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An improved self-supervised learning for EEG classification


  • Motor Imagery EEG (MI-EEG) classification plays an important role in different Brain-Computer Interface (BCI) systems. Recently, deep learning has been widely used in the MI-EEG classification tasks, however this technology requires a large number of labeled training samples which are difficult to obtain, and insufficient labeled training samples will result in a degradation of the classification performance. To address the degradation problem, we investigate a Self-Supervised Learning (SSL) based MI-EEG classification method to reduce the dependence on a large number of labeled training samples. The proposed method includes a pretext task and a downstream classification one. In the pretext task, each MI-EEG is rearranged according to the temporal characteristic. A network is pre-trained using the original and rearranged MI-EEGs. In the downstream task, a MI-EEG classification network is firstly initialized by the network learned in the pretext task and then trained using a small number of the labeled training samples. A series of experiments are conducted on Data sets 1 and 2b of BCI competition IV and IVa of BCI competition III. In the case of one third of the labeled training samples, the proposed method can obtain an obvious improvement compared to the baseline network without using SSL. In the experiments under different percentages of the labeled training samples, the results show that the designed SSL strategy is effective and beneficial to improving the classification performance.

    Citation: Yanghan Ou, Siqin Sun, Haitao Gan, Ran Zhou, Zhi Yang. An improved self-supervised learning for EEG classification[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 6907-6922. doi: 10.3934/mbe.2022325

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  • Motor Imagery EEG (MI-EEG) classification plays an important role in different Brain-Computer Interface (BCI) systems. Recently, deep learning has been widely used in the MI-EEG classification tasks, however this technology requires a large number of labeled training samples which are difficult to obtain, and insufficient labeled training samples will result in a degradation of the classification performance. To address the degradation problem, we investigate a Self-Supervised Learning (SSL) based MI-EEG classification method to reduce the dependence on a large number of labeled training samples. The proposed method includes a pretext task and a downstream classification one. In the pretext task, each MI-EEG is rearranged according to the temporal characteristic. A network is pre-trained using the original and rearranged MI-EEGs. In the downstream task, a MI-EEG classification network is firstly initialized by the network learned in the pretext task and then trained using a small number of the labeled training samples. A series of experiments are conducted on Data sets 1 and 2b of BCI competition IV and IVa of BCI competition III. In the case of one third of the labeled training samples, the proposed method can obtain an obvious improvement compared to the baseline network without using SSL. In the experiments under different percentages of the labeled training samples, the results show that the designed SSL strategy is effective and beneficial to improving the classification performance.



    In numerous engineering and scientific domains, nonlinear fractional partial differential equations (NFDEs) have gained significant importance in the description and modelling of intricate processes. The memory and inherited characteristics of the system being modelled can be precisely captured via NFPDEs. These equations are particularly helpful for explaining anomalous diffusion phenomena and have a wide range of applications in disciplines like chemistry, physics, engineering, finance, biology, and economics [1,2,3,4,5,6]. As a result, in many application domains, the exploration of NFPDEs is crucial to comprehending and forecasting the behavior of complex systems. Nonetheless, there are special difficulties in resolving and interpreting fractional derivatives due to their non-locality and non-linearity. The solution and analysis of these equations may require the development of new methods and tools, as traditional analytical and numerical approaches might not be directly relevant. Notwithstanding these difficulties, research on NFPDEs has produced significant advances in a number of scientific and engineering domains [7,8,9]. Research opportunities have increased and our understanding of complicated processes has improved as a result of the development of new analytical and numerical techniques for solving and analyzing NFPDEs [10,11,12]. Exact solutions to nonlinear fractional problems can currently be obtained by a variety of effective methods, such as the generalized projective Riccati equation approach [13], sine-Gordon expansion method [14], generalized Riccati method and the auxiliary ordinary differential equation method [15], Lie symmetry approach [16], first integral method [17], modified Kudryashov method [18], extended exp(ϕ(ξ))-expansion method [19], modified auxiliary equation method [20], unified method [21], (GG)-expansion methods [22,23,24], Khater methods [25,26], Poincaré-Lighthill-Kuo method [27], Riccati-Bernoulli Sub-ODE [28], exp-function method [29], fractional Sin-Gordon method [30], sub-equation method [31], tanh-method [32], extended direct algebraic method [33,34,35,36], Sardar sub-equation method [37], exponential rational function method [38], and so on [39,40,41,42,43,44].

    Kundu et al. [45] introduced the Kundu-Mukherjee-Naskar equation (KMNE) for the first time in 2014 and found optical soliton solutions for it. The KMNE has been the subject of extensive research recently. Its wave events are crucial for accurately modelling propagation pulses in high-speed data transfer in communication systems, optical fibers and the ocean currents of rogue waves [46,47,48]. One common application for optical fibers is the transmission of light between their two ends. Their long-distance and faster data transmission rate compared to wires makes them commonly employed in optical fiber communications. The goal of the current study is to construct and analyze optical soliton solutions for the gFKMNE, a fractional generalization of KMNE, which exhibits the following dimensionless display [49]:

    iDδtz+aDβy(Dαxz)+ibz(zDαxzzDαx)=0, (1.1)

    wherein 0<α,β,δ1, z=z(t,x,y) is the quantity that represents a complex wave envelope, z is the complex conjugate of z. Furthermore, the dispersion term and the nonlinearity term are indicated by the two parameters a, b respectively. The wave's temporal history is represented by the first term in Eq (1), which is succeeded by the dispersion term aDβy(Dαxz). Ultimately, the coefficient of b that is changed from the standard Kerr law nonlinearity represents the nonlinear portion. In optical fibers and communication systems, the pulse propagation is described by this equation. The fractional derivatives Dαx(), Dβy(), and Dδt() are the generalized conformable fractional derivatives (gCFDs) of order α, β, and δ respectively.

