twenty-seven different solutions available. Type I: when φ = s2 − 4rk > 0 and sk ̸= 0 or rk ̸= 0, we obtained H1(β) = − 1 2k ( s+ √ φ tanh (√ φ 2 β )) , (13) H2(β) = − 1 2k ( s+ √ φ coth (√ φ 2 β )) , (14) H3(β) = − 1 2k (s+ √ φ (tanh ( √ φβ)± i sech ( √ φβ))) , (15) H4(β) = − 1 2k (s+ √ φ (coth ( √ φβ)± csch ( √ φβ))) , (16) H5(β) = − 1 4k ( 2s+ √ φ ( tanh (√ φ 4 β ) + coth (√ φ 4 β ))) , (17) H6(β) = − 1 2k ( s− √ φ(µ2 + σ2)− µ √ φ cosh (√ φβ ) µ sinh (√ φβ ) + σ ) , (18) H7(β) = − 1 2k ( s− √ φ(σ2 − µ2) + µ √ φ sinh (√ φβ ) µ cosh (√ φβ ) + σ ) , (19) where µ and σ are two real constants that are not zero, and they meet the condition that σ2 − µ2. H8(β) = 2r cosh (√ φ 2 β ) √ φ sinh (√ φ 2 β ) − s cosh (√ φ 2 β ) , (20) H9(β) = −2r sinh (√ φ 2 β ) s sinh (√ φ 2 β ) −√ φ cosh (√ φ 2 β ) , (21) H10(β) = 2r cosh (√ φβ ) √ φ sinh (√ φβ ) − s cosh (√ φβ ) ± i √ φ , (22) H11(β) = 2r sinh (√ φβ ) −s sinh (√ φβ ) + √ φ cosh (√ φβ ) ± i √ φ , (23) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 6 of 24 H12(β) = 4r sinh (√ φ 4 β ) cosh (√ φ 4 β ) −2s sinh (√ φ 4 β ) cosh (√ φ 4 β ) + 2 √ φ cosh2 (√ φ 4 β ) −√ φ , (24) Type II: when φ = s2 − 4rk < 0 and sk ̸= 0 or rk ̸= 0, we obtain H13(β) = − 1 2k ( s− √ −φ tan (√ −φ 2 β )) , (25) H14(β) = − 1 2k ( s+ √ −φ cot (√ −φ 2 β )) , (26) H15(β) = − 1 2k ( s− √ −φ ( tan (√ −φβ ) ± sec (√ −φβ ))) , (27) H16(β) = − 1 2k ( s+ √ −φ ( cot (√ −φβ ) ± csc (√ −φβ ))) , (28) H17(β) = − 1 4k ( 2s− √ −φ ( tan (√ −φ 4 β ) − cot (√ −φ 4 β ))) , (29) H18(β) = − 1 2k ( s− ± √ −φ(µ2 − σ2)− µ √ −φ cos ( √ −φβ) µ sin ( √ −φβ) + σ ) , (30) H19(β) = − 1 2k ( s+ ± √ −φ(µ2 − σ2) + µ √ −φ sin ( √ −φβ) µ cos ( √ −φβ) + σ ) , (31) where µ and σ are two real constants that are not zero, and they meet the condition that µ2 − σ2 > 0. H20(β) = −2r cos (√ −φ 2 β ) √ −φ sin (√ −φ 2 β ) + s cos (√ −φ 2 β ) , (32) H21(β) = 2r sin (√ −φ 2 β ) −s sin (√ −φ 2 β ) + √ −φ cos (√ −φ 2 β ) , (33) H22(β) = −2r cos ( √ −φβ)√ −φ sin ( √ −φβ) + s cos ( √ −φβ)± √ −φ , (34) H23(β) = 2r sin ( √ −φβ) −s sin ( √ −φβ) + √ −φ cos ( √ −φβ)± √ −φ , (35) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 7 of 24 H24(β) = 4r sin (√ −φ 4 β ) cos (√ −φ 4 β ) −2s sin (√ −φ 4 β ) cos (√ −φ 4 β ) + 2 √ −φ cos2 (√ −φ 4 β ) − √ −φ , (36) Type III: when r = 0 and sk ̸= 0, we get H25(β) = −sψ k(ψ + cosh(sβ)− sinh(sβ)) , (37) H26(β) = −s(cosh(sβ) + sinh(sβ)) k(ψ + cosh(sβ)− sinh(sβ)) , (38) for any constant ψ that is arbitrary. Type IV: when r = s = 0 and k ̸= 0, we get H27(β) = − 1 kβ + ζ . (39) In this case, ζ is a constant that may be selected with no limitations. Step 3. Determining the integer For the purpose of obtaining the number N in Equation (11), it is necessary to achieve a balance between the highest-order derivative and the nonlinear parts. Step 4. Obtaining the solutions The parameters ρi, (i = 0, 1, 2, 3, . . . , N) and c can be determined by first gathering the coefficients of all terms that have the same order of H i, (j = 0, 1, 2, 3, . . .) and then setting those coefficients to zero. Therefore, we are the analytical answers to Equation (10), which we have constructed. S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 8 of 24 List of Symbols u(x, y, t) Dependent variable (wave function or solution). x, y, t Independent spatial and temporal variables. α Fractional order parameter, 0 < α ≤ 1. Γ(·) Gamma function. Dα x , D α t , D α xx, D α tt, . . . Jumarie’s modified Riemann–Liouville fractional derivatives with respect to x or t. a, b, c Constants in the traveling wave transformation; c denotes wave velocity. β Traveling wave variable defined in the transformation. U(β) Reduced ordinary