Research article

The new soliton solution types to the Myrzakulov-Lakshmanan-XXXII-equation

  • Received: 13 December 2023 Revised: 19 January 2024 Accepted: 23 January 2024 Published: 02 February 2024
  • MSC : 35C07, 35Q51, 83C15

  • Our attention concenters on deriving diverse forms of the soliton arising from the Myrzakulov-Lakshmanan XXXII (M-XXXII) that describes the generalized Heisenberg ferromagnetic equation. This model has been solved numerically only using the N-fold Darboux Transformation method, not solved analytically before. We will derive new types of the analytical soliton solutions that will be constructed for the first time in the framework of three impressive schemas that are prepared for this target. These three techniques are the Generalized Kudryashov scheme (GKS), the (G'/G)-expansion scheme and the extended direct algebraic scheme (EDAS). Moreover, we will establish the 2D, 3D graphical simulations that clear the new dynamic properties of our achieved solutions.

    Citation: Emad H. M. Zahran, Ahmet Bekir, Reda A. Ibrahim, Ratbay Myrzakulov. The new soliton solution types to the Myrzakulov-Lakshmanan-XXXII-equation[J]. AIMS Mathematics, 2024, 9(3): 6145-6160. doi: 10.3934/math.2024300

    Related Papers:

    [1] Chun Huang, Zhao Li . Soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation. AIMS Mathematics, 2022, 7(8): 14460-14473. doi: 10.3934/math.2022797
    [2] Ghazala Akram, Saima Arshed, Maasoomah Sadaf, Hajra Mariyam, Muhammad Nauman Aslam, Riaz Ahmad, Ilyas Khan, Jawaher Alzahrani . Abundant solitary wave solutions of Gardner's equation using three effective integration techniques. AIMS Mathematics, 2023, 8(4): 8171-8184. doi: 10.3934/math.2023413
    [3] Elsayed M. E. Zayed, Mona El-Shater, Khaled A. E. Alurrfi, Ahmed H. Arnous, Nehad Ali Shah, Jae Dong Chung . Dispersive optical soliton solutions with the concatenation model incorporating quintic order dispersion using three distinct schemes. AIMS Mathematics, 2024, 9(4): 8961-8980. doi: 10.3934/math.2024437
    [4] Abdulah A. Alghamdi . Analytical discovery of dark soliton lattices in (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation. AIMS Mathematics, 2024, 9(8): 23100-23127. doi: 10.3934/math.20241123
    [5] Raed Qahiti, Naher Mohammed A. Alsafri, Hamad Zogan, Abdullah A. Faqihi . Kink soliton solution of integrable Kairat-X equation via two integration algorithms. AIMS Mathematics, 2024, 9(11): 30153-30173. doi: 10.3934/math.20241456
    [6] Emad H. M. Zahran, Omar Abu Arqub, Ahmet Bekir, Marwan Abukhaled . New diverse types of soliton solutions to the Radhakrishnan-Kundu-Lakshmanan equation. AIMS Mathematics, 2023, 8(4): 8985-9008. doi: 10.3934/math.2023450
    [7] Abdulghani R. Alharbi . Traveling-wave and numerical solutions to a Novikov-Veselov system via the modified mathematical methods. AIMS Mathematics, 2023, 8(1): 1230-1250. doi: 10.3934/math.2023062
    [8] Mohammed Aly Abdou, Loubna Ouahid, Saud Owyed, A. M. Abdel-Baset, Mustafa Inc, Mehmet Ali Akinlar, Yu-Ming Chu . Explicit solutions to the Sharma-Tasso-Olver equation. AIMS Mathematics, 2020, 5(6): 7272-7284. doi: 10.3934/math.2020465
    [9] Naeem Ullah, Muhammad Imran Asjad, Azhar Iqbal, Hamood Ur Rehman, Ahmad Hassan, Tuan Nguyen Gia . Analysis of optical solitons solutions of two nonlinear models using analytical technique. AIMS Mathematics, 2021, 6(12): 13258-13271. doi: 10.3934/math.2021767
    [10] Muhammad Imran Asjad, Waqas Ali Faridi, Adil Jhangeer, Maryam Aleem, Abdullahi Yusuf, Ali S. Alshomrani, Dumitru Baleanu . Nonlinear wave train in an inhomogeneous medium with the fractional theory in a plane self-focusing. AIMS Mathematics, 2022, 7(5): 8290-8313. doi: 10.3934/math.2022462
  • Our attention concenters on deriving diverse forms of the soliton arising from the Myrzakulov-Lakshmanan XXXII (M-XXXII) that describes the generalized Heisenberg ferromagnetic equation. This model has been solved numerically only using the N-fold Darboux Transformation method, not solved analytically before. We will derive new types of the analytical soliton solutions that will be constructed for the first time in the framework of three impressive schemas that are prepared for this target. These three techniques are the Generalized Kudryashov scheme (GKS), the (G'/G)-expansion scheme and the extended direct algebraic scheme (EDAS). Moreover, we will establish the 2D, 3D graphical simulations that clear the new dynamic properties of our achieved solutions.