    In many applications, such as irrigation and river flows, tidal waves, weather research and tsunami prediction, Eq (1.1) is used. Due to such significance, many researchers have developed an interest in examining Eq (1.1). For instance, Eq (1.1) was solved with new exact solutions by Günerhan et al. [50] through the use of an enhanced direct algebraic approach. Using the csch-approach, extended tanh-coth method, and extended rational sinh-cosh method, Rizvi et al. [51] were able to get the singular soliton, dark soliton, combination dark-singular soliton, and other hyperbolic solutions for Eq (1.1). Talarposhti et al. [52] derived optical soliton solutions for the KMNE using the exp-function approach. Onder et al. [53] used the novel Kudryashov techniques along with the Sardar sub-equation to introduce optical soliton solutions for the KMNE. Zafar et al. [54] found new soliton solutions for the KMNE by employing the exp-function approach and the extended Jacobi's elliptic expansion function. Kumar et al. [55] used the new auxiliary equation approach with the generalized Kudryashov method to find dark, bright, periodic U-shaped, and singular soliton solutions for Eq (1.1). Furthermore, this equation was investigated by other authors through the use of new precise solution techniques such the semi-inverse approach [56], the extended trial function method [57], the sine-Gordon and sinh-Gordon expansion methods [58], the Hamiltonian-based algorithm [59], Laplace-Adomian decomposition method [60], and other methods [61,62,63,64,65].

    Inspired by current research, we seek to address gFKMNE expressed in (1.1) through two enhanced implementations of the (GG)-expansion technique: the extended (GG)-expansion method and the generalized (r+GG)-expansion method, which are based on the reduction of NFPDEs to integer-order NODEs produced by the model's wave transformation. The strategic techniques further transform the resultant NODEs into an algebraic system of nonlinear equations by assuming a series-form solution. These equations may then be solved using the Maple tool to produce dark soliton lattices for gFKMNE. The important solitons in these dark soliton lattices are organized regularly in an optical medium. Dark soliton formations are collections of dark solitaires arranged regularly inside a medium. They are frequently shaped like wave channels or optical fibers. Similar to the lattice of a crystal, these solitons are restricted to areas of decreased optical intensity within a larger field that retain their structure and amplitude while moving. Not only can dark soliton lattices be used to precisely control and manipulate optical waves, but they can also be used to process information and communications networks, provide insight into nonlinear optics phenomena, provide platforms for basic research on the behavior of wave propagation, and have potential applications in signal processing, optical communication, and sensing. Thus, dark soliton lattices offer a stimulating new topic for nonlinear optics research that will enable both basic and applied investigations [66,67,68].

    The format of the present article is as follows: Section 1 provides an introduction. The operational mechanism for the (GG)-expansion methods and the description of gCFD are detailed in Section 2. We build many additional sets of optical soliton solutions for Eq (1.1) in Section 3. A few illustrations and a graphical representations of dark soliton lattices are presented and discussed in Section 4. Our study excursion comes to an end with the provided conclusion in Section 5.

    In this section, the basic definition of used fractional derivative gCFD and the operational procedure of the proposed methods are presented.

    Explicit soliton solutions to nonlinear FPDEs can be obtained by taking use of the advantages that gCFDs have over conventional fractional derivative operators. Notably, alternate formulations of fractional derivatives do not provide the optical soliton solutions of Eq (1.1) because they violate the chain rule [69,70]. Consequently, gCFDs were added to Eq (1.1). [71] defined this derivative operator of order as follows:

    gCFDDφz(φ)=limϖ0z((Γ(n)Γ(n+1))ϖφ1+φ)z(φ)ϖ,(0,1]n1,nR. (2.1)

    We utilize the following characteristics of this derivative in this study:

    gCFDDφφγ=γΓ(n)Γ(n+1)φγ,γR, (2.2)
    gCFDDφ(γ1ρ(φ)±γ2η(φ))=γ1gCFDDφ(ρ(φ))±γ2gCFDDφ(η(φ)), (2.3)
    gCFDDφω[ζ(φ)]=ωζ(ζ(φ))gCFDDφζ(φ), (2.4)

    where ρ(φ), η(φ), ω(φ), and ζ(φ) are arbitrary differentiable functions, whereas γ, γ1 and γ2 signify constants.

    In this phase of our study, we discuss the processes of the (GG)-expansion approach, with a focus on solving the given general NFPDE:

    R(z,Dδtz,Dαx1z,Dβx1z,zDαx1z,)=0,0<α,β,δ1, (2.5)

    where z=z(t,x1,x2,x3,,xj).

    Equation (2.5) is solved using the following strategy:

    (ⅰ). Starting with a variable transformation of the framework z(t,x1,x2,x3,,xj)=Z(φ), where φ can be stated in a number of ways. Equation (2.5) progresses via this transformation, resulting in the NODE that follows:

    Q(Z,ZZ,Z,)=0, (2.6)

    where Z=dZdφ. On rare occasions, the NODE may become vulnerable to the homogeneous balancing principle due to the integration of Eq (2.6).

    (ⅱ). Next, based on the method's version, we assume the following series-form solution for (2.6):

    Version 1. For the extended (GG)-expansion method, we assume the following (GG) solution:

    Z(φ)=ρl=ρτl(G(φ)G(φ))l, (2.7)

    Version 2. For the generalized (r+GG)-expansion method, we assume the following (GG) solution:

    Z(φ)=ρl=ρτl(r+G(φ)G(φ))l,rR, (2.8)

    where it is subsequently necessary to compute the values of τls(l=ρ...ρ). The balance number ρ, a positive integer in Eqs (2.7) and (2.8), can be found by homogeneously balancing the nonlinear terms and the derivative with the highest order terms in Eq (2.6). Greater accuracy can be achieved in determining the balance number ρ by applying the following mathematical formulas:

    D(dpZdφp)=ρ+p,  and  D(Zq(dpZdφp)s)=ρq+p(s+ρ), (2.9)

    where D expresses the degree of Z(φ), whereas p,q, and s are positive integers.

    Moreover, the function G(φ) in Eq (2.8) satisfies the subsequent second-order linear ODE:

    G(φ)+λG(φ)+μG(φ)=0, (2.10)

    where λ,μ are constants.

    Moreover, with Eq (2.10)'s general solution, we have:

    (G(φ)G(φ))={12κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ)12λ,κ<0,12κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ)12λ,κ>0,ν2ν1+ν2φ,κ=0, (2.11)

    where κ=λ24μ, while ν1 and ν2 in (2.11) are arbitrary constants.

    (ⅲ). Next, we insert Eqs (2.8) or (2.7) into Eq (2.6) or into the Equation that we acquired from the integration of Eq (2.6), and we gather all the terms that have an analogous power of (G(φ)G(φ)).

    (ⅳ). The derived polynomial (G(φ)G(φ))i has all of its coefficients equal to zero, resulting in a collection of nonlinear algebraic equations in τl(l=ρ...,ρ), λ, μ, and other necessary parameters.

    (ⅴ). The unresolved parameters are discovered by using the Maple tool to solve the resulting system.

    (ⅵ). To generate sets of soliton solutions for Eq (2.5), the values predicted from step 5 are subsequently substituted in Eqs (2.7) or (2.8), depending on the version.

    We use the suggested methods in this section to build families of optical soliton solutions for (1.1). Using the subsequent wave modification as a component of the procedure, we can (1.1):

    z(x,y,t)=eiξZ(φ),ξ=ξ(t,x,y),φ=φ(t,x,y),φ=p1Γ(nα+1)xααΓ(n)+p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n),ξ=q1Γ(nα+1)xααΓ(n)q2Γ(nβ+1)yββΓ(n)+ωΓ(nδ+1)tδδΓ(n)+θ0, (3.1)

    where Z(φ) symbolizes the amplitude part, q1 and q2 indicate wave numbers in the x and y directions, respectively, ω represents the frequency of the wave and θ0 is a constant, the parameters p1 and p2 indicate inverse width along the x and y directions, respectively, and ϱ is used for the velocity, delivering the afterwards NODE from the real part of (1.1):

    ap1p2Z(ω+aq1q2)Z2bq1Z3=0, (3.2)

    with the constraint from the imaginary part:

    ϱ=a(q1p2+q2p1). (3.3)

    It is established that ρ=1 when a homogeneous balancing condition between Z and Z3 (given in (3.2)) has been created.

    First, we wish to address (3.2) with the help of the extended (GG)-expansion method. Equation (2.7) yields the following closed form solution for (3.2) when ρ=1 is substituted in it:

    U(φ)=1l=1τl(G(φ)G(φ))l. (3.4)

    By including (3.4) in (3.2) and bringing together terms with a comparable power of (G(φ)G(φ)), we get an expression based on the terms (G(φ)G(φ)). By setting the coefficients to zero, this formula can be converted into a set of algebraic nonlinear equations. The system can be solved using Maple in the following three cases of solutions:

    Case 1.1.

    τ0=12τ1λμ,τ1=0,τ1=τ1,p1=p1,p2=p2,q1=ap1p2μ2bτ12,q2=q2,ω=12ap1p2(4μbτ12+2aμ2q2+λ2bτ12)bτ12. (3.5)

    Case 1.2.

    τ0=12τ1λ,τ1=τ1,τ1=0,p1=p1,p2=p2,q1=ap1p2bτ12,q2=q2,ω=12ap1p2(4μbτ12+2aq2+λ2bτ12)bτ12. (3.6)

    Case 1.3.

    τ0=τ0,τ1=τ1,τ1=τ1,p1=0,p2=p2,q1=0,q2=q2,ω=0. (3.7)

    Considering case 1.1 and utilizing Eq (3.1), (3.4) and the corresponding general solution of (2.10), we construct the subsequent plethora of optical soliton solutions for (1.1):

    Family 1.1.1. When τ>0,

    (ⅰ) When ν10, ν20

    u1,1,1(t,x,y)=eiξ(τ1(12κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ)12λ)1+12τ1λμ). (3.8)

    (ⅱ) When ν1=0, ν20

    u1,1,2(t,x,y)=eiξ(τ112κcoth(12κφ)12λ+12τ1λμ). (3.9)

    (ⅲ) When ν10, ν2=0

    u1,1,3(t,x,y)=eiξ(τ112κtanh(12κφ)12λ+12τ1λμ). (3.10)

    Family 1.1.2. When τ<0,

    (ⅰ) When ν10, ν20

    u1,1,4(t,x,y)=eiξ(τ1(12κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ)12λ)1+12τ1λμ). (3.11)

    (ⅱ) When ν1=0, ν20

    u1,1,5(t,x,y)=eiξ(τ112κcot(12κφ)12λ+12τ1λμ). (3.12)

    (ⅲ) When ν10, ν2=0

    u1,1,6(t,x,y)=eiξ(τ112κtan(12κφ)12λ+12τ1λμ). (3.13)

    Family 1.1.3. When τ=0,

    (ⅰ) When ν10, ν20

    u1,1,7(t,x,y)=eiξ(τ1(ν1+ν2φ)ν2+12τ1λμ). (3.14)

    (ⅱ) When ν1=0, ν20

    u1,1,8(t,x,y)=eiξ(τ1φ112λ+12τ1λμ). (3.15)

    (ⅲ) When ν10, ν2=0

    u1,1,9(t,x,y)=eiξ(2τ1λ+12τ1λμ), (3.16)

    where φ=p1Γ(nα+1)xααΓ(n)+p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n) ξ=q1Γ(nα+1)xααΓ(n)q2Γ(nβ+1)yββΓ(n)+ωΓ(nδ+1)tδδΓ(n)+θ0 for q1 and ω defined in Eq (3.5).

    Considering case 1.2 and utilizing Eq (3.1), (3.4) and the corresponding general solution of (2.10), we construct the subsequent plethora of optical soliton solutions for (1.1):

    Family 1.2.1. When τ>0,

    (ⅰ) When ν10, ν20

    u1,2,1(t,x,y)=eiξ(12τ1λ+τ1(12κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ)12λ)). (3.17)

    (ⅱ) When ν1=0, ν20

    u1,2,2(t,x,y)=eiξ(12τ1λ+τ1(12κcoth(12κφ)12λ)). (3.18)

    (ⅲ) When ν10, ν2=0

    u1,2,3(t,x,y)=eiξ(12τ1λ+τ1(12κtanh(12κφ)12λ)). (3.19)

    Family 1.2.2. When τ<0,

    (ⅰ) When ν10, ν20

    u1,2,4(t,x,y)=eiξ(12τ1λ+τ1(12κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ)12λ)). (3.20)

    (ⅱ) When ν1=0, ν20

    u1,2,5(t,x,y)=eiξ(12τ1λ+τ1(12κcot(12κφ)12λ)). (3.21)

    (ⅲ) When ν10, ν2=0

    u1,2,6(t,x,y)=eiξ(12τ1λ+τ1(12κtan(12κφ)12λ)). (3.22)

    Family 1.2.3. When τ=0,

    (ⅰ) When ν10, ν20

    u1,2,7(t,x,y)=eiξ(12τ1λ+τ1ν2ν1+ν2φ). (3.23)

    (ⅱ) When ν1=0, ν20

    u1,2,8(t,x,y)=eiξ(12τ1λ+τ1(φ112λ)), (3.24)

    where φ=p1Γ(nα+1)xααΓ(n)+p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n)ξ=q1Γ(nα+1)xααΓ(n)q2Γ(nβ+1)yββΓ(n)+ωΓ(nδ+1)tδδΓ(n)+θ0 for q1 and ω defined in Eq (3.6).

    Considering case 1.3 and utilizing Eq (3.1), (3.4) and the corresponding general solution of (2.10), we construct the subsequent plethora of optical soliton solutions for (1.1)

    Family 1.3.1. When τ>0,

    (ⅰ) When ν10, ν20

    u1,3,1(t,x,y)=eiξ(τ1(12κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ)12λ)1+τ0+τ1(12κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ)12λ)). (3.25)

    (ⅱ) When ν1=0, ν20

    u1,3,2(t,x,y)=eiξ(τ112κcoth(12κφ)12λ+τ0+τ1(12κcoth(12κφ)12λ)). (3.26)

    (ⅲ) When ν10, ν2=0

    u1,3,3(t,x,y)=eiξ(τ112κtanh(12κφ)12λ+τ0+τ1(12κtanh(12κφ)12λ)). (3.27)

    Family 1.3.2. When τ<0,

    (ⅰ) When ν10, ν20

    u1,3,4(t,x,y)=eiξ(τ1(12κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ)12λ)1+τ0+τ1(12κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ)12λ)). (3.28)

    (ⅱ) When ν1=0, ν20

    u1,3,5(t,x,y)=eiξ(τ112κcot(12κφ)12λ+τ0+τ1(12κcot(12κφ)12λ)). (3.29)

    (ⅲ) When ν10, ν2=0

    u1,3,6(t,x,y)=eiξ(τ112κtan(12κφ)12λ+τ0+τ1(12κtan(12κφ)12λ)). (3.30)

    Family 1.3.3. When τ=0,

    (ⅰ) When ν10, ν20

    u1,3,7(t,x,y)=eiξ(τ1(ν1+ν2φ)ν2+τ0+τ1ν2ν1+ν2φ). (3.31)

    (ⅱ) When ν1=0, ν20

    u1,3,8(t,x,y)=eiξ(τ1φ112λ+τ0+τ1(φ112λ)). (3.32)

    (ⅲ) When ν10, ν2=0

    u1,3,9(t,x,y)=eiξ(2τ1λ+τ012τ1λ), (3.33)

    where φ=p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n)ξ=q2Γ(nβ+1)yββΓ(n)+θ0.

    Now, we wish to address (3.2) with the help of the generalized (r+GG)-expansion method. Equation (2.8) yields the following closed-form solution for (3.2) when ρ=1 is substituted in it:

    U(φ)=1l=1τl(r+G(φ)G(φ))l. (3.34)

    We obtain an expression based on the terms (r+G(φ)G(φ)) by including (3.34) in (3.2) and combining terms with similar powers of (r+G(φ)G(φ)). By setting the coefficients to zero, this formula can be converted into a set of algebraic nonlinear equations. Following Maple's solution to this system, the following six cases of solutions are generated:

    Case 2.1.

    τ0=τ0,τ1=0,τ1=τ0κ,p1=p1,p2=p2,q1=0,q2=q2,ω=κap1p2,r=12λ+12κ. (3.35)

    Case 2.2.

    τ0=τ0,τ1=0,τ1=2τ0(r2λr+μ)2rλ,p1=p1,q1=14(4r24λr+λ2)ap1p2τ02b,q2=q2,ω=14(4aq2r24aq2λr8τ02bμ+2τ02bλ2+aq2λ2)ap1p2τ02b,p2=p2,r=r. (3.36)

    Case 2.3.

    τ0=12τ1λτ1r,τ1=τ1,τ1=0,p1=p1,p2=p2,q1=ap1p2bτ12,q2=q2,ω=12ap1p2(λ2bτ12+2aq24μbτ12)bτ12,r=r. (3.37)

    Case 2.4.

    τ0=0,τ1=τ1,τ1=μτ1+14τ1λ2,p1=p1,p2=p2,q1=ap1p2bτ12,q2=q2,ω=ap1p2(8μbτ12+2λ2bτ12+aq2)bτ12,r=12λ. (3.38)

    Case 2.5.

    τ0=τ0,τ1=0,τ1=2τ0(12λ+12κ)+λτ0,p1=p1,p2=p2,q1=14κap1p2τ02b,q2=q2,ω=14(8τ02bμ+2τ02bλ2+4aq2μaq2λ2)ap1p2τ02b,r=12λ+12κ. (3.39)

    Case 2.6.

    τ0=0,τ1=τ1,τ1=μτ114τ1λ2,p1=p1,p2=p2,q1=ap1p2bτ12,q2=q2,ω=ap1p2(4μbτ12λ2bτ12+aq2)bτ12,r=12λ. (3.40)

    Considering case 2.1 and utilizing Eq (3.1), (3.34) and the corresponding general solution of Eq (2.10), we construct the subsequent plethora of optical soliton solutions for (1.1):

    Family 2.1.1. When τ>0,

    (ⅰ) When ν10, ν20

    u2,1,1(t,x,y)=eiξ(τ0κ(12κ+12κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ))1+τ0). (3.41)

    (ⅱ) When ν1=0, ν20

    u2,1,2(t,x,y)=eiξ(τ0κ12κ+12κcoth(12κφ)+τ0). (3.42)

    (ⅲ) When ν10, ν2=0

    u2,1,3(t,x,y)=eiξ(τ0κ12κ+12κtanh(12κφ)+τ0). (3.43)

    Family 2.1.2. When τ<0,

    (ⅰ) When ν10, ν20

    u2,1,4(t,x,y)=eiξ(τ0κ(12κ+12κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ))1+τ0). (3.44)

    (ⅱ) When ν1=0, ν20

    u2,1,5(t,x,y)=eiξ(τ0κ12κ+12κcot(12κφ)+τ0). (3.45)

    (ⅲ) When ν10, ν2=0

    u2,1,6(t,x,y)=eiξ(τ0κ12κ12κtan(12κφ)+τ0). (3.46)

    Family 2.1.3. When τ=0,

    (ⅰ) When ν10, ν20

    u2,1,7(t,x,y)=eiξ(τ0κ(12λ+12κ+ν2ν1+ν2φ)1+τ0). (3.47)

    (ⅱ) When ν1=0, ν20

    u2,1,8(t,x,y)=eiξ(τ0κ12κ+φ1+τ0). (3.48)

    (ⅲ) When ν10, ν2=0

    u2,1,9(t,x,y)=τ0eiξ, (3.49)

    where φ=p1Γ(nα+1)xααΓ(n)+p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n) ξ=q1Γ(nα+1)xααΓ(n)q2Γ(nβ+1)yββΓ(n)+ωΓ(nδ+1)tδδΓ(n)+θ0 for q1 and ω defined in Eq (3.35).

    Considering case 2.2 and utilizing Eq (3.1), (3.34) and the corresponding general solution of Eq (2.10), we construct the subsequent plethora of optical soliton solutions for (1.1):

    Family 2.2.1. When τ>0,

    (ⅰ) When ν10, ν20

    u2,2,1(t,x,y)=eiξ(2τ0(r2λr+μ)(2rλ)(r+12κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ)12λ)+τ0). (3.50)

    (ⅱ) When ν1=0, ν20

    u2,2,2(t,x,y)=eiξ(2τ0(r2λr+μ)(2rλ)(r+12κcoth(12κφ)12λ)+τ0). (3.51)

    (ⅲ) When ν10, ν2=0

    u2,2,3(t,x,y)=eiξ(2τ0(r2λr+μ)(2rλ)(r+12κtanh(12κφ)12λ)+τ0). (3.52)

    Family 2.2.2. When τ<0,

    (ⅰ) When ν10, ν20

    u2,2,4(t,x,y)=eiξ(2τ0(r2λr+μ)(2rλ)(r+12κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ)12λ)+τ0). (3.53)

    (ⅱ) When ν1=0, ν20

    u2,2,5(t,x,y)=eiξ(2τ0(r2λr+μ)(2rλ)(r+12κcot(12κφ)12λ)+τ0). (3.54)

    (ⅲ) When ν10, ν2=0

    u2,2,6(t,x,y)=eiξ(2τ0(r2λr+μ)(2rλ)(r12κtan(12κφ)12λ)+τ0). (3.55)

    Family 2.2.3. When τ=0,

    (ⅰ) When ν10, ν20

    u2,2,7(t,x,y)=eiξ(2τ0(r2λr+μ)(2rλ)1(r+ν2ν1+ν2φ)1+τ0). (3.56)

    (ⅱ) When ν1=0, ν20

    u2,2,8(t,x,y)=eiξ(2τ0(r2λr+μ)(2rλ)(r+φ112λ)+τ0). (3.57)

    (ⅲ) When ν10, ν2=0

    u2,2,9(t,x,y)=eiξ(2τ0(r2λr+μ)(2rλ)(r12λ)+τ0), (3.58)

    where φ=p1Γ(nα+1)xααΓ(n)+p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n) ξ=q1Γ(nα+1)xααΓ(n)q2Γ(nβ+1)yββΓ(n)+ωΓ(nδ+1)tδδΓ(n)+θ0 for q1 and ω defined in Eq (3.36).

    Considering case 2.3 and utilizing Eq (3.1), (3.34) and the corresponding general solution of Eq (2.10), we construct the subsequent plethora of optical soliton solutions for (1.1):

    Family 2.3.1. When τ>0,

    (ⅰ) When ν10, ν20

    u2,3,1(t,x,y)=eiξ(12τ1λτ1r+τ1(r+12κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ)12λ)). (3.59)

    (ⅱ) When ν1=0, ν20

    u2,3,2(t,x,y)=eiξ(12τ1λτ1r+τ1(r+12κcoth(12κφ)12λ)). (3.60)

    (ⅲ) When ν10, ν2=0

    u2,3,3(t,x,y)=eiξ(12τ1λτ1r+τ1(r+12κtanh(12κφ)12λ)). (3.61)

    Family 2.3.2. When τ<0,

    (ⅰ) When ν10, ν20

    u2,3,4(t,x,y)=eiξ(12τ1λτ1r+τ1(r+12κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ)12λ)). (3.62)

    (ⅱ) When ν1=0, ν20

    u2,3,5(t,x,y)=eiξ(12τ1λτ1r+τ1(r+12κcot(12κφ)12λ)). (3.63)

    (ⅲ) When ν10, ν2=0

    u2,3,6(t,x,y)=eiξ(12τ1λτ1r+τ1(r12κtan(12κφ)12λ)). (3.64)

    Family 2.3.3. When τ=0,

    (ⅰ) When ν10, ν20

    u2,3,7(t,x,y)=eiξ(12τ1λτ1r+τ1(r+ν2ν1+ν2φ)). (3.65)

    (ⅱ) When ν1=0, ν20

    u2,3,8(t,x,y)=eiξ(12τ1λτ1r+τ1(r+φ112λ)). (3.66)

    (ⅲ) When ν10, ν2=0

    u2,3,9(t,x,y)=eiξ(12τ1λτ1r+τ1(r12λ)), (3.67)

    where φ=p1Γ(nα+1)xααΓ(n)+p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n) ξ=q1Γ(nα+1)xααΓ(n)q2Γ(nβ+1)yββΓ(n)+ωΓ(nδ+1)tδδΓ(n)+θ0 for q1 and ω defined in Eq (3.37).

    Considering case 2.4 and utilizing Eq (3.1), (3.34) and the corresponding general solution of Eq (2.10), we construct the subsequent plethora of optical soliton solutions for (1.1):

    Family 2.4.1. When τ>0,

    (ⅰ) When ν10, ν20

    u2,4,1(t,x,y)=eiξ(2(μτ1+14τ1λ2)(ν1cosh(12κφ)+ν2sinh(12κφ))κ(ν1sinh(12κφ)+ν2cosh(12κφ))+12τ1κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ)). (3.68)

    (ⅱ) When ν1=0, ν20

    u2,4,2(t,x,y)=eiξ(2μτ1+14τ1λ2κcoth(12κφ)+12τ1κcoth(12κφ)). (3.69)

    (ⅲ) When ν10, ν2=0

    u2,4,3(t,x,y)=eiξ(2μτ1+14τ1λ2κtanh(12κφ)+12τ1κtanh(12κφ)). (3.70)

    Family 2.4.2. When τ<0,

    (ⅰ) When ν10, ν20

    u2,4,4(t,x,y)=eiξ(2(μτ1+14τ1λ2)(ν1cos(12κφ)+ν2sin(12κφ))κ(ν1sin(12κφ)+ν2cos(12κφ))+12τ1κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ)). (3.71)

    (ⅱ) When ν1=0, ν20

    u2,4,5(t,x,y)=eiξ(2μτ1+14τ1λ2κcot(12κφ)+12τ1κcot(12κφ)). (3.72)

    (ⅲ) When ν10, ν2=0

    u2,4,6(t,x,y)=eiξ(2μτ1+14τ1λ2κtan(12κφ)12τ1κtan(12κφ)). (3.73)

    Family 2.4.3. When τ=0,

    (ⅰ) When ν10, ν20

    u2,4,7(t,x,y)=eiξ((μτ1+14τ1λ2)(12λ+ν2ν1+ν2φ)1+τ1(12λ+ν2ν1+ν2φ)). (3.74)

    (ⅱ) When ν1=0, ν20

    u2,4,8(t,x,y)=eiξ((μτ1+14τ1λ2)φ+τ1φ), (3.75)

    where φ=p1Γ(nα+1)xααΓ(n)+p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n) ξ=q1Γ(nα+1)xααΓ(n)q2Γ(nβ+1)yββΓ(n)+ωΓ(nδ+1)tδδΓ(n)+θ0 for q1 and ω defined in Eq (3.38).

    Considering case 2.5 and utilizing Eq (3.1), (3.34) and the corresponding general solution of Eq (2.10), we construct the subsequent plethora of optical soliton solutions for (1.1):

    Family 2.5.1. When τ>0,

    (ⅰ) When ν10, ν20

    u2,5,1(t,x,y)=eiξ((2τ0(12λ+12κ)+λτ0)(12κ+12κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ))+τ0). (3.76)

    (ⅱ) When ν1=0, ν20

    u2,5,2(t,x,y)=eiξ(2τ0(12λ+12κ)+λτ012κ+12κcoth(12κφ)+τ0). (3.77)

    (ⅲ) When ν10, ν2=0

    u2,5,3(t,x,y)=eiξ(2τ0(12λ+12κ)+λτ012κ+12κtanh(12κφ)+τ0). (3.78)

    Family 2.5.2. When τ<0,

    (ⅰ) When ν10, ν20

    u2,5,4(t,x,y)=eiξ((2τ0(12λ+12κ)+λτ0)(12κ+12κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ))+τ0). (3.79)

    (ⅱ) When ν1=0, ν20

    u2,5,5(t,x,y)=eiξ(2τ0(12λ+12κ)+λτ012κ+12κcot(12κφ)+τ0). (3.80)

    (ⅲ) When ν10, ν2=0

    u2,5,6(t,x,y)=eiξ(2τ0(12λ+12κ)+λτ012κ12κtan(12κφ)+τ0). (3.81)

    Family 2.5.3. When τ=0,

    (ⅰ) When ν10, ν20

    u2,5,7(t,x,y)=eiξ((2τ0(12λ+12κ)+λτ0)(12λ+12κ+ν2ν1+ν2φ)1+τ0). (3.82)

    (ⅱ) When ν1=0, ν20

    u2,5,8(t,x,y)=eiξ(2τ0(12λ+12κ)+λτ012κ+φ1+τ0). (3.83)

    (ⅲ) When ν10, ν2=0

    u2,5,9(t,x,y)=eiξ(22τ0(12λ+12κ)+λτ0κ+τ0), (3.84)

    where φ=p1Γ(nα+1)xααΓ(n)+p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n) ξ=q1Γ(nα+1)xααΓ(n)q2Γ(nβ+1)yββΓ(n)+ωΓ(nδ+1)tδδΓ(n)+θ0 for q1 and ω defined in Eq (3.39).

    Considering case 2.6 and utilizing Eq (3.1), (3.34) and the corresponding general solution of Eq (2.10), we construct the subsequent plethora of optical soliton solutions for (1.1):

    Family 2.6.1. When τ>0,

    (ⅰ) When ν10, ν20

    u2,6,1(t,x,y)=eiξ(2(μτ114τ1λ2)(ν1cosh(12κφ)+ν2sinh(12κφ))κ(ν1sinh(12κφ)+ν2cosh(12κφ))+12τ1κ(ν1sinh(12κφ)+ν2cosh(12κφ))ν1cosh(12κφ)+ν2sinh(12κφ)). (3.85)

    (ⅱ) When ν1=0, ν20

    u2,6,2(t,x,y)=eiξ(2μτ114τ1λ2κcoth(12κφ)+12τ1κcoth(12κφ)). (3.86)

    (ⅲ) When ν10, ν2=0

    u2,6,3(t,x,y)=eiξ(2μτ114τ1λ2κtanh(12κφ)+12τ1κtanh(12κφ)). (3.87)

    Family 2.6.2. When τ<0,

    (ⅰ) When ν10, ν20

    u2,6,4(t,x,y)=eiξ(2(μτ114τ1λ2)(ν1cos(12κφ)+ν2sin(12κφ))κ(ν1sin(12κφ)+ν2cos(12κφ))+12τ1κ(ν1sin(12κφ)+ν2cos(12κφ))ν1cos(12κφ)+ν2sin(12κφ)). (3.88)

    (ⅱ) When ν1=0, ν20

    u2,6,5(t,x,y)=eiξ(2μτ114τ1λ2κcot(12κφ)+12τ1κcot(12κφ)). (3.89)

    (ⅲ) When ν10, ν2=0

    u2,6,6(t,x,y)=eiξ(2μτ114τ1λ2κtan(12κφ)12τ1κtan(12κφ)). (3.90)

    Family 2.6.3. When τ=0,

    (ⅰ) When ν10, ν20

    u2,6,7(t,x,y)=eiξ((μτ114τ1λ2)(12λ+ν2ν1+ν2φ)1+τ1(12λ+ν2ν1+ν2φ)). (3.91)

    (ⅱ) When ν1=0, ν20

    u2,6,8(t,x,y)=eiξ((μτ114τ1λ2)φ+τ1φ), (3.92)

    where φ=p1Γ(nα+1)xααΓ(n)+p2Γ(nβ+1)yββΓ(n)ϱΓ(nδ+1)tδδΓ(n) ξ=q1Γ(nα+1)xααΓ(n)q2Γ(nβ+1)yββΓ(n)+ωΓ(nδ+1)tδδΓ(n)+θ0 for q1 and ω defined in Eq (3.40).

    We present graphic representations of several wave patterns observed in the (2+1)-dimensional gFKMNE under study in this section. In contour and 3D visuals, we found and visualized wave patterns using the extended (GG)-expansion method and generalized (r+GG)-expansion method. The depictions show that the created soliton solutions, often referred to as dark soliton lattices, are cyclically structured in optical media. Dark soliton formations are collections of dark solitaires arranged regularly inside a medium. They are frequently shaped like wave channels or optical fibers. Similar to the lattice of a crystal, these solitons are restricted to areas of decreased optical intensity within a larger field that retain their structure and amplitude while moving. Not only can dark soliton lattices be used to precisely control and manipulate optical waves, but also to process information and communications networks, provide insight into nonlinear optics phenomena, provide platforms for basic research on the behavior of wave propagation, and have potential applications in signal processing, optical communication, and sensing. Thus, dark soliton lattices offer a stimulating new topic for nonlinear optics research that will enable both basic and applied investigations.

    Figure 1.  The 3D and contour visuals of the periodic soliton solution u1,1,2 articulated in (3.9) are graphed for λ=4,μ=3,n=2,α=1,β=1,δ=1,τ1=5,p1=2,p2=1,q2=5,a=10,b=3,θ0=0,t=0.
    Figure 2.  The 3D and contour visuals of the periodic soliton solution u1,1,6 articulated in (3.13) are graphed for λ=1,μ=2,n=2,α=1,β=1,δ=0.9,τ1=10,p1=3,p2=4,q2=1,a=2,b=4,θ0=1,t=10.
    Figure 3.  The 3D and contour visuals of the periodic soliton solution u1,1,8 articulated in (3.15) are graphed for λ=4,μ=4,n=2,α=0.7,β=0.8,δ=0.9,τ1=5,p1=1,p2=2,q2=3,a=5,b=1,θ0=2,t=20.
    Figure 4.  The 3D and contour visuals of the periodic soliton solution u1,2,3 articulated in (3.19) are graphed for λ=4,μ=1,n=3,α=0.3,β=0.5,δ=0.2,τ1=6,p1=3,p2=1,q2=7,a=10,b=10,θ0=1,t=50.
    Figure 5.  The 3D and contour visuals of the periodic soliton solution u2,2,3 articulated in (3.52) are graphed for λ=3,μ=1,n=1,α=1,β=1,δ=1,τ0=6,p1=10,p2=5,q2=6,a=2,b=2,θ0=0,t=100,r=1.
    Figure 6.  The 3D and contour visuals of the periodic soliton solution u2,2,8 articulated in (3.57) are graphed for λ=6,μ=9,n=2,α=1,β=1,δ=0.5,τ0=1,p1=20,p2=1,q2=8,a=1,b=4,θ0=100,t=30,r=10.
    Figure 7.  The 3D and contour visuals of the periodic soliton solution u2,4,2 articulated in (3.69) are graphed for λ=5,μ=0,n=1,α=1,β=0.9,δ=0.5,τ1=1,p1=5,p2=3,q2=1,a=2,b=5,θ0=0,t=50.
    Figure 8.  The 3D and contour visuals of the periodic soliton solution u2,4,6 articulated in (3.73) are graphed for λ=0,μ=1,n=2,α=1,β=1,δ=1,τ1=15,p1=2,p2=7,q2=3,a=5,b=10,θ0=1,t=100.

    In summary, this study derived optical soliton solutions for the (2+1)-dimensional gFKMNE, a nonlinear model that explains pulse transmission in communication structures and optical fibers. The two improved versions of the (GG)-expansion method were the extended (GG)-expansion method and the generalized (r+GG)-expansion method. These techniques relied on the model's wave translation into integer-order NODEs. By assuming a series-form solution, the strategic techniques further turned the resulting NODEs into a system of nonlinear algebraic equations. Using the Maple program, these equations were solved to produce the gFKMNE's optical soliton solutions. In comparison with the extended (GG)-expansion method, it was observed that the generalized (r+GG)-expansion method produced more families of optical soliton solutions. Furthermore, all of the results obtained by the extended (GG)-expansion method could be obtained from the results of the generalized (r+GG)-expansion method with r=0. As a result, the generalized (r+GG)-expansion method was found to be more effective and superior. Furthermore, using 3D and contour visuals, it was demonstrated that the generated soliton solutions are periodically structured in optical media, which are called dark soliton lattices. Such dark soliton lattices have applications in optical communications, optical signal processing, and nonlinear optics, among other areas. Moreover, the effectiveness and dependability of the methods used in this study show how widely relevant they are to nonlinear issues in a range of scientific fields. The soliton dynamics and their implications for the models studied have been greatly improved by the (GG)-expansion approaches; however, it is important to recognize that this technique has limitations, especially when the highest derivative is not uniformly balanced with the nonlinear term. Notwithstanding this shortcoming, the paper highlights the amount of unaddressed concerns regarding nonlinear behavior and soliton dynamics and provides fresh avenues for future research in the area.



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