differential equation (ODE) solution of u(x, y, t). ρi Real coefficients in the finite series solution (Equation (11)). r, s, k Parameters in the generalized Riccati equation (Equation (12)). ϕ Discriminant parameter ϕ = s2 − 4rk in Riccati equation cases. µ, σ Auxiliary constants for special Riccati solutions (Types I–II). ψ, ζ Arbitrary constants in Riccati solutions (Types III–IV). δ, λ Nonzero physical constants in the cKG equation. ξ Nonzero constant parameter in the AKNS equation. N Degree of finite series expansion in Equation (11). H(β) Riccati function satisfying Equation (12). ui(x, y, t) Exact analytic solutions of the governing equations (i = 1, . . . , 27). 3. Applications In this paper, we show the traveling wave effects of the nonlinear space and time frac- tional (2+1)-dimensional cKG equation as well as the nonlinear space and time fractional (2+1)-dimensional AKNS equation. 3.1. The Space-Time Fractional Cubic Klein-Gordon Equation The following statement provides a definition of the second-order nonlinear space-time fractional cKG equation: D2α xxu+D2α yyu−D2α tt u+ δu+ λu3 = 0, t > 0, 0 < α ≤ 1, (40) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 9 of 24 where the value of δ, λ are constants and u = u(x, y, t). Taking into consideration the fact that the solution u(x, y, t) = U(β) and implementing the transformation β = axα Γ(α+ 1) + byα Γ(α+ 1) − ctα Γ(α+ 1) , (41) where a, b, and c are constants that are not equal to zero. There was a transformation of Equation (40) into an ODE. a2 d2U dβ2 + b2 d2U dβ2 + c2 d2U dβ2 + δU + λU3 = 0. (42) The solution is represented by Equation (11) and is constructed by the use of the gener- alized Riccati equation mapping method. After that, the nonlinear term in Equation (42) and the highest-order derivative are brought into balance. Now, N = 1. Equation (11), this is what we have: U(β) = ρ0 + ρ1H(β). (43) It is necessary to substitute Equation (43) for Equation (42). According to the following, we gathered all of the terms that were of the same power of H(β) and set each coefficient to zero as shown below: H0(β) : a2ρ1rs+ b2ρ1rs+ c2ρ1rs+ δρ0 + λρ30 = 0, (44) H1(β) : a2ρ1s 2 + 2a2ρ1rk + b2ρ1s 2 + 2b2ρ1rk +c2ρ1s 2 + 2c2ρ1rk + δρ1 + 3λρ20ρ1 = 0, (45) H2(β) : a2ρ1sk + 2a2ρ1sk + b2ρ1sk + 2b2ρ1sk +c2ρ1sk + 2c2ρ1sk + 3λρ0ρ 2 1 = 0, (46) H3(β) : 2a2ρ1k 2 + 2b2ρ1k 2 + 2c2ρ1k 2 + λρ31 = 0. (47) Attempting to solve the system of Equations (44) - (47), we obtain the following: case I: we obtain ρ0 = ∓ √ δs√ −λ(s2 − 4rk) , ρ1 = ∓ 2 √ δk√ −λ(s2 − 4rk) , and c = √ −2δ + (a2 + b2)(s2 − 4rk)√ −(s2 − 4rk) , (48) case II: we get ρ0 = ∓ √ δs√ −λ(s2 − 4rk) , ρ1 = ∓ 2 √ δk√ −λ(s2 − 4rk) , and c = − √ −2δ + (a2 + b2)(s2 − 4rk)√ −(s2 − 4rk) . (49) In the nonlinear space and time fractional (2+1)-dimensional cKG equation, which is represented by Equations (13) - (39), Equations (41) and (48), there are 27 kinds of analytical solutions with an arbitrary constant. These solutions are described by the S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 10 of 24 equations. The following is a list of the several solutions: Type I: when φ = s2 − 4rk > 0 and sk ̸= 0 or rk ̸= 0, we obtained u1(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s+ √ φ tanh (√ φ 2 β ))) , (50) u2(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s+ √ φ coth (√ φ 2 β ))) , (51) u3(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k (s+ √ φ (tanh ( √ φβ)± i sech ( √ φβ))) ) , (52) u4(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k (s+ √ φ (coth ( √ φβ)± csch ( √ φβ))) ) , (53) u5(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 4k ( 2s+ √ φ ( tanh (√ φ 4 β ) + coth (√ φ 4 β )))) , (54) u6(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s− √ φ(µ2 + σ2)− µ √ φ cosh (√ φβ ) µ sinh (√ φβ ) + σ )) , (55) u7(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s− √ φ(σ2 − µ2) + µ √ φ sinh (√ φβ ) µ cosh (√ φβ ) + σ )) , (56) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 11 of 24 where µ and σ are two real constants that are not zero, and they meet the condition that σ2 − µ2. u8(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk)  2r cosh (√ φ 2 β ) √ φ sinh (√ φ 2 β ) − s cosh (√ φ 2 β )  , (57) u9(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk)  −2r sinh (√ φ 2 β ) s sinh (√ φ 2 β ) −√ φ cosh (√ φ 2 β )  , (58) u10(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( 2r cosh (√ φβ ) √ φ sinh (√ φβ ) − s cosh (√ φβ ) ± i √ φ ) , (59) u11(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( 2r sinh (√ φβ ) −s sinh (√ φβ ) + √ φ cosh (√ φβ ) ± i √ φ ) , (60) u12(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk)  4r sinh (√ φ 4 β ) cosh (√ φ 4 β ) −2s sinh (√ φ 4 β ) cosh (√ φ 4 β ) + 2 √ φ cosh2 (√ φ 4 β ) −√ φ  , (61) Type II: when φ = s2 − 4rk < 0 and sk ̸= 0 or rk ̸= 0, we obtain u13(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s− √ −φ tan (√ −φ 2 β ))) , (62) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 12 of 24 u14(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s+ √ −φ cot (√ −φ 2 β ))) , (63) u15(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s− √ −φ ( tan (√ −φβ ) ± sec (√ −φβ )))) , (64) u16(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s+ √ −φ ( cot (√ −φβ ) ± csc (√ −φβ )))) , (65) u17(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 4k ( 2s− √ −φ ( tan (√ −φ 4 β ) − cot (√ −φ 4 β )))) , (66) u18(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s− ± √ −φ(µ2 − σ2)− µ √ −φ cos ( √ −φβ) µ sin ( √ −φβ) + σ )) , (67) u19(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 2k ( s+ ± √ −φ(µ2 − σ2) + µ √ −φ sin ( √ −φβ) µ cos ( √ −φβ) + σ )) , (68) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 13 of 24 where µ and σ are two real constants that are not zero, and they meet the condition that µ2 − σ2 > 0. u20(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk)  −2r cos (√ −φ 2 β ) √ −φ sin (√ −φ 2 β ) + s cos (√ −φ 2 β )  , (69) u21(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk)  2r sin (√ −φ 2 β ) −s sin (√ −φ 2 β ) + √ −φ cos (√ −φ 2 β )  , (70) u22(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( −2r cos ( √ −φβ)√ −φ sin ( √ −φβ) + s cos ( √ −φβ)± √ −φ ) , (71) u23(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( 2r sin ( √ −φβ) −s sin ( √ −φβ) + √ −φ cos ( √ −φβ)± √ −φ ) , (72) u24(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk)  4r sin (√ −φ 4 β ) cos (√ −φ 4 β ) −2s sin (√ −φ 4 β ) cos (√ −φ 4 β ) + 2 √ −φ cos2 (√ −φ 4 β ) − √ −φ  , (73) Type III: when r = 0 and sk ̸= 0, we get u25(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( −sψ k(ψ + cosh(sβ)− sinh(sβ)) ) , (74) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 14 of 24 u26(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( −s(cosh(sβ) + sinh(sβ)) k(ψ + cosh(sβ)− sinh(sβ)) ) , (75) for any constant ψ that is arbitrary. Type IV: when r = s = 0 and k ̸= 0, we get u27(x, y, t) = ∓ √ δs√ −λ(s2 − 4rk) ∓ 2 √ δk√ −λ(s2 − 4rk) ( − 1 kβ + ζ ) . (76) In this case, ζ is a constant that may be selected with no limitations, where β = axα Γ(α+ 1) + byα Γ(α+ 1) ∓ √ −2δ + (a2 + b2)(s2 − 4rk)tα√ −(s2 − 4rk)Γ(α+ 1) . 