    The NLPDEs play important roles in different fields of science such as [1], who introduced a book for the Nonlinear Partial Differential Equations for Scientists and Engineers. The dispersion-less equations include the integrable hydrodynamic equations that are a constitute branch of the integrable partial differential equations which are crucial to applications for its extensive application in various branches as condensed matter theory, string theory, quantum arena theories, and conformal field theory; particularly, [2] proposed CTE solvability, nonlocal symmetry, and interaction solutions of coupled integrable dispersion-less system; complexity, [3] who investigated the Dispersion-less Toda hierarchy and two-dimensional string theory; and [4], who introduced the Topological conformal field theory with a rational W potential and the dispersion less KP hierarchy. Here, we will focus on one of the famous models that plays significant role in the magnetism theory. The suggested model is a general form of the well-known Heisenberg ferromagnetic equation, which is considered one of the fundamental integrable systems in soliton theory and has been studied by some authors. The study of the (1+1)-dimensions: Integrable generalized Heisenberg ferromagnetic equations: Reductions and gauge equivalence [5]. Furthermore, Ratbay Myrzakulov [6] introduced this model as a sigma model with potentials, which is one of the coupled integrable dispersion-less equations that covers generalized Heisenberg ferromagnetic equations and [7] who studied the Coupled dispersion less and generalized Heisenberg ferromagnetic equations with self-consistent sources: Geometry and equivalence. Moreover, there are many authors that have extracted studies of high quality and are qualitative studying the soliton solutions for the (2+1)-dimensional HFSC equation, which has a significant role in the phenomena and processes in various fields that relate to the ferromagnetic materials, nonlinear optics and optical fibers [8,9,10,11,12]. In the same way, various magnetic interactions of the (2+1) dimensional HFSC equation that have integral behaviors and split into classical and semi-classical limit have been studied in [13,14,15,16,17]. Furthermore, the soliton solutions for various nonlinear problems in different fields have been published [18,19,20,21,22,23,24].

    The suggested model has been introduced in reference [12] to be

    Qxt+2Px4mQ=0,mx=σ(12(|Q|2)x+PQ+QP),nx=σ(PQ+QP),Px=2iτP2nQ, (1)

    where P(x,y,t),Q(x,y,t) are complex functions in the normalized spatial variables x and y and temporal variable t that are appropriate continuum approximation of the coherent magnetism amplitude to the bosonic operators at spin-lattice sites, while m,n are real functions, τ is real parameter and σ=±1. The main target is transforming this system to one equation that we will name the modified equation of the Myrzakulov-Lakshmanan XXXII, by differentiation the first and the last part of Eq (1) with respect to x we get

    Qxxt+2Pxx4mxQ4mQx=0.Pxx=2iτPx2nxQ2nQx. (2)

    By inserting the second part of Eq (2) into the first part of Eq (2) we get

    Qxxt4iτPx4nxQ4nQx4mxQ4mQx=0. (3)

    By inserting the second, third parts of Eq (1) into Eq (3) we get

    Qxxt4iτPx+4Qσ(PQ+QP)4nQx4Qσ(12(|Q|2)x+PQ+QP)4mQx=0. (4)

    The above equation can be reduced to

    Qxxt4iτPx4Qσ(12(|Q|2)x)4(m+n)Qx=0. (5)

    Now, let us suppose these complex transformations Q(ζ)=R1(ζ1)eiψ(x,t), ζ1=x+w1t, P(ζ)=R2(ζ2)eiψ(x,t), ζ2=x+w2t, where R1,R2 denote to the wave amplitudes, while w1,w2 are the soliton speeds and ψ=x+δt+θ0 denotes to the phase portion, δ is the wave number, θ0 is the phase constant.