3.2. The Space-Time Fractional Ablowitz-Kaup-Newell-Segur Equation According to the following definition, the nonlinear space and time fractional (2+1)- dimensional AKNS equation is determined as follows: 4Dα t D α xu+Dα t D 3α xxxu+ 8Dα xuD α yD α xu+ 4D2α xxD α y u− ξD2α xxu = 0, t > 0, 0 < α ≤ 1, (77) in which u = u(x, y, t) and ξ is a constant. Comparable to the equation that came before it, we determine the solution and then become Equation (77) by Equation (41), which results in the following: −4ac d2U dβ2 − a3c d4U dβ4 + 8a2b dU dβ d2U dβ2 + 4a2b dU dβ d2U dβ2 − a2ξ d2U dβ2 = 0, (78) −(4c+ aξ) d2U dβ2 − a2c d4U dβ4 + 12ab dU dβ d2U dβ2 = 0. (79) In order to integrate Equation (79), we must first set the constant to zero. −(4c+ aξ) dU dβ − a2c d3U dβ3 + 6ab ( dU dβ ) = 0. (80) Equation (80) for balancing, we obtained the value of N = 1. Consequently, Equation (11) turned out to be U(β) = ρ0 + ρ1H(β). (81) The Equation (80) is being replaced by the Equation (81). The process of gathering all of the terms that have the same power of H(β). When we set each of their coefficients to zero, we get the following: H0(β) : −4cρ1r − aξρ1r − a2cρ1rs 2 − 2a2cρ1r 2k + 6abρ21r 2 = 0, (82) H1(β) : −4cρ1s− aξρ1s− a2cρ1s 3 − 8a2cρ1rsk + 12abρ21r 2 = 0, (83) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 15 of 24 H2(β) : −4cρ1k − aξρ1k − a2cρ1s 2k − 6a2cρ1s 2k − 8a2cρ1rk 2 +12abρ21rk + 6abρ21s 2 = 0, (84) H3(β) : −12a2cρ1sk 2 + 12abρ21sk = 0, (85) H4(β) : −6a2cρ1k 3 + 6abρ21k 2 = 0. (86) By solving Equations (82)–(86), we are able to get ρ1 = a2kξ b(−4 + 4a2kr − a2s2) and c = aξ −4 + 4a2kr − a2s2 . (87) In 27 distinct scenarios, it is feasible to write the precise traveling wave solutions of the nonlinear space and time fractional (2+1)-dimensional AKNS equation. This is a possibility. Type I: when φ = s2 − 4rk > 0 and sk ̸= 0 or rk ̸= 0, we obtained u1(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k ( s+ √ φ tanh (√ φ 2 β ))) , (88) u2(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k ( s+ √ φ coth (√ φ 2 β ))) , (89) u3(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k (s+ √ φ (tanh ( √ φβ)± i sech ( √ φβ))) ) , (90) u4(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k (s+ √ φ (coth ( √ φβ)± csch ( √ φβ))) ) , (91) u5(x, y, t) = ρ0+ a2kξ b(−4 + 4a2kr − a2s2) ( − 1 4k ( 2s+ √ φ ( tanh (√ φ 4 β ) + coth (√ φ 4 β )))) , (92) u6(x, y, t) = ρ0+ a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k ( s− √ φ(µ2 + σ2)− µ √ φ cosh (√ φβ ) µ sinh (√ φβ ) + σ )) , (93) u7(x, y, t) = ρ0+ a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k ( s− √ φ(σ2 − µ2) + µ √ φ sinh (√ φβ ) µ cosh (√ φβ ) + σ )) , (94) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 16 of 24 where µ and σ are two real constants that are not zero, and they meet the condition that σ2 − µ2. u8(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2)  2r cosh (√ φ 2 β ) √ φ sinh (√ φ 2 β ) − s cosh (√ φ 2 β )  , (95) u9(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2)  −2r sinh (√ φ 2 β ) s sinh (√ φ 2 β ) −√ φ cosh (√ φ 2 β )  , (96) u10(x, y, t) = ρ0+ a2kξ b(−4 + 4a2kr − a2s2) ( 2r cosh (√ φβ ) √ φ sinh (√ φβ ) − s cosh (√ φβ ) ± i √ φ ) , (97) u11(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( 2r sinh (√ φβ ) −s sinh (√ φβ ) + √ φ cosh (√ φβ ) ± i √ φ ) , (98) u12(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2)  4r sinh (√ φ 4 β ) cosh (√ φ 4 β ) −2s sinh (√ φ 4 β ) cosh (√ φ 4 β ) + 2 √ φ cosh2 (√ φ 4 β ) −√ φ  , (99) Type II: when φ = s2 − 4rk < 0 and sk ̸= 0 or rk ̸= 0, we obtain u13(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k ( s− √ −φ tan (√ −φ 2 β ))) , (100) u14(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k ( s+ √ −φ cot (√ −φ 2 β ))) , (101) u15(x, y, t) = ρ0+ a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k ( s− √ −φ ( tan (√ −φβ ) ± sec (√ −φβ )))) , (102) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 17 of 24 H16(β) = − 1 2k ( s+ √ −φ ( cot (√ −φβ ) ± csc (√ −φβ ))) , (103) H17(β) = − 1 4k ( 2s− √ −φ ( tan (√ −φ 4 β ) − cot (√ −φ 4 β ))) , (104) u18(x, y, t) = ρ0+ a2kξ b(−4 + 4a2kr − a2s2) ( − 1 2k ( s− ± √ −φ(µ2 − σ2)− µ √ −φ cos ( √ −φβ) µ sin ( √ −φβ) + σ )) , (105) H19(β) = − 1 2k ( s+ ± √ −φ(µ2 − σ2) + µ √ −φ sin ( √ −φβ) µ cos ( √ −φβ) + σ ) , (106) where µ and σ are two real constants that are not zero, and they meet the condition that µ2 − σ2 > 0. u20(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2)  −2r cos (√ −φ 2 β ) √ −φ sin (√ −φ 2 β ) + s cos (√ −φ 2 β )  , (107) u21(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2)  2r sin (√ −φ 2 β ) −s sin (√ −φ 2 β ) + √ −φ cos (√ −φ 2 β )  , (108) u22(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( −2r cos ( √ −φβ)√ −φ sin ( √ −φβ) + s cos ( √ −φβ)± √ −φ ) , (109) u23(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( 2r sin ( √ −φβ) −s sin ( √ −φβ) + √ −φ cos ( √ −φβ)± √ −φ ) , (110) u24(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2)  4r sin (√ −φ 4 β ) cos (√ −φ 4 β ) −2s sin (√ −φ 4 β ) cos (√ −φ 4 β ) + 2 √ −φ cos2 (√ −φ 4 β ) − √ −φ  , (111) S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 18 of 24 Figure 1: Three-dimensional, two-dimensional, and contour plots of the kink-type solution given by Equation (88) with parameters (a, b, s, r, k, ξ, α) = (1, 1, 3, 2, 1, 2, 0.5). The figure illustrates the formation of a localized traveling wave structure that remains stable along both x-direction and y-direction. Figure 2: Three-dimensional, two-dimensional, and contour plots of the kink-type solution corresponding to Equation (94). The selected parameters highlight the sensitivity of the wave profile to changes in r and k, resulting in a steeper kink structure compared to Figure 1. Type III: when r = 0 and sk ̸= 0, we get u25(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( −sψ k(ψ + cosh(sβ)− sinh(sβ)) ) , (112) u26(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( −s(cosh(sβ) + sinh(sβ)) k(ψ + cosh(sβ)− sinh(sβ)) ) , (113) for any constant ψ that is arbitrary. Type IV: when r = s = 0 and k ̸= 0, we get u27(x, y, t) = ρ0 + a2kξ b(−4 + 4a2kr − a2s2) ( − 1 kβ + ζ ) . (114) In this case, ζ is a constant that may be selected with no limitations, where β = axα Γ(α+ 1) + byα Γ(α+ 1) − aξtα (−4 + 4a2kr − a2s2)Γ(α+ 1) . 4. Results and Discussions This section examines the precise solutions derived for the space-time fractional cKG and AKNS equations by the application of the generalized Riccati equation mapping S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 19 of 24 Figure 3: Three-dimensional, two-dimensional, and contour plots of the periodic solution corresponding to Equation (100) with parameters s = 2, r = 2 and k = 2. The oscillatory structure confirms the trigonometric nature of this class of exact solutions. Figure 4: Three-dimensional, two-dimensional, and contour plots of the periodic solution derived from Equation (106). The parameter choice enhances the oscillatory pattern, demonstrating the flexibility of the generalized Riccati equation mapping method in capturing recurrent nonlinear wave dynamics. method. A total of twenty-seven analytical solutions were obtained, comprising fourteen hyperbolic functions, twelve trigonometric functions, and one rational function. These solutions demonstrate kink-type and periodic behaviors contingent upon the parameter combinations. The parameter values used in the graphical simulations were carefully cho- sen to highlight different