    Qx=(R1+iR1)eiψ(x,t). (6)
    Px=(R2+iR2)eiψ(x,t). (7)
    Qxx=(R1+2iR1R1)eiψ(x,t). (8)
    Qxxt=(w1R1+i(δ+2w1)R1(w1+2δ)R1iδR1)eiψ(x,t). (9)
    (|Q|2)x=2R1R1. (10)

    By inserting the relations (6)–(10) into Eq (5) we get

    (w1R1+i(δ+2w1)R1(w1+2δ)R1iδR1)eiψ(x,t)4iτ(R2+iR2)eiψ(x,t)4(m+n)(R1+iR1)eiψ(x,t)4σR21R1eiψ(x,t)=0. (11)

    That can be divided into the following real and imaginary parts:

    w1R1(w1+2δ+4(m+n))R14σR21R1+4τR2=0. (12)
    (δ+2w1)R14τR2(4(m+n)+δ)R1=0. (13)

    By integrating Eq (13) with respect to ζ we get

    (δ+2w1)R14τR2(2(m+n)+δ2)R21=0. (14)

    By emerging Eqs (12) and (14) we obtain

    w1R1+(w1δ4(m+n))R14σR21R1(2(m+n)+δ2)R21=0. (15)

    Now, our aim is to extract the exact solutions of the Eq (15) in the framework of three various schemas namely the GKS [25,26,27,28,29], the (G'/G)-expansion scheme [30,31] and the EDAS [32,33,34].

    According to the GKS, any NLPDE which is

    Φ(R,Rx,Rt,Rxx,Rtt,...)=0. (16)

    With the transformation R(x,t)=R(ζ),ζ=kx±wt,where(k&w) are the wave number and traveling wave speed can be transformed to the ODE

    E(R,R,R,R,...)=0. (17)

    The Kudryashov scheme assumed the solution of Eq (17) in the form

    R(ζ)=Ni=1AiSi(ζ)Mj=1BjSj(ζ)=A0+A1S(ζ)+A2S2(ζ)+...B0+B1S(ζ)+B2S2(ζ)+..., (18)

    where Ai(i=0,1,...,N)&Bj(j=0,1,...,M) are constants which will be determined later such that AN0&BM0 and the function S(ζ) is the solution of the nonlinear equation

    dS(ζ)dζ=S2(ζ)S(ζ). (19)

    Integrating Eq (19) then solution takes the form

    S(ζ)=11+Ceζ, (20)

    here C is the integration constsnt.

    The next step in the Kudryashov scheme is determining the positive numbers N and M in solution Eq (18) by implementing the balance rule for the suggested model Eq (15) it implies M=1, hence according to the GKS N=M+1=2 thus the solution is

    R1(ζ)=A0+A1S(ζ)+A2S2(ζ)B0+B1S(ζ). (21)

    By evaluation R1,R1,R1,R1 and substituting into Eq (15), collecting and equating the coefficients of like powers of Si(ζ) to zero; one can obtain a system of equations from which the following two results will be appeared.

    (1)A0(1+3)A11283,A212(3+3)A1,δ4(m+n),B12(323)B0,w1(2+3)σA2112B20. (22)
    (2)A0(1+3)A112+83,A212(3+3)A1,δ4(m+n),B12(3+23)B0,w1(3+23)σA21123B20. (23)

    Now, let us deduce the corresponding solutions to the first result (22) that can be simplified to be

    A1=B0=σ=δ=n=1,A0=0.4,A2=0.634,B1=0.928,w1=0.3,m=1.25. (24)

    Substituting from (24) into (21) the solution will be

    R1(ζ)=0.4+S(ζ)0.634S2(ζ)10.928S(ζ). (25)

    Using Eq (20) with C=1 the solution after simplifying becomes

    R1(ζ)=0.034+0.2eξ0.4e2ξ(0.072+eξ)(1.+eξ). (26)

    Since Q(x,t)=R1(ζ1)eiψ(x,t),ζ1=x+y+w1t=x+y+0.3t,ψ=x+y+δt+θ0, then

    Q(x,t)=0.034+0.2e(x+0.3t)0.4e2(x+0.3t)(0.072+e(x+0.3t))(1+e(x+0.3t))ei(x+t+0.1).
    Q(x,t)=0.034+0.2e(x+0.3t)0.4e2(x+0.3t)(0.072+e(x+0.3t))(1+e(x+0.3t))(cos(x+t+0.1)+isin(x+t+0.1)). (27)
    ReQ(x,t)=0.034+0.2e(x+0.3t)0.4e2(x+0.3t)(0.072+e(x+0.3t))(1+e(x+0.3t))cos(x+t+0.1). (28)
    ImQ(x,t)=0.034+0.2e(x+0.3t)0.4e2(x+0.3t)(0.072+e(x+0.3t))(1+e(x+0.3t))sin(x+t+0.1). (29)

    By the same way for the second solution (23)

    A1=B0=σ=δ=n=1,A0=0.1,A2=2.366,B1=12.93,w1=0.02,m=1.25. (30)

    Substituting from (30) into (21) the solution will be

    R1(ζ)=0.1+S(ζ)2.366S2(ζ)1+12.93S(ζ). (31)

    Using Eq (20) with C=1 the solution after simplifying becomes

    R1(ζ)=1.466+0.8eξ0.1e2ξ(1+eξ)(13.93+eξ). (32)

    Then

    Q(x,t)=1.466+0.8e(x+0.02t)0.1e2(x+0.02t)(1+e(x+0.02t))(13.93+e(x+0.02t))ei(x+t+0.1).
    Q(x,t)=1.466+0.8e(x+0.02t)0.1e2(x+0.02t)(1+e(x+0.02t))(13.93+e(x+0.02t))(cos(x+t+0.1)+isin(x+t+0.1). (33)
    ReQ(x,t)=1.466+0.8e(x+0.02t)0.1e2(x+0.02t)(1+e(x+0.02t))(13.93+e(x+0.02t))cos(x+t+0.1). (34)
    ImQ(x,t)=1.466+0.8e(x+0.02t)0.1e2(x+0.02t)(1+e(x+0.02t))(13.93+e(x+0.02t))sin(x+t+0.1). (35)

    The (G'/G)-expansion scheme introduces the solution of Eq (17) to be in the form

    R(ζ)=A0+Mk=1Ak[GG]k,AM0, (36)

    where the function G(ζ) achieves the 2nd-order differential equation G+μG+λG=0, that admits the following forms of solutions according to the discriminate of this equation which is either any one of these inequalities μ24λ0, μ24λ0, and μ24λ=0.

    (I) When μ24λ0, the solution is

    (GG)=μ24λ2(l1sinh(μ24λ2)ζ+l2cosh(μ24λ2)ζl1cosh(μ24λ2)ζ+l2sinh(μ24λ2)ζ)μ2. (37)

    (II) When μ24λ0, the solution is

    (GG)=μ24λ2(l1sin(μ24λ2)ζ+l2cos(μ24λ2)ζl1cos(μ24λ2)ζ+l2sin(μ24λ2)ζ)μ2. (38)

    (III) When μ24λ=0, the solution is

    (GG)=(l2l1+l2ζ)μ2, (39)

    where M appearing in Eq (36) has been calculated before to be M=1, hence the solution is

    R1(ζ)=A0+A1(GG). (40)

    By substituting R1,R21R1,R1 into Eq (15), collecting and equating the coefficients of various powers of (GG)i to zero, this leads to a system of equations from which the following two results will be emerged

    (1)A0=0,μ=0,w=2δ3+2λ,A1=1.7δ2λσ3σ,m=0.01{72n54δ81δ2σ(3σ+2λσ)2+54δ2λσ(3σ+2λσ)2+432δλσ2(3σ+2λσ)2288δλ2σ2(3σ+2λσ)2+94δ32σ(3σ+2λσ)3262δ32λσ(3σ+2λσ)3227δ2(3σ+2λσ)108δσ(3σ+2λσ)+216δσλ(3σ+2λσ)94δ323σ+2λσ}. (41)
    (2)A0=0,μ=0,w=2δ3+2λ,A1=1.7δ2λσ3σ,m=0.01{72n54δ81δ2σ(3σ+2λσ)2+54δ2λσ(3σ+2λσ)2+432δλσ2(3σ+2λσ)2288δλ2σ2(3σ+2λσ)2+94δ32σ(3σ+2λσ)3262δ32λσ(3σ+2λσ)3227δ2(3σ+2λσ)108δσ(3σ+2λσ)+216δσλ(3σ+2λσ)94δ323σ+2λσ}. (42)
    (3)w=0.2(3δ+2σA20),λ=0,m=0.3(4nδ),μ=3.5A0σ3δ2σA203δ+2σA20,A1=0.63δ2σA20σ. (43)
    (4)w=0.2(3δ+2σA20),λ=0,m=0.3(4nδ),μ=3.5A0σ3δ2σA20,A1=0.63δ2σA20σ. (44)

    Let us now derive the corresponding solutions to the first result that can be simplified to be