wave phenomena. Specifically, the constants a = b = 1 and α = 0.5 were selected to simplify the traveling wave transformation while still reflecting the effect of fractional order. The Riccati parameters s, r, and k were tuned to control the balance between nonlinear and dispersive effects, thereby allowing a transition be- tween kink-type and periodic solutions. Additional constants such as µ, σ, ψ, and ζ were assigned small integer values to generate nontrivial structures without introducing unnec- essary complexity. The complete set of parameter values corresponding to each figure is summarized in Table 1, ensuring the reproducibility of the results. Figures 1–5 depict representative three-dimensional, two-dimensional, and contour plots of the obtained solutions. Figure 1 pertains to Equation (88), wherein the selection (a, b, s, r, k, ξ, α) = (1, 1, 3, 2, 1, 2, 0.5) yields a confined kink configuration that maintains stability in both spatial dimensions. Figure 2, corresponding to Equation (94), illustrates an additional kink effect with a marginally modified steepness resulting from the impact of the Riccati constants r and k. Figure 3, aligned with Equation (100), distinctly illus- trates periodic oscillations resulting from the parameters s = 2 and k = 2, so validating S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 20 of 24 Figure 5: Three-dimensional, two-dimensional, and contour plots of the rational-type kink solution corresponding to Equation (114) under the conditions s = r = 0, k = 2, and ζ = 1. The solution exhibits a decaying kink profile, showing the adaptability of the method to rationally structured waveforms. Table 1: Configuration of parameters for Eqs. (88)-(114) Equations Parameters Fig. Wave behaviors 88 ρ0 = 1, a = 1, b = 1, s = 3, r = 2, k = 1, ξ = 2, 1 kink t = 10, α = 0.5, 1 ≤ x ≤ 20, 1 ≤ y ≤ 20 89, 91-92, 95-96, ρ0 = 1, a = 1, b = 1, s = 3, r = 2, k = 1, ξ = 2, - kink 98-99 t = 10, α = 0.5, 1 ≤ x ≤ 20, 1 ≤ y ≤ 20 93 ρ0 = 1, a = 1, b = 1, s = 3, r = 2, k = 1, ξ = 2, t = 10, - kink α = 0.5, µ = 1, σ = 2, 1 ≤ x ≤ 20, 1 ≤ y ≤ 20 94 ρ0 = 1, a = 1, b = 1, s = 3, r = 2, k = 1, ξ = 2, t = 10, 2 kink α = 0.5, µ = 1, σ = 2, 1 ≤ x ≤ 20, 1 ≤ y ≤ 20 100 ρ0 = 1, a = 1, b = 1, s = 2, r = 2, k = 2, ξ = 2, 3 periodic t = 10, α = 0.5, 1 ≤ x ≤ 30, 1 ≤ y ≤ 30 101-104, 107-111 ρ0 = 1, a = 1, b = 1, s = 2, r = 2, k = 2, ξ = 2, - periodic t = 10, α = 0.5, 1 ≤ x ≤ 30, 1 ≤ y ≤ 30 105 ρ0 = 1, a = 1, b = 1, s = 2, r = 2, k = 2, ξ = 2, t = 10, - periodic α = 0.5, µ = 2, σ = 1, 1 ≤ x ≤ 30, 1 ≤ y ≤ 30 106 ρ0 = 1, a = 1, b = 1, s = 2, r = 2, k = 2, ξ = 2, t = 10, 4 periodic α = 0.5, µ = 2, σ = 1, 1 ≤ x ≤ 30, 1 ≤ y ≤ 30 112-113 ρ0 = 1, a = 1, b = 1, s = 2, r = 0, k = 2, ξ = 2, t = 10, - kink α = 0.5, ψ = 1, 1 ≤ x ≤ 30, 1 ≤ y ≤ 30 114 ρ0 = 1, a = 1, b = 1, s = 0, r = 0, k = 2, ξ = 2, t = 10, 5 kink α = 0.5, ζ = 1, 1 ≤ x ≤ 30, 1 ≤ y ≤ 30 the trigonometric characteristics of this solution family. Likewise, Figure 4 illustrates the periodic pattern of Equation (106), wherein the oscillatory profile is enhanced through a distinct parameter modification. Figure 5 depicts the rational-type kink solution derived from Equation (114), achieved under the parameters s = r = 0, k = 2, and ζ = 1, demon- strating the versatility of the Riccati mapping approach for rationally decaying waves. In comparison to alternative analytical methods, the generalized Riccati equation map- ping method offers a wider array of exact solutions, encompassing kink, periodic, and ra- S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 21 of 24 tional families. The acquired solutions exhibit greater variability than