    A0=0,A1=1.7,μ=0,w=1,σ=n=δ=1,λ=2,m=2.14,l1=1,l2=2. (45)

    The solution in the framework of this result can be extracted as follow:

    (GG)=μ24λ2(l1sin(μ24λ2)ζ+l2cos(μ24λ2)ζl1cos(μ24λ2)ζ+l2sin(μ24λ2)ζ)μ2.
    (GG)=2i(sin2iζ+2cos2iζcos2iζ+2sin2iζ). (46)
    R1(ζ)=A0+A1(GG).
    R1(ζ)=8i(sin2iζ+2cos2iζcos2iζ+2sin2iζ).
    R1(ζ)=8i(isinh2ζ+2cosh2ζcosh2ζ+2isinh2ζ).
    R1(ζ)=(8sinh2ζ+16icosh2ζcosh2ζ+2isinh2ζ). (47)

    Hence, Q(ζ)=R1(ζ1)eiψ(x,t),ζ1=x+w1t,,ψ=x+δt+θ0.

    Q(x,t)=(40sinh2ζcosh2ζ+16icosh22ζ+4sinh22ζ)ei(x+t+0.1). (48)
    Q(x,t)=(40sinh2ζcosh2ζ+16icosh22ζ+4sinh22ζ)(cos(x+t+0.1)+isin(x+t+0.1)). (49)
    ReQ(x,t)=((40sinh2ζcosh2ζcos(x+t+0.1)16sin(x+t+0.1))cosh22ζ+4sinh22ζ). (50)
    ImQ(x,t)=((40sinh2ζcosh2ζsin(x+t+0.1)+16cos(x+t+0.1))cosh22ζ+4sinh22ζ). (51)

    This technique proposes the solution of Eq (17) to be

    R1(ζ)=Mi=0biφi(ζ),φ2=αφ2+βφ3+γφ4. (52)

    The solution according to this technique for the suggested model whose balance M=1 is

    R1=b0+b1φ. (53)

    Consequently

    R1=b1φ=b1αφ2+βφ3+γφ4. (54)
    R1=b1φ=b1(αφ+1.5βφ2+2γφ3). (55)
    R1=b1φ=b1(α+3βφ+6γφ2)αφ2+βφ3+γφ4. (56)

    By inserting the relations (53)–(56) into Eq (15) we get

    w1R1(α+3βφ+6γφ2)+(w1δ4(m+n))R14σ(b0+b1φ)2R1(2(m+n)+δ2)R21=0.

    Collecting and equating the coefficients of various powers of φi to zero we get a system whose unique solution is:

    α=(4σb21+6wγ)wb21(0.5δ+2η+2μ){b20(0.5δ+2η+2μ)+b21(0.5δ+2η+2μ)4σb21+6wγ(3w+3δ+4μ+4η+4σb20)},β=0.3(4σb21+6wγ)wb21(0.5δ+2η+2μ){b0b1(δ+4η+4μ)+8σb0b31(0.5δ+2η+2μ)4σb21+6wγ}. (57)

    This result can be simplified to be

    α=20,β=3,γ=1,σ=w=δ=μ=η=b0=b1=1. (58)

    Thus the solution is R1=b0+b1φ, where φ can be derived from the relation

    φ=20φ2+3φ3+γφ4.

    To be

    φ=220(e20ζ1.520)1(e20ζ1.520)2. (59)
    R1=1+220(e20ζ1.520)1(e20ζ1.520)2. (60)

    This result generates two results; we will implement the identical solution for one of them which is:

    R1=1+1+1+2(e22ζ2e2ζ1.5)(e22ζ2e2ζ1.5).
    Q(x,t)={1+220(e20ζ1.520)1(e20ζ1.520)2}ei(x+t+0.1). (61)
    ReQ(x,t)={1+220(e20ζ1.520)1(e20ζ1.520)2}cos(x+t+0.1). (62)
    ImQ(x,t)={1+220(e20ζ1.520)1(e20ζ1.520)2}sin(x+t+0.1). (63)

    The other solutions corresponding to the other results can be extracted in the same connection.