those documented in earlier studies, broadening the application of analytical methods to higher-dimensional and fractional-order contexts [20, 28]. The solutions to the fractional cKG equation are complex-valued and not readily visualizable; yet, their algebraic structures yield significant insights into wave propagation in Josephson junctions. Conversely, the AKNS equation produces real-valued kink and periodic solutions, which hold physical significance in nonlin- ear optics. The results indicate that the figures serve as both mathematical representations and physically significant depictions of nonlinear wave phenomena. 5. Conclusions In this study, we utilized the generalized Riccati equation mapping method alongside Jumarie’s Riemann–Liouville fractional derivative to derive 27 unique analytic solutions for two essential nonlinear space-time fractional (2+1)-dimensional models: the cKG equa- tion and the AKNS equation. The resulting solutions span a wide range of wave behaviors, including kink-type and periodic structures, thereby illustrating the adaptability and effi- cacy of the suggested method in comparison to traditional analytical techniques. In addition to their mathematical originality, the results possess immediate physical significance. The cKG equation is esteemed for its application in modeling fluxion propa- gation in Josephson junctions, crystal dislocations, and particle dynamics, while the AKNS equation is significant in nonlinear optics, particularly in the study of soliton propagation and optical wave interactions. The solutions derived herein offer significant insights into nonlinear wave dynamics pertinent to both physics and engineering domains. Nonetheless, the current study possesses specific limitations. The study is confined to precise analytic solutions, with no consideration given to stability, perturbation effects, or resilience against external influences. Subsequent study should concentrate on numerical validation, employing techniques such as spectrum simulations or finite-difference meth- ods, to evaluate the precision and physical relevance of the results. Furthermore, the methodology can be used to additional categories of fractional nonlinear evolution equa- tions, encompassing higher-order and coupled systems, which may exhibit more complex dynamics. In conclusion, the extended Riccati equation mapping approach provides a systematic, adaptable, and robust framework for generating various solution families of fractional non- linear equations. This paper integrates analytic theory with prospective physical applica- tions, establishing a basis for further exploration of the dynamics of nonlinear fractional systems and facilitating future research in mathematical physics and engineering. Acknowledgements The authors wish to express their appreciation to all reviewers for their perceptive comments and to the editorial staff for their essential support in refining this work. S. Phoosree et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6624 22 of 24 Conflicts of Interest The authors declare that there are no conflicts of interest to be disclosed with this publication. References [1] W Thadee, A Chankaew, and S Phoosree. Effects of wave solutions on shallow-water equation, optical fibre equation and electric-circuit equation. Maejo International Journal of Science and Technology, 16(3):262–274, 2022. [2] H K Barman, M A Akbar, M S Osman, K S Nisar, M Zakarya, A. Abdel-Aty, and H Eleuch. Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions. Results in Physics, 24:104092, 2021. [3] F L Hasan. 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