    Throughout this work, three various schemas have been introduced to obtain new perceptions of the soliton solutions for the Myrzakulov-Lakshmanan XXXII-equation. These three schemas are the GKS, the (G'/G)-expansion schema and the EDAS. These suggested schemas have been implemented in the same way and are parallel. All these schemas have been used for the first time for this target. Each schema archives many forms of results from which we choose only one result and design the corresponding solution. The designed solutions using these three schemas have been established. The obtained solutions appear in many forms as M-shaped soliton solutions, W-shaped soliton solutions Figures 14, kink soliton solution Figure 5, bright soliton solution Figure 6 and hyperbolic function Figures 7 and 8. Our achieved soliton solutions in the framework of any other methods have not been achieved before. The realized soliton solutions are new compared with [12], who solved the suggested model numerically using the N-fold Darboux Transformation method; hence, the novelty of our obtained solutions is clear. The 2D, 3D soliton behaviors that describe the dynamic properties for all achieved soliton solutions that have emerged from the suggested model have been configured.

    Figure 1.  The soliton behavior of the Re. Part Eq (28) in 2D and 3D with values: A1=B0=σ=δ=n=1,A0=0.4,A2=0.634,B1=0.928,w1=0.3,m=1.25.
    Figure 2.  The soliton behavior of the Im. Part Eq (29) in 2D and 3D with values:A1=B0=σ=δ=n=1,A0=0.4,A2=0.634,B1=0.928,w1=0.3,m=1.25.
    Figure 3.  The soliton behavior of the Re. Part Eq (34) in 2D and 3D with values:A1=B0=σ=δ=n=1,A0=0.1,A2=2.366,B1=12.93,w1=0.02,m=1.25.
    Figure 4.  The soliton behavior of the Im. Part Eq (35) in 2D and 3D with values: A1=B0=σ=δ=n=1,A0=0.1,A2=2.366,B1=12.93,w1=0.02,m=1.25.
    Figure 5.  The soliton behavior of the Re. Part Eq (50) in 2D and 3D with values: A0=0,A1=1.7,μ=0,w=1,σ=n=δ=1,λ=2,m=2.14,l1=1,l2=2.
    Figure 6.  The soliton behavior of the Im. Part Eq (51) in 2D and 3D with values: A0=0,A1=1.7,μ=0,w=1,σ=n=δ=1,λ=2,m=2.14,l1=1,l2=2.
    Figure 7.  The soliton behavior of the Re. Part Eq (62) in 2D and 3D with values: α=20,β=3,γ=1,σ=w=δ=μ=η=b0=b1=1.
    Figure 8.  The soliton behavior of the Im. Part Eq (63) in 2D and 3D with values: α=20,β=3,γ=1,σ=w=δ=μ=η=b0=b1=1.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by the Ministry of Science and Higher Education of the Republic of Kazakhstan, Grant AP14870191.

    The authors declare no conflict of interest.



    [1] L. Debnath, Nonlinear partial differential equations for scientists and engineers, Massachusetts: Birkhäuser Boston, 2005. https://doi.org/10.1007/b138648
    [2] J. Yu, B. Ren, P. Liu, J. Zhou, CTE solvability, nonlocal symmetry, and interaction solutions of coupled integrable dispersion-less system, Complexity, 2022 (2022), 32211447. https://doi.org/10.1155/2022/3221447
    [3] K. Takasaki, Dispersionless Toda hierarchy and two-dimensional string theory, Commun. Math. Phys., 170 (1995), 101–116. https://doi.org/10.1007/BF02099441 doi: 10.1007/BF02099441
    [4] S. Aoyama, Y. Kodama, Topological conformal field theory with a rational W potential and the dispersionless KP hierarchy, Mod. Phys. Lett. A, 9 (1994), 2481–2492. https://doi.org/10.1142/S0217732394002355 doi: 10.1142/S0217732394002355
    [5] Z. Sagidullayeva, K. Yesmakhanova, R. Myrzakulov, Z. Myrzakulova, N. Serikbayev, G. Nugmanova, et al., Integrable generalized Heisenberg ferromagnet equations in 1+1 dimensions: reductions and gauge equivalence, arXiv: 2205.02073.
    [6] R. Myrzakulov, On some sigma models with potentials and the Klein-Gordon type equations, arXiv: hep-th/9812214.
    [7] K. Yesmakhanova, G. Nugmanova, G. Shaikhova, G. Bekova, R. Myrzakulov, Coupled dispersionless and generalized Heisenberg ferromagnet equations with self-consistent sources: geometry and equivalence, Int. J. Geom. Methods M., 17 (2020), 2050104. https://doi.org/10.1142/S0219887820501042 doi: 10.1142/S0219887820501042
    [8] M. Latha, C. Christal Vasanthi, An integrable model of (2+1)-dimensional Heisenberg ferromagnetic spin chain and soliton excitations, Phys. Scr., 89 (2014), 065204. https://doi.org/10.1088/0031-8949/89/6/065204
    [9] H. Triki, A. Wazwaz, New solitons and periodic wave solutions for the (2+1) dimensional Heisenberg ferromagnetic spin chain equation, J. Electromagnet. Wave., 30 (2016), 788–794. https://doi.org/10.1080/09205071.2016.1153986 doi: 10.1080/09205071.2016.1153986
    [10] M. Inc, A. Aliyu, A. Yusuf, D. Baleanu, Optical solitons and modulation instability analysis of an integrable model of (2+1)-Dimensional Heisenberg ferromagnetic spin chain equation, Micro Nanostructures, 112 (2017), 628–638. https://doi.org/10.1016/j.spmi.2017.10.018 doi: 10.1016/j.spmi.2017.10.018
    [11] S. Rayhanul Islam, M. Bashar, N. Muhammad, Immeasurable soliton solutions and enhanced (G'/G)-expansion method, Physics Open, 9 (2021), 100086. https://doi.org/10.1016/j.physo.2021.100086 doi: 10.1016/j.physo.2021.100086
    [12] B. Deng, H. Hao, Breathers, rogue waves and semi-rational solutions for a generalized Heisenberg ferromagnetic equation, Appl. Math. Lett., 140 (2023), 108550. https://doi.org/10.1016/j.aml.2022.108550
    [13] M. Daniel, L. Kavitha, R. Amuda, Soliton spin excitations in an anisotropic Heisenberg ferromagnet with octupole-dipole interaction, Phys. Rev. B, 59 (1999), 13774. https://doi.org/10.1103/PhysRevB.59.13774 doi: 10.1103/PhysRevB.59.13774
    [14] H. Triki, A. Wazwaz, New solitons and periodic wave solutions for the (2+1) dimensional Heisenberg ferromagnetic spin chain equation, J. Electromagnet. Wave., 30 (2016), 788–794. https://doi.org/10.1080/09205071.2016.1153986 doi: 10.1080/09205071.2016.1153986
    [15] M. Bashar, S. Rayhanul Islam, D. Kumar, Construction of traveling wave solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation, Partial Differential Equations in Applied Mathematics, 4 (2021), 100040. https://doi.org/10.1016/j.padiff.2021.100040 doi: 10.1016/j.padiff.2021.100040
    [16] M. Bashar, S. Rayhanul Islam, Exact solutions to the (2+1)-Dimensional Heisenberg ferromagnetic spin chain equation by using modified simple equation and improve F-expansion methods, Physics Open, 5 (2020), 100027. https://doi.org/10.1016/j.physo.2020.100027
    [17] C. Christal Vasanthi, M. Latha, Heisenberg ferromagnetic spin chain with bilinear and biquadratic interactions in (2+1)-dimensions, Commun. Nonlinear Sci., 28 (2015), 109–122. https://doi.org/10.1016/j.cnsns.2015.04.012 doi: 10.1016/j.cnsns.2015.04.012
    [18] E. Zahran, A. Bekir, New unexpected variety of solitons arising from spatio-temporal dispersion (1+1) dimensional Ito-equation, Mod. Phys. Lett. B, 38 (2024), 2350258. https://doi.org/10.1142/S0217984923502585 doi: 10.1142/S0217984923502585
    [19] E. Zahran, A. Bekir, Optical soliton solutions to the perturbed Biswas-Milovic equation with Kudryashov's law of refractive index, Opt. Quant. Electron., 55 (2023), 1211. https://doi.org/10.1007/s11082-023-05453-w doi: 10.1007/s11082-023-05453-w
    [20] S. Kumar, R. Jiwari, R. Mittal, J. Awrejcewicz, Dark and bright soliton solutions and computational modeling of nonlinear regularized long wave model, Nonlinear Dyn., 104 (2021), 661–682. https://doi.org/10.1007/s11071-021-06291-9 doi: 10.1007/s11071-021-06291-9
    [21] E. Zahran, A. Bekir, New unexpected soliton solutions to the generalized (2+1) Schrödinger equation with its four mixing waves, Int. J. Mod. Phys. B, 36 (2022), 2250166. https://doi.org/10.1142/S0217979222501661 doi: 10.1142/S0217979222501661
    [22] M. Younis, T. Sulaiman, M. Bilal, S. Ur Rehman, U. Younas, Modulation instability analysis optical and other solutions to the modified nonlinear Schrödinger equation, Commun. Theor. Phys., 72 (2020), 065001. https://doi.org/10.1088/1572-9494/ab7ec8 doi: 10.1088/1572-9494/ab7ec8
    [23] E. Zahran, A. Bekir, R. Ibrahim, New optical soliton solutions of the popularized anti-cubic nonlinear Schrödinger equation versus its numerical treatment, Opt. Quant. Electron., 55 (2023), 377. https://doi.org/10.1007/s11082-023-04624-z doi: 10.1007/s11082-023-04624-z
    [24] E. Zahran, A. Bekir, M. Shehata, New diverse variety analytical optical soliton solutions for two various models that are emerged from the perturbed nonlinear Schrödinger equation, Opt. Quant. Electron., 55 (2023), 190. https://doi.org/10.1007/s11082-022-04423-y doi: 10.1007/s11082-022-04423-y
    [25] M. Ali Akbar, A. Wazwaz, F. Mahmud, D. Baleanu, R. Roy, H. Barman, et al., Dynamical behavior of solitons of the perturbed nonlinear Schrödinger equation and microtubules through the generalized Kudryashov scheme, Results Phys., 43 (2022), 106079. https://doi.org/10.1016/j.rinp.2022.106079
    [26] L. Ouahid, S. Owyed, M. Abdou, N. Alshehri, S. Elagan, New optical soliton solutions via generalized Kudryashov's scheme for Ginzburg-Landau equation in fractal order, Alex. Eng. J., 60 (2021), 5495–5510. https://doi.org/10.1016/j.aej.2021.04.030 doi: 10.1016/j.aej.2021.04.030
    [27] G. Genc, M. Ekici, A. Biswas, M. Belic, Cubic-quartic optical solitons with Kudryashov's law of refractive index by F-expansions schemes, Results Phys., 18 (2020), 103273. https://doi.org/10.1016/j.rinp.2020.103273 doi: 10.1016/j.rinp.2020.103273
    [28] D. Kumar, A. Seadawy, A. Joardar, Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese J. Phys., 56 (2018), 75–85. https://doi.org/10.1016/j.cjph.2017.11.020 doi: 10.1016/j.cjph.2017.11.020
    [29] C. Gomez S, H. Roshid, M. Inc, L. Akinyemi, H. Rezazadeh, On soliton solutions for perturbed Fokas-Lenells equation, Opt. Quant. Electron., 54 (2022), 370. https://doi.org/10.1007/s11082-022-03796-4 doi: 10.1007/s11082-022-03796-4
    [30] E. Zahran, A. Bekir, New variety diverse solitary wave solutions to the DNA Peyrard-Bishop model, Mod. Phys. Lett. B, 37 (2023), 2350027. https://doi.org/10.1142/S0217984923500276 doi: 10.1142/S0217984923500276
    [31] E. Zahran, A. Bekir, New solitary solutions to the nonlinear Schrödinger equation under the few-cycle pulse propagation property, Opt. Quant. Electron., 55 (2023), 696. https://doi.org/10.1007/s11082-023-04916-4 doi: 10.1007/s11082-023-04916-4
    [32] E. Zahran, A. Bekir, New diverse soliton solutions for the coupled Konno-Oono equations, Opt. Quant. Electron., 55 (2023), 112. https://doi.org/10.1007/s11082-022-04376-2 doi: 10.1007/s11082-022-04376-2
    [33] E. Zahran, H. Ahmad, T. Saeed, T. Botmart, New diverse variety for the exact solutions to Keller-Segel-Fisher system, Results Phys., 35 (2022), 105320. https://doi.org/10.1016/j.rinp.2022.105320 doi: 10.1016/j.rinp.2022.105320
    [34] A. Hyder, M. Barakat, General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics, Phys. Scr., 95 (2020), 045212. https://doi.org/10.1088/1402-4896/ab6526 doi: 10.1088/1402-4896/ab6526
  • This article has been cited by:

    1. Emad H M Zahran, Ahmet Bekir, Reda A Ibrahim, New dynamics performance for established dark solitons in polariton condensate, 2025, 77, 0253-6102, 035004, 10.1088/1572-9494/ad88f7
    2. Reda A. Ibrahim, Ahmet Bekir, Emad H. M. Zahran, On the exact traveling wave solutions to the Akbota-Gudekli-Kairat-Zhaidary equation and its numerical solutions, 2025, 140, 2190-5444, 10.1140/epjp/s13360-025-06289-x
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1305) PDF downloads(73) Cited by(2)

Figures and Tables

Figures(8)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog