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Review

Sustainable A2BBX6 based lead free perovskite solar cells: The challenges and research roadmap for power conversion efficiency improvement

  • The stability issues in the widely known CH3NH3PbI3, lead to the development of alternative halide double perovskite materials, which has received great attention in recent times. Although the stability issue of double halide perovskite seems promising, their device performance remains far inferior to CH3NH3PbI3 and with challenges for further improvements. Furthermore, the power conversion efficiency of single junction organic-inorganic halide perovskite is now 24.2% and 29.15% for the textured monolithic perovskite/silicon tandem solar cell; however, for the all-inorganic halide perovskite solar cell, it is 7.11%, and halide double perovskite solar cells are based on A2BBX6 (A = monocation, B = cation or vacancy, X = halide) such as Cs2AgBiBr6, Cs2TiBr6, Cs2AgTlBr6 and Cs2Ag(Bi1−xInx)Br6, being 2.8% and 3.3%, respectively. This creates big questions and concerns about the performance improvement of A2BBX6-based perovskite solar cells. Not only is this a concern, but there are many other big challenges faced by halide double perovskite solar cells. Such big challenges include: (a) geometric constraints and limited integration with interfacial materials; (b) dynamic disorder, a wide band gap, and a localized conduction band caused by a cubic unit cell that restrains the interactions of orbitals; (c) high processing temperature which may limit the diverse applications; and (d) low electronic dimensionality that makes them less appropriate for single junction solar cell purpose, etc. Moreover, the origin of electronic and optical properties such as the polarizability, the presence of molecular dipoles, and their influence on the dynamics of the photo-excitations remain bottleneck concerns that need to be elucidated. We roadmap performance sustainable improvement, which is suggested with a particular focus on engineering material surface and bulk, band gap, interfacial, composition, doping, device architectural, polar, and domain order. The reason that this review was developed was to forward great contributions to the readers and commercial ventures.

    Citation: Etsana Kiros Ashebir, Berhe Tadese Abay, Taame Abraha Berhe. Sustainable A2BⅠBⅢX6 based lead free perovskite solar cells: The challenges and research roadmap for power conversion efficiency improvement[J]. AIMS Materials Science, 2024, 11(4): 712-759. doi: 10.3934/matersci.2024036

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  • The stability issues in the widely known CH3NH3PbI3, lead to the development of alternative halide double perovskite materials, which has received great attention in recent times. Although the stability issue of double halide perovskite seems promising, their device performance remains far inferior to CH3NH3PbI3 and with challenges for further improvements. Furthermore, the power conversion efficiency of single junction organic-inorganic halide perovskite is now 24.2% and 29.15% for the textured monolithic perovskite/silicon tandem solar cell; however, for the all-inorganic halide perovskite solar cell, it is 7.11%, and halide double perovskite solar cells are based on A2BBX6 (A = monocation, B = cation or vacancy, X = halide) such as Cs2AgBiBr6, Cs2TiBr6, Cs2AgTlBr6 and Cs2Ag(Bi1−xInx)Br6, being 2.8% and 3.3%, respectively. This creates big questions and concerns about the performance improvement of A2BBX6-based perovskite solar cells. Not only is this a concern, but there are many other big challenges faced by halide double perovskite solar cells. Such big challenges include: (a) geometric constraints and limited integration with interfacial materials; (b) dynamic disorder, a wide band gap, and a localized conduction band caused by a cubic unit cell that restrains the interactions of orbitals; (c) high processing temperature which may limit the diverse applications; and (d) low electronic dimensionality that makes them less appropriate for single junction solar cell purpose, etc. Moreover, the origin of electronic and optical properties such as the polarizability, the presence of molecular dipoles, and their influence on the dynamics of the photo-excitations remain bottleneck concerns that need to be elucidated. We roadmap performance sustainable improvement, which is suggested with a particular focus on engineering material surface and bulk, band gap, interfacial, composition, doping, device architectural, polar, and domain order. The reason that this review was developed was to forward great contributions to the readers and commercial ventures.



    Magnetohydrodynamic (MHD) equations are composed of Euler (or Navier-Stokes) equations and Maxwells equations, which are mainly used to describe the complex interactions between conductive fluids and electromagnetic fields. They are widely applied in plasma [1], astrophysical research [2,3], controlled thermonuclear fusion [4], and new industrial technologies [5]. The study of exact solutions for magnetohydrodynamics systems can provide possible ideas for finding the global smooth solutions of the Navier-Stokes equation. However, compared with the Navier-Stokes equation, MHD equations contain additional nonlinear and coupling terms for velocity and magnetic fields, which makes their research more challenging.

    The qualitative stability analysis of MHD systems has been widely studied. Qin et al. [6] investigated the exponential stability of the global solution of (1 + 1)-dimensional compressible MHD equations. Suo et al. [7] studied the well-posedness of (2 + 1)-dimensional incompressible MHD equations with horizontal dissipation. Wang et al. [8] proved the stability of the global weak solution of (3 + 1)-dimensional incompressible MHD equations when the norms of the initial data are bounded by the minimal value of the viscosity coefficients. Li et al. [9] studied the convergence stability of local solutions for (3 + 1)-dimensional compressible viscous MHD equations. Xu et al. [10] studied the stability of local solutions to (3 + 1)-dimensional barotropic compressible MHD equations with vacuum. In the quantitative analysis, the complex nonlinearity and strong coupling of MHD equations make it difficult to seek the analytical solutions by some classical methods, such as the bilinear method [11], Darboux transformation method [12,13], Backlund transformation method [14], Lie symmetry analysis method [15,16,17], non-local symmetry analysis method [18,19], and Riemann-Hilbert method [20,21]. The quantitative calculations on MHD equations mainly focused on constructing specific forms of exact solutions or numerical solutions. Nevertheless, analytical solutions can provide an accurate mathematical description and theoretical basis for analysis and regulation of MHD systems, which has aroused widespread research interest. Donato et al. [22] studied exact solutions of (1 + 1)-dimensional MHD equations by Lie group analysis. Dorodnitsyn et al. [23] explored symmetries of plane one-dimensional MHD flows in the mass Lagrangian coordinates. Liu et al. [24] derived exact solutions of (2 + 1)-dimensional incompressible and barotropic MHD equations by Lie symmetry analysis. Xia et al. [25] studied group invariant solutions of (2 + 1)-dimensional incompressible ideal MHD equations by Lie symmetry method. Picard et al. [26] obtained some exact solutions of (3 + 1)-dimensional ideal MHD equations based on Lie group theory. Considering the physical significance of MHD equations and importance of analytical calculation, more diverse forms of exact solutions of MHD equations deserve to be further studied.

    As powerful tools for solving nonlinear equations, symmetry analysis [15,16,17,18,19] and the simplest equation methods [27] demonstrate special advantages in handling nonlinear terms in dynamical systems. For instance, Zhao et al. [15] studied the Heisenberg equation from the perspective of statistical physics by Lie symmetry analysis. Ali et al. [16] obtained new exact invariant solutions of (3 + 1)-dimensional variable coefficients Kudryashov-Sinelshchikov equation by Lie symmetry analysis. Adeyemo et al. [17] explored closed-form solutions of integrable (2 + 1)-dimensional Boussinesq equation by Lie symmetry reductions. Ren et al. [18] derived interaction solutions of modified Kadomtsev-Petviashvili equation by nonlocal symmetry reductions. Vitanov et al. [27] investigated the role of the simplest equations in obtaining exact and approximate solutions of nonlinear partial differential equations. The Lie symmetry analysis method simplifies problems by finding the invariance of differential equations, and transforms the original equations into a more easily solvable form through symmetry transformations. This method provides powerful tools for solving nonlinear problems with complex structures. The generalized Riccati equation is an important auxiliary equation with rich special solutions. This makes the generalized Riccati equation mapping method an effective direct method for constructing the solitary wave solutions, the periodic solutions and the rational solutions for MHD equations. In this paper, using the Lie symmetry analysis method and generalized Riccati equation expansion method, we obtain new solutions with various forms of MHD equations. The major contributions of this article are listed as follows:

    (1) Based on symmetry analysis and generalized Riccati equation expansion methods, the complex nonlinear and strongly coupled terms in MHD equations are technically handled. Different forms of new solutions are derived, which can describe various wave behaviors for MHD flows. Some of the solutions can be reduced to exact solutions for Euler or Navier-Stokes equations when magnetic fields vanish, which may provide references for the research on global solutions for Navier-Stokes equations.

    (2) The stability of solutions for MHD equations is analyzed from both qualitative and quantitative perspectives based on the obtained solutions.

    (3) The new solutions, wave behaviors, and stability analysis provide accurate mathematical descriptions and theoretical basis for numerical analysis and regulation of MHD systems.

    The rest of the paper is organized as follows: The transformations for MHD equations are given in Section 2. In Section 3, the exact solutions of inviscid and viscous (2 + 1)-dimensional MHD equations are obtained by the Lie symmetry analysis method and generalized Riccati equation expansion method. In Section 4, inviscid and viscous (3 + 1)-dimensional MHD equations are further studied. In Section 5, the stability of MHD equations is studied from qualitative and quantitative perspectives. Finally, some conclusions are drawn in Section 6.

    The flow of conducting fluid in a magnetic field is governed by the following incompressible MHD equations [28], which are a combination of Euler (or Navier-Stokes) equations of fluid dynamics and Maxwell's equations of electromagnetism. The set of equations express the conservation of mass, momentum and the interaction of the flow with the magnetic field. Consider (2 + 1)- and (3 + 1)-dimensional incompressible MHD equations [28]

    {UtνΔU+(U)U+p+κB×curlB=0,Bt+ηcurlcurlBcurl(U×B)+r=0,divU=0, divB=0, (2.1)

    where U is fluid velocity, p is hydrodynamic pressure, B is magnetic induction, r is magnetic pressure. The physical parameters ν, μ and σ represent kinematic viscosity, magnetic permeability and electric conductivity, respectively. η=1μσ, κ=1μ. Substituting equations

    B×curlB=12(|B|2)(B)B, curlcurlB=ΔB,curl(U×B)=(B)U(U)B,

    into (2.1), the incompressible MHD equations (2.1) can be rewritten as

    {UtνΔU+(U)U+p+κ[12(|B|2)(B)B]=0,BtηΔB(B)U+(U)B+r=0,divU=0, divB=0. (2.2)

    Denote x=(x,y), U=(u1(t,x),u2(t,x)), B=(b1(t,x),b2(t,x)) in (2.2). (2 + 1)-dimensional MHD equations can be given as

    {u1tν(u1xx+u1yy)+(u1u1x+u2u1y)+κ(b2b2xb2b1y)+px=0,u2tν(u2xx+u2yy)+(u1u2x+u2u2y)+κ(b1b1yb1b2x)+py=0,b1tη(b1xx+b1yy)(b1u1x+b2u1y)+(u1b1x+u2b1y)+rx=0,b2tη(b2xx+b2yy)(b1u2x+b2u2y)+(u1b2x+u2b2y)+ry=0,u1x+u2y=0, b1x+b2y=0. (3.1)

    The vector field of system (3.1) can be expressed as

    V_=ζ1t+ζ2x+ζ3y+ϕ1u1+ϕ2u2+φ1b1+φ2b2+ψ1p+ψ2r, (3.2)

    where ζi (i=1,2,3), ϕj, φj, ψj (j=1,2) are undetermined coefficients about t, x, U, B, p, r. It follows from second-order prolongation pr(2)V_(Δ)|Δ=0=0 that

    ϕt1ν(ϕxx1+ϕyy1)+ϕ1u1x+u1ϕx1+ϕ2u1y+u2ϕy1+κ(φ2b2x+b2φx2φ2b1yb2φy1)+ψx1=0,ϕt2ν(ϕxx2+ϕyy2)+ϕ1u2x+u1ϕx2+ϕ2u2y+u2ϕy2+κ(φ1b1y+b1φy1φ1b2xb1φx2)+ψy1=0,φt1η(φxx1+φyy1)φ1u1xb1ϕx1φ2u1yb2ϕy1+ϕ1b1x+u1φx1+ϕ2b1y+u2φy1+ψx2=0,φt2η(φxx2+φyy2)φ1u2xb1ϕx2φ2u2yb2ϕy2+ϕ1b2x+u1φx2+ϕ2b2y+u2φy2+ψy2=0,ϕx1+ϕy2=0, φx1+φy2=0. (3.3)

    Choosing ν=η=0 and κ=1 in Eq (3.1), the inviscid MHD equations can be obtained as

    {u1t+(u1u1x+u2u1y)+(b2b2xb2b1y)+px=0,u2t+(u1u2x+u2u2y)+(b1b1yb1b2x)+py=0,b1t(b1u1x+b2u1y)+(u1b1x+u2b1y)+rx=0,b2t(b1u2x+b2u2y)+(u1b2x+u2b2y)+ry=0,u1x+u2y=0, b1x+b2y=0. (3.4)

    Solving (3.3) with ν=η=0 and κ=1, the coefficient functions of vector field V_ can be obtained as

    ζ1=2C1t+C2, ζ2=C0xC12y+f1(t)+C3, ζ3=C12x+C0y+f2(t)+C4,ϕ1=(C02C1)u1C12u2+f1(t), ϕ2=C12u1+(C02C1)u2+f2(t),φ1=(C02C1)b1C12b2, φ2=C12b1+(C02C1)b2,ψ1=2(C02C1)pxf1(t)yf2(t)+α(t), ψ2=2(C02C1)r+β(t), (3.5)

    where C0,C1,C2,C3,C4 and C12 are arbitrary constants. f1(t),f2(t),α(t) and β(t) are arbitrary functions related to t only. When C2=1, C3=ˉv1, C4=ˉv2, C0=C1=C12=f1(t)=f2(t)=0,

    V_=(C2t+C3x+C4y)+α(t)p+β(t)r=t+ˉv1x+ˉv2y+α(t)p+β(t)r. (3.6)

    The characteristic equation is

    dt1=dxˉv1=dyˉv2=du10=du20=db10=db20=dpα(t)=drβ(t). (3.7)

    It follows from (3.7) that corresponding invariants are

    ˉζ1=xˉv1t, ˉζ2=yˉv2t, F1(ˉζ1,ˉζ2)=u1, F2(ˉζ1,ˉζ2)=u2, G1(ˉζ1,ˉζ2)=b1,G2(ˉζ1,ˉζ2)=b2, Q(ˉζ1,ˉζ2)=p+α(t)dt, R(ˉζ1,ˉζ2)=r+β(t)dt. (3.8)

    Substituting (3.8) into (3.4), reduced equations can be obtained as

    {ˉv1F1ˉζ1+ˉv2F1ˉζ2+F1F1ˉζ1+F2F1ˉζ2+G2G2ˉζ1G2G1ˉζ2Qˉζ1=0,ˉv1F2ˉζ1+ˉv2F2ˉζ2+F1F2ˉζ1+F2F2ˉζ2+G1G1ˉζ2G1G2ˉζ1Qˉζ2=0,ˉv1G1ˉζ1+ˉv2G1ˉζ2G1F1ˉζ1G2F1ˉζ2+F1G1ˉζ1+F2G1ˉζ2Rˉζ1=0,ˉv1G2ˉζ1+ˉv2G2ˉζ2G1F2ˉζ1G2F2ˉζ2+F1G2ˉζ1+F2G2ˉζ2Rˉζ2=0,F1ˉζ1+F2ˉζ2=0, G1ˉζ1+G2ˉζ2=0. (3.9)

    It can be obtained that (3.10)–(3.12) are three kinds of solutions for (3.9).

    Case 1. Sin/cos-type solution.

    {F1(ˉζ1,ˉζ2)=cos2(ˉζ1ˉζ2)ˉv1, F2(ˉζ1,ˉζ2)=cos2(ˉζ1ˉζ2)ˉv2,G1(ˉζ1,ˉζ2)=sin(ˉζ1ˉζ2)cos(ˉζ1ˉζ2)ˉv1, G2(ˉζ1,ˉζ2)=sin(ˉζ1ˉζ2)cos(ˉζ1ˉζ2)ˉv1,Q(ˉζ1,ˉζ2)=ˉv1sin(2ˉζ1+2ˉζ2)cos(4ˉζ1+4ˉζ2)8+m, R(ˉζ1,ˉζ2)=n, (3.10)

    where m and n are arbitrary constants.

    Case 2. Sech-type solution.

    {F1(ˉζ1,ˉζ2)=sech2(ˉζ1ˉζ2)ˉv1, F2(ˉζ1,ˉζ2)=sech2(ˉζ1ˉζ2)ˉv2,G1(ˉζ1,ˉζ2)=c1, G2(ˉζ1,ˉζ2)=c1, Q(ˉζ1,ˉζ2)=m, R(ˉζ1,ˉζ2)=n, (3.11)

    where c1 is arbitrary constant.

    Case 3. Rational solution.

    {F1(ˉζ1,ˉζ2)=c2ˉζ2ˉζ21+ˉζ22, F2(ˉζ1,ˉζ2)=c2ˉζ1ˉζ21+ˉζ22,G1(ˉζ1,ˉζ2)=c3ˉζ2ˉζ21+ˉζ22, G2(ˉζ1,ˉζ2)=c3ˉζ1ˉζ21+ˉζ22,Q(ˉζ1,ˉζ2)=c2(2ˉζ1ˉv22ˉζ2ˉv1+c2)2(ˉζ21+ˉζ22)+m, R(ˉζ1,ˉζ2)=c3(ˉζ1ˉv2ˉζ2ˉv1)ˉζ21+ˉζ22+n, (3.12)

    where c2 and c3 are arbitrary constants. Substituting (3.8) into (3.10)–(3.12), respectively, we obtain that (3.13)–(3.15) are three kinds of solutions for (2+1)-dimensional MHD equations (3.4).

    Case 1. Sin/cos-type solution.

    u1=cos2[xy(ˉv1ˉv2)t]+ˉv1, u2=cos2[xy(ˉv1ˉv2)t]+ˉv2,b1=sin[xy(ˉv1ˉv2)t]cos[xy(ˉv1ˉv2)t]+ˉv1,b2=sin[xy(ˉv1ˉv2)t]cos[xy(ˉv1ˉv2)t]+ˉv1, r=n+β(t)dt,p=ˉv1sin[(2ˉv12ˉv2)t2x+2y]+cos[(4ˉv14ˉv2)t4x+4y]8m+α(t)dt. (3.13)

    Setting ˉv1=3,ˉv2=4 and x=6 for u1 in (3.13), we obtain Figure 1 of periodic solution u1 as follows.

    Figure 1.  (a) The evolution of periodic solution via (3.13), (b) u1(t = 1, 3, 5).

    From solution (3.13) and Figure 1, it can be seen that the solution exhibits periodic characteristics over time and space. The physical significance of the solution mainly includes the following points:

    (ⅰ) Periodic solution can be used to analyze the stability of MHD system. If the MHD system can reach periodic solutions, it usually means that the system can achieve stability under certain conditions.

    (ⅱ) Periodic solution can describe oscillatory phenomena in the MHD system, such as periodic changes in magnetic fields, periodic fluctuations in fluid velocity, etc.

    (ⅲ) In industry, such as magnetohydrodynamic power generation, periodic flow can improve power generation efficiency. By optimizing the periodic solution, more efficient power generation equipment can be designed.

    Case 2. Sech-type solution.

    u1=sech2[xy(ˉv1ˉv2)t]+ˉv1, u2=sech2[xy(ˉv1ˉv2)t]+ˉv2,b1=c1, b2=c1, p=m+α(t)dt, r=n+β(t)dt. (3.14)

    Setting ˉv1=1 and ˉv2=2 for u1 in (3.14), we obtain Figure 2 of single soliton solution u1 as follows.

    Figure 2.  The evolution of a single soliton solution via (3.14) (t=1, 4, 7, respectively).

    From Figure 2, it can be seen that the velocity is constant in certain domains of space. Moreover, the velocity is induced to a sudden rise until it reaches a maximum value. As a stable wave form, the characteristics of solitons emerge from the collective behavior of nonlinear media. Solitons play an important role in the study of MHD waves due to their unique properties and applications in various physical contexts. The importance of solitons in the main problem mostly includes the following points:

    (ⅰ) As a special wave phenomenon, solitons can form stable wave structures in plasmas. In controlled thermonuclear fusion research, soliton waves can be used to describe some wave phenomena in plasma, which has potential application value for achieving and maintaining the stability of fusion plasma.

    (ⅱ) Solitons can maintain their shape and amplitude is unchanged during propagation. This property is important for understanding and predicting some wave propagations in MHD flow.

    (ⅲ) Solitons can help explain some phenomena in MHD flow, such as the localized structure of magnetic fields and the dynamic behavior of magnetic domain walls.

    Case 3. Rational solution.

    u1=c2(yˉv2t)(xˉv1t)2+(yˉv2t)2, u2=c2(xˉv1t)(xˉv1t)2+(yˉv2t)2,b1=c3(yˉv2t)(xˉv1t)2+(yˉv2t)2, b2=c3(xˉv1t)(xˉv1t)2+(yˉv2t)2,p=c2[2(xˉv1t)ˉv22(yˉv2t)ˉv1+c2]2[(xˉv1t)2+(yˉv2t)2]m+α(t)dt,r=c3[(xˉv1t)ˉv2(yˉv2t)ˉv1](xˉv1t)2+(yˉv2t)2n+β(t)dt. (3.15)

    Setting ˉv1=1,ˉv2=1,c2=1 and x=2 for u1 in (3.15), we obtain lump (c.f. Figure 3 for solution u1 as follows.

    Figure 3.  (a) The evolution of lump solution via (3.15), (b) Overview of u1.

    From Figure 3, it can be seen the flow have the characteristics of spatial and temporal localization. Lump solution corresponds to the emergent phenomenon of energy focusing in a specific region or time point. The amplitude of peak and valley is several times higher than the surrounding background height. The scale transformation of the lump has already been processed in mathematics. Actually, shock wave may be seen and local instability may occur in reality.

    Remark 3.1. (1) If b1=b2=0 and r=0 in (3.14) and (3.15), then (3.14) and (3.15) reduce to exact solutions for (2 + 1)-dimensional Euler equation.

    (2) Since ω=u2xu1y0 in (3.13) and (3.14) and ω=u2xu1y=0 in (3.15), it can be concluded that (3.13) and (3.14) correspond to rotational flow. Additionally, (3.15) corresponds to inrotational flow.

    Without loss of generality, choosing ν=η=κ=1 in Eq (3.1), the viscous MHD equations can be obtained as

    {u1t(u1xx+u1yy)+(u1u1x+u2u1y)+(b2b2xb2b1y)+px=0,u2t(u2xx+u2yy)+(u1u2x+u2u2y)+(b1b1yb1b2x)+py=0,b1t(b1xx+b1yy)(b1u1x+b2u1y)+(u1b1x+u2b1y)+rx=0,b2t(b2xx+b2yy)(b1u2x+b2u2y)+(u1b2x+u2b2y)+ry=0,u1x+u2y=0, b1x+b2y=0. (3.16)

    Solving (3.3) with ν=η=κ=1, the coefficient functions of vector field V_ can be obtained as

    ζ1=2C1t+C2, ζ2=C1xC12y+f1(t)+C3, ζ3=C12x+C1y+f2(t)+C4,ϕ1=C1u1C12u2+f1(t), ϕ2=C12u1C1u2+f2(t),φ1=C1b1C12b2, φ2=C12b1C1b2,ψ1=2C1pxf1(t)yf2(t)+α(t), ψ2=2C1r+β(t). (3.17)

    Case 1. When C1=C2=C3=C4=C12=f1(t)=f2(t)=0, V_=α(t)p+β(t)r.

    The corresponding invariants are

    ˉζ0=t, ˉζ1=x, ˉζ2=y, F1(ˉζ0,ˉζ2)=u1, F2(ˉζ0,ˉζ1)=u2,G1(ˉζ0, ˉζ2)=b1, G2(ˉζ0,ˉζ1)=b2. (3.18)

    Substituting invariants (3.18) into (3.16), and solving the reduced equations,

    u1=g1etcosy,  u2=g2etcosx, b1=g3etcosy, b2=g4etcosx,p=[cos(2y)g23g4+2g34sin2x+4g3(g22g24)sinxsiny]e2t4g4+m(t), r=n(t), (3.19)

    is a sin/cos-type solution for MHD equations (3.16), where g1g4g2g3=0. m(t) and n(t) are arbitrary functions related to t only. Setting g1=1 for u1 in (3.19), we obtain Figure 4 of solution u1 as follows.

    Figure 4.  (a) The evolution of periodic solution via (3.19), (b) u1(t = 0.2, 0.5, 0.8).

    From Figure 4, it can be seen that as time increases, the shape and direction of the velocity remain unchanged, but the amplitude decreases.

    Case 2. When C2=1, Ci=C12=f1(t)=f2(t)=α(t)=β(t)=0 (i=0,1,3,4), V_=t.

    The characteristic equation is

    dt1=dx0=dy0=du10=du20=db10=db20=dp0=dr0. (3.20)

    The corresponding invariants are

    ˉζ1=x, ˉζ2=y, F1(ˉζ1,ˉζ2)=u1, F2(ˉζ1,ˉζ2)=u2,G1(ˉζ1,ˉζ2)=b1, G2(ˉζ1,ˉζ2)=b2, Q(ˉζ1,ˉζ2)=p, R(ˉζ1,ˉζ2)=r. (3.21)

    Substituting invariants (3.21) into (3.16), and solving the reduced equations,

    u1=sech2(x+iy), u2=isech2(x+iy),b1=sech2(x+iy), b2=isech2(x+iy), p=m, r=n, (3.22)

    is a sech-type solution for MHD equations (3.16). Using symmetry

    V_=tx+ty+u1+u2,

    solution (3.22) can further generate the following invariant solution

    u1=sech2(xεt+i(yεt))+ε, u2=isech2(xεt+i(yεt))+ε,b1=sech2(xεt+i(yεt)), b2=isech2(xεt+i(yεt)), p=m(t), r=n(t), (3.23)

    where ε is arbitrary constant.

    Remark 3.2. (1) The lump solution (3.15) for inviscid MHD equations (3.4) also satisfies the viscous MHD equations (3.16).

    (2) If b1=b2=0 and r=0 in (3.23), then (3.23) reduces to exact solutions for (2 + 1)-dimensional Navier-Stokes equation.

    (3) Since ω=u2xu1y0 in (3.19) and ω=u2xu1y=0 in (3.23), it can be concluded that (3.19) corresponds to rotational flow. Moreover, (3.23) corresponds to inrotational flow.

    As an important method of simplest equation methods, the generalized Riccati equation method [29,30] provides a powerful mathematical tool to deal with the complex nonlinear and strong coupling terms in MHD equations. Using traveling wave transformation,

    ζ=k2x+k3yk1t, (3.24)

    equations (3.1) are transformed into following ordinary differential equations (ODEs) as

    {k1u1ζν(k22u1ζζ+k23u1ζζ)+(k2u1u1ζ+k3u2u1ζ)+κ(k2b2b2ζk3b2b1ζ)+k2pζ=0,k1u2ζν(k22u2ζζ+k23u2ζζ)+(k2u1u2ζ+k3u2u2ζ)+κ(k3b1b1ζk2b1b2ζ)+k3pζ=0,k1b1ζη(k22b1ζζ+k23b1ζζ)(k2b1u1ζ+k3b2u1ζ)+k2u1b1ζ+k3u2b1ζ+k2rζ=0,k1b2ζη(k22b2ζζ+k23b2ζζ)(k2b1u2ζ+k3b2u2ζ)+k2u1b2ζ+k3u2b2ζ+k3rζ=0,k2u1ζ+k3u2ζ=0, k2b1ζ+k3b2ζ=0. (3.25)

    Suppose that the solution of ODEs (3.25) can be expressed as a polynomial of ϕ(ζ) as

    u1=N1i=0aiϕi(ζ), u2=N2i=0miϕi(ζ), b1=N3i=0niϕi(ζ),b2=N4i=0siϕi(ζ), p=N5i=0liϕi(ζ)+l(t), r=N6i=0qiϕi(ζ)+q(t), (3.26)

    where ai,mi,ni,si,li,qi are undetermined constants and aN1,mN2,nN3,sN40. l(t) and q(t) are arbitrary functions related to t only. ϕ(ζ) satisfies the generalized Riccati equation

    ϕ(ζ)=ξ0+ξ1ϕ(ζ)+ξ2ϕ2(ζ), (3.27)

    where ξ0,ξ1 and ξ2 are arbitrary constants with ξ20. We choose N1=N2=N3=N4=N5=N6=2 with can balance the highest order of the derivative and nonlinear terms in ODEs.

    When ν=η=0 and κ=1 in ODEs (3.25), substituting (3.26) and (3.27) into (3.25), collecting the coefficients of ϕi(ζ) and setting them to be zeros, we obtain

    a0=a0,a1=a1,a2=a2,k1=k1,k2=k2,k3=k3,l1=l1,l2=n21(k22+k23)2k23,m0=a0k2+k1k3,m1=a1k2k3,m2=k2a2k3,n0=k23l1n1(k22+k23),n1=n1,n2=0,q1=0,q2=0,s0=k2k3l1n1(k22+k23),s1=k2n1k3,s2=0. (3.28)

    Substituting (3.28) and the general solutions of (3.27) (c.f. [29]) into (3.26), it can be obtained following four kinds of solutions for the (2 + 1)-dimensional inviscid MHD equations.

    Case 1. When ξ214ξ2ξ0>0 and ξ1ξ20 (or ξ0ξ20), the tanh-type solution can be obtained as follows.

    u1=4a0ξ222a1ξ1ξ2+a2ξ214ξ22+a2ξ1a1ξ22ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)+a2(ξ214ξ2ξ0)4ξ22tanh2(ξ214ξ2ξ02ζ),u2=4ξ22(a0k2+k1)+2a1k2ξ1ξ2k2a2ξ214k3ξ22k2a2(ξ124ξ2ξ0)4k3ξ22tanh2(ξ214ξ2ξ02ζ)+k2(a1ξ2a2ξ1)2k3ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ),b1=2k23l1ξ2+n21(k22+k23)ξ12n1(k22+k23)ξ2n1ξ214ξ2ξ02ξ2tanh(ξ214ξ2ξ02ζ),b2=2k2k23l1ξ2+k2n21(k22+k23)ξ12k3n1(k22+k23)ξ2+k2n1ξ214ξ2ξ02k3ξ2tanh(ξ214ξ2ξ02ζ), (3.29)

    where ζ=k2x+k3yk1t. Setting a0=1,a1=7,a2=1,k1=3,k2=1,k3=1,ξ0=1,ξ1=3 and ξ2=1 for u1 in (3.29), we obtain Figure 5 of kink solution u1 as follows.

    Figure 5.  u1(t=2, 4, 6, respectively).

    In particular, when ξ0=0 and ξ1ξ20, the sinh-cosh-type solution can be obtained as follows

    u1=a0a1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+a2ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,u2=a0k2+k1k3+a1k2ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]k3ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]k2a2ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2k3ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,b1=k23l1n1(k22+k23)n1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C],b2=k2k3l1n1(k22+k23)+k2n1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]k3ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C], (3.30)

    where C is arbitrary constant. Setting a0=4,a1=10,a2=8,k1=4,k2=16,k3=2,ξ0=0, ξ1=1,ξ2=1,C=1 and x=1 for u1 in (3.30), we obtain Figure 6 of anti-kink-like solution u1 as follows.

    Figure 6.  (a) u1, (b) u1(t = 1, 2, 3).

    The kink and kink-like solutions can be understood as a macroscopic stable structure generated from the field dynamics at the microscale. They manifest as a rapid change or discontinuity in some field at the macro level.

    Case 2. When ξ214ξ2ξ0<0 and ξ1ξ20 (or ξ0ξ20), the tan-type solution can be obtained as follows.

    u1=4a0ξ222a1ξ1ξ2+a2ξ214ξ22+a1ξ2a2ξ12ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)+a2(4ξ2ξ0ξ21)4ξ22tan2(4ξ2ξ0ξ212ζ),u2=4ξ22(a0k2+k1)+2a1k2ξ1ξ2k2a2ξ214k3ξ22k2a2(4ξ2ξ0ξ21)4k3ξ22tan2(4ξ2ξ0ξ212ζ)+k2(a2ξ1a1ξ2)2k3ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ),b1=2k23l1ξ2+n21(k22+k23)ξ12n1(k22+k23)ξ2+n14ξ2ξ0ξ212ξ2tan(4ξ2ξ0ξ212ζ),b2=2k2k23l1ξ2+k2n21(k22+k23)ξ12k3n1(k22+k23)ξ2k2n14ξ2ξ0ξ212k3ξ2tan(4ξ2ξ0ξ212ζ).

    Case 3. When ξ1=ξ0=0 and ξ20, the rational solution can be obtained as follows

    u1=a0a1ξ2ζ+C+a2(ξ2ζ+C)2, u2=a0k2+k1k3+a1k2k3(ξ2ζ+C)a2k2k3(ξ2ζ+C)2,b1=k23l1n1(k22+k23)n1ξ2ζ+C, b2=k2k3l1n1(k22+k23)+k2n1k3(ξ2ζ+C).  (3.31)

    When ν=η=κ=1 in ODEs (3.25), Substituting (3.26) and (3.27) into (3.25), we obtain

    a0=a0,a1=a1,a2=a2,k1=k1,k2=k2,k3=ik2,l1=((a0+im0)k2+k1)a1k2,l2=((a0+im0)k2+k1)a2k2,m0=m0,m1=ia1,m2=ia2,n0=n0,n1=n1,s2=0,n2=0,q1=((a0+im0)n1(is0n0)a1)k2+n1k1k2,q2=a2(is0n0),s0=s0, s1=in1. (3.32)

    Substituting (3.32) and general solutions of (3.27) (c.f. [29]) into (3.26), it can be obtained that (3.33)–(3.40) are four kinds of solutions for (2 + 1)-dimensional viscous MHD equations.

    Case 1. When ξ214ξ2ξ0>0 and ξ1ξ20 (or ξ0ξ20), the following tanh-type solution can be obtained.

    u1=4a0ξ222a1ξ1ξ2+a2ξ214ξ22+a2ξ1a1ξ22ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)+a2(ξ214ξ2ξ0)4ξ22tanh2(ξ214ξ2ξ02ζ),u2=m0+i(2a1ξ1ξ2a2ξ21)4ξ22+i(a1ξ2a2ξ1)2ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)ia2(ξ214ξ2ξ0)4ξ22tanh2(ξ214ξ2ξ02ζ),b1=2n0ξ2n1ξ12ξ2n1ξ214ξ2ξ02ξ2tanh(ξ214ξ2ξ02ζ),b2=s0+in1ξ12ξ2+in1ξ214ξ2ξ02ξ2tanh(ξ214ξ2ξ02ζ). (3.33)

    In particular, when ξ0=0 and ξ1ξ20, the following sinh-cosh-type solution can be obtained.

    u1=a0a1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+a2ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,u2=m0+ia1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]ia2ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,b1=n0n1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C], b2=s0+in1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]. (3.34)

    Setting a1=2,a2=2,m0=1,k1=1,k2=1,ξ0=0,ξ1=2,ξ2=14,C=1 and x=1 for u2 in (3.34), we obtain breather (c.f. Figure 7) for solution u2 as follows.

    Figure 7.  (a) The evolution of Akhmediev breaher solution via |u2|, (b) Overview of |u2|, (c) |u2|(t=1,2,3).

    From Figure 7, the breather appears to be localized in the t-axis direction, and periodic in the y-axis direction. It corresponds to a type of nonlinear wave where energy is concentrated in a local oscillation manner. The breather solutions can serve as a carrier of energy transfer during the propagation process, and the characteristics of this energy transfer are related to the macroscopic behavior in emergent phenomena.

    Case 2. When ξ214ξ2ξ0<0 and ξ1ξ20 (or ξ0ξ20), the following tan-type solution can be obtained.

    u1=4a0ξ222a1ξ1ξ2+a2ξ214ξ22+a1ξ2a2ξ12ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)+a2(4ξ2ξ0ξ21)4ξ22tan2(4ξ2ξ0ξ212ζ), (3.35)
    u2=m0+i(2a1ξ1ξ2a2ξ21)4ξ22+i(a2ξ1a1ξ2)2ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)ia2(4ξ2ξ0ξ21)4ξ22tan2(4ξ2ξ0ξ212ζ), (3.36)
    b1=2n0ξ2n1ξ12ξ2+n14ξ2ξ0ξ212ξ2tan(4ξ2ξ0ξ212ζ), (3.37)
    b2=s0+in1ξ12ξ2in14ξ2ξ0ξ212ξ2tan(4ξ2ξ0ξ212ζ). (3.38)

    Setting a1=12,a2=2,m0=1,k1=6,k2=1,ξ0=2,ξ1=2,ξ2=1 and x=1 in (3.36), we obtain Figure 8 for breather solution u2 as follows.

    Figure 8.  (a) The evolution of Kuznetsov-Ma breaher solution via |u2|, (b) Overview of |u2|, (c) |u2|(y=1,1.5,2).

    From Figure 8, the breather appears to be localized in the y-axis direction, and periodic in the t-axis direction, which corresponds to a nonlinear local wave that oscillates periodically in time.

    Case 3. When ξ1=ξ0=0 and ξ20, the following rational solution can be obtained.

    u1=a0a1ξ2ζ+C+a2(ξ2ζ+C)2, u2=m0+ia1ξ2ζ+Cia2(ξ2ζ+C)2, (3.39)
    b1=n0n1ξ2ζ+C, b2=s0+in1ξ2ζ+C. (3.40)

    Denote x=(x,y,z), U=(u1(t,x),u2(t,x),u3(t,x)), B=(b1(t,x),b2(t,x),b3(t,x)) in (2.2), the component form of the (3 + 1)-dimensional MHD equations can be obtained as

    {u1tν(u1xx+u1yy+u1zz)+(u1u1x+u2u1y+u3u1z)+κ(b2b2x+b3b3xb2b1yb3b1z)+px=0,u2tν(u2xx+u2yy+u2zz)+(u1u2x+u2u2y+u3u2z)+κ(b1b1y+b3b3yb1b2xb3b2z+py=0,u3tν(u3xx+u3yy+u3zz)+(u1u3x+u2u3y+u3u3z)+κ(b1b1z+b2b2zb1b3xb2b3y)+pz=0,b1tη(b1xx+b1yy+b1zz)(b1u1x+b2u1y+b3u1z)+(u1b1x+u2b1y+u3b1z)+rx=0,b2tη(b2xx+b2yy+b2zz)(b1u2x+b2u2y+b3u2z)+(u1b2x+u2b2y+u3b2z)+ry=0,b3tη(b3xx+b3yy+b3zz)(b1u3x+b2u3y+b3u3z)+(u1b3x+u2b3y+u3b3z)+rz=0,u1x+u2y+u3z=0, b1x+b2y+b3z=0. (4.1)

    The vector field of the system (4.1) can be expressed as

    V_=ζ1t+ζ2x+ζ3y+ζ4z+ϕ1u1+ϕ2u2+ϕ3u3+φ1b1+φ2b2+φ3b3+ψ1p+ψ2r, (4.2)

    where ζi (i=1,2,3,4), ϕj, φj (j=1,2,3) and ψk (k=1,2) are undetermined coefficients about t, x, U, B, p and r. It follows from the second-order prolongation pr(2)V_(Δ)|Δ=0=0 that

    ϕt1ν(ϕxx1+ϕyy1+ϕzz1)+ϕ1u1x+u1ϕx1+ϕ2u1y+u2ϕy1+ϕ3u1z+u3ϕz1+κ(φ2b2x+b2φx2+φ3b3x+b3φx3φ2b1yb2φy1φ3b1zb3φz1)+ψx1=0, (4.3)
    ϕt2ν(ϕxx2+ϕyy2+ϕzz2)+ϕ1u2x+u1ϕx2+ϕ2u2y+u2ϕy2+ϕ3u2z+u3ϕz2+κ(φ1b1y+b1φy1+φ3b3y+b3φy3φ1b2xb1φx2φ3b2zb3φz2)+ψy1=0, (4.4)
    ϕt3ν(ϕxx3+ϕyy3+ϕzz3)+ϕ1u3x+u1ϕx3+ϕ2u3y+u2ϕy3+ϕ3u3z+u3ϕz3+κ(φ1b1z+b1φz1+φ2b2z+b2φz2φ1b3xb1φx3φ2b3yb2φy3)+ψz1=0, (4.5)
    φt1η(φxx1+φyy1+φzz1)φ1u1xb1ϕx1φ2u1yb2ϕy1φ3u1zb3ϕz1+ϕ1b1x+u1φx1+ϕ2b1y+u2φy1+ϕ3b1z+u3φz1+ψx2=0, (4.6)
    φt2η(φxx2+φyy2+φzz2)φ1u2xb1ϕx2φ2u2yb2ϕy2φ3u2zb3ϕz2+ϕ1b2x+u1φx2+ϕ2b2y+u2φy2+ϕ3b2z+u3φz2+ψy2=0, (4.7)
    φt3η(φxx3+φyy3+φzz3)φ1u3xb1ϕx3φ2u3yb2ϕy3φ3u3zb3ϕz3+ϕ1b3x+u1φx3+ϕ2b3y+u2φy3+ϕ3b3z+u3φz3+ψz2=0, (4.8)
    ϕx1+ϕy2+ϕz3=0, φx1+φy2+φz3=0. (4.9)

    Choosing ν=η=0 and κ=1 in equations (4.1), the inviscid MHD equations can be obtained as

    {u1t+(u1u1x+u2u1y+u3u1z)+(b2b2x+b3b3xb2b1yb3b1z)+px=0,u2t+(u1u2x+u2u2y+u3u2z)+(b1b1y+b3b3yb1b2xb3b2z)+py=0,u3t+(u1u3x+u2u3y+u3u3z)+(b1b1z+b2b2zb1b3xb2b3y)+pz=0,b1t(b1u1x+b2u1y+b3u1z)+(u1b1x+u2b1y+u3b1z)+rx=0,b2t(b1u2x+b2u2y+b3u2z)+(u1b2x+u2b2y+u3b2z)+ry=0,b3t(b1u3x+b2u3y+b3u3z)+(u1b3x+u2b3y+u3b3z)+rz=0,u1x+u2y+u3z=0, b1x+b2y+b3z=0. (4.10)

    Solving (4.3)–(4.9) with ν=η=0 and κ=1, the coefficient functions of vector field V_ can be obtained as

    ζ1=2C1t+C2, ζ2=C0xC12yC13z+f1(t)+C3,ζ3=C12x+C0yC23z+f2(t)+C4, ζ4=C13x+C23y+C0z+f3(t)+C5,ϕ1=(C02C1)u1C12u2C13u3+f1(t), ϕ2=C12u1+(C02C1)u2C23u3+f2(t),ϕ3=C13u1+C23u2+(C02C1)u3+f3(t), φ1=(C02C1)b1C12b2C13b3,φ2=C12b1+(C02C1)b2C23b3, φ3=C13b1+C23b2+(C02C1)b3,ψ1=2(C02C1)pxf1(t)yf2(t)zf3(t)+α(t), ψ2=2(C02C1)r+β(t).

    When C2=1, C3=ˉv1, C4=ˉv2, C5=ˉv3, C0=C1=C12=C13=C23=0, f1(t)=f2(t)=0,

    V_=(C2t+C3x+C4y+C5z)+α(t)p+β(t)r=t+ˉv1x+ˉv2y+ˉv3z+α(t)p+β(t)r. (4.11)

    The characteristic equation is

    dt1=dxˉv1=dyˉv2=dzˉv3=du10=du20=du30=db10=db20=db30=dpα(t)=drβ(t). (4.12)

    It follows from (4.12) that corresponding invariants are

    ˉζ1=xˉv1t, ˉζ2=yˉv2t, ˉζ3=zˉv3t, F1(ˉζ1,ˉζ2,ˉζ3)=u1, F2(ˉζ1,ˉζ2,ˉζ3)=u2,F3(ˉζ1,ˉζ2,ˉζ3)=u3, G1(ˉζ1,ˉζ2,ˉζ3)=b1, G2(ˉζ1,ˉζ2,ˉζ3)=b2, G3(ˉζ1,ˉζ2,ˉζ3)=b3,Q(ˉζ1,ˉζ2,ˉζ3)=p+α(t)dt, R(ˉζ1,ˉζ2,ˉζ3)=r+β(t)dt. (4.13)

    Substituting (4.13) into (4.10), reduced equations can be obtained as

    {ˉv1F1ˉζ1+ˉv2F1ˉζ2+ˉv3F1ˉζ3+F1F1ˉζ1+F2F1ˉζ2+F3F1ˉζ3+G2G2ˉζ1+G3G3ˉζ1G2G1ˉζ2G3G1ˉζ3Qˉζ1=0,ˉv1F2ˉζ1+ˉv2F2ˉζ2+ˉv3F2ˉζ3+F1F2ˉζ1+F2F2ˉζ2+F3F2ˉζ3+G1G1ˉζ2+G3G3ˉζ2G1G2ˉζ1G3G2ˉζ3Qˉζ2=0,ˉv1F3ˉζ1+ˉv2F3ˉζ2+ˉv3F3ˉζ3+F1F3ˉζ1+F2F3ˉζ2+F3F3ˉζ3+G1G1ˉζ3+G2G2ˉζ3G1G3ˉζ1G2G3ˉζ2Qˉζ3=0,ˉv1G1ˉζ1+ˉv2G1ˉζ2+ˉv3G1ˉζ3G1F1ˉζ1G2F1ˉζ2G3F1ˉζ3+F1G1ˉζ1+F2G1ˉζ2+F3G1ˉζ3Rˉζ1=0,ˉv1G2ˉζ1+ˉv2G2ˉζ2+ˉv3G2ˉζ3G1F2ˉζ1G2F2ˉζ2G3F2ˉζ3+F1G2ˉζ1+F2G2ˉζ2+F3G2ˉζ3Rˉζ2=0,ˉv1G3ˉζ1+ˉv2G3ˉζ2+ˉv3G3ˉζ3G1F3ˉζ1G2F3ˉζ2G3F3ˉζ3+F1G3ˉζ1+F2G3ˉζ2+F3G3ˉζ3Rˉζ3=0,F1ˉζ1+F2ˉζ2+F3ˉζ3=0, G1ˉζ1+G2ˉζ2+G3ˉζ3=0. (4.14)

    It can be obtained that (4.15)–(4.17) are three kinds of solutions for (4.14).

    Case 1. Sin-cos-type solution.

    {F1(ˉζ1,ˉζ2,ˉζ3)=cos2(2ˉζ1ˉζ2ˉζ3)ˉv1,F2(ˉζ1,ˉζ2,ˉζ3)=cos2(2ˉζ1ˉζ2ˉζ3)ˉv2,F3(ˉζ1,ˉζ2,ˉζ3)=cos2(2ˉζ1ˉζ2ˉζ3)ˉv3,G1(ˉζ1,ˉζ2,ˉζ3)=sin(2ˉζ1ˉζ2ˉζ3)cos(2ˉζ1ˉζ2ˉζ3)ˉv1,G2(ˉζ1,ˉζ2,ˉζ3)=sin(2ˉζ1ˉζ2ˉζ3)cos(2ˉζ1ˉζ2ˉζ3)ˉv1,G3(ˉζ1,ˉζ2,ˉζ3)=sin(2ˉζ1ˉζ2ˉζ3)cos(2ˉζ1ˉζ2ˉζ3)ˉv1,Q(ˉζ1,ˉζ2,ˉζ3)=3ˉv1sin(4ˉζ1+2ˉζ2+2ˉζ3)23cos(8ˉζ1+4ˉζ2+4ˉζ3)16+m, R(ˉζ1,ˉζ2,ˉζ3)=n, (4.15)

    where m and n are arbitrary constants.

    Case 2. Sech-type solution.

    {F1(ˉζ1,ˉζ2,ˉζ3)=sech2(2ˉζ1ˉζ2ˉζ3)ˉv1, F2(ˉζ1,ˉζ2,ˉζ3)=sech2(2ˉζ1ˉζ2ˉζ3)ˉv2,F3(ˉζ1,ˉζ2,ˉζ3)=sech2(2ˉζ1ˉζ2ˉζ3)ˉv3, G1(ˉζ1,ˉζ2,ˉζ3)=12(c1+c2),G2(ˉζ1,ˉζ2,ˉζ3)=c1, G3(ˉζ1,ˉζ2,ˉζ3)=c2, Q(ˉζ1,ˉζ2,ˉζ3)=m, R(ˉζ1,ˉζ2,ˉζ3)=n, (4.16)

    where c1 and c2 are arbitrary constants.

    Case 3. Rational solution.

    {F1(ˉζ1,ˉζ2,ˉζ3)=c3ˉζ2ˉζ21+ˉζ22, F2(ˉζ1,ˉζ2,ˉζ3)=c3ˉζ1ˉζ21+ˉζ22, F3(ˉζ1,ˉζ2,ˉζ3)=c5,G1(ˉζ1,ˉζ2,ˉζ3)=c4ˉζ2ˉζ21+ˉζ22, G2(ˉζ1,ˉζ2,ˉζ3)=c4ˉζ1ˉζ21+ˉζ22, G3(ˉζ1,ˉζ2,ˉζ3)=c6,Q(ˉζ1,ˉζ2,ˉζ3)=c3(2ˉζ1ˉv22ˉζ2ˉv1+c3)2(ˉζ21+ˉζ22)+m, R(ˉζ1,ˉζ2,ˉζ3)=c4(ˉζ1ˉv2ˉζ2ˉv1)ˉζ21+ˉζ22+n, (4.17)

    where c3,c4,c5 and c6 are arbitrary constants. Substituting (4.13) into (4.15)–(4.17), respectively, we obtain that (4.18)–(4.20) are three kinds of solutions for (3 + 1)-dimensional MHD equations (4.10).

    Case 1. Sin-cos-type solution.

    u1=cos2[2xyz(2ˉv1ˉv2ˉv3)t]+ˉv1,u2=cos2[2xyz(2ˉv1ˉv2ˉv3)t]+ˉv2,u3=cos2[2xyz(2ˉv1ˉv2ˉv3)t]+ˉv3,b1=sin[2xyz(2ˉv1ˉv2ˉv3)t]cos[2xyz(2ˉv1ˉv2ˉv3)t]+ˉv1,b2=sin[2xyz(2ˉv1ˉv2ˉv3)t]cos[2xyz(2ˉv1ˉv2ˉv3)t]+ˉv1,b3=sin[2xyz(2ˉv1ˉv2ˉv3)t]cos[2xyz(2ˉv1ˉv2ˉv3)t]+ˉv1,p=3ˉv1sin[(4ˉv12ˉv22ˉv3)t4x+2y+2z]2+3cos[(8ˉv14ˉv24ˉv3)t8x+4y+4z]16m+α(t)dt,r=n+β(t)dt. (4.18)

    Case 2. Sech-type solution.

    u1=sech2[2xyz(2ˉv1ˉv2ˉv3)t]+ˉv1,u2=sech2[2xyz(2ˉv1ˉv2ˉv3)t]+ˉv2,u3=sech2[2xyz(2ˉv1ˉv2ˉv3)t]+ˉv3,b1=12(c1+c2), b2=c1, b3=c2, p=m+α(t)dt, r=n+β(t)dt. (4.19)

    Setting ˉv1=2,ˉv2=4,ˉv3=1,y=2 and x=2 for u1 in (4.19), we obtain Figure 9 of solution u1 as follows.

    Figure 9.  (a) The evolution of single soliton solution u1, (b) u1(z = 1, 2, 3).

    Case 3. Rational solution.

    u1=c3(yˉv2t)(xˉv1t)2+(yˉv2t)2, u2=c3(xˉv1t)(xˉv1t)2+(yˉv2t)2, u3=c5,b1=c4(yˉv2t)(xˉv1t)2+(yˉv2t)2, b2=c4(xˉv1t)(xˉv1t)2+(yˉv2t)2, b3=c6,p=c3[2(xˉv1t)ˉv22(yˉv2t)ˉv1+c3]2[(xˉv1t)2+(yˉv2t)2]m+α(t)dt,r=c4[(xˉv1t)ˉv2(yˉv2t)ˉv1](xˉv1t)2+(yˉv2t)2n+β(t)dt. (4.20)

    Remark 4.1. (1) If b1=b2=b3=0 and r=0 in (4.19) and (4.20), then (4.19) and (4.20) reduce to exact solutions for (3 + 1)-dimensional Euler equation.

    (2) Since ω=×U0 in (4.18) and (4.19) and ω=×U=0 in (4.20), it can be concluded that (4.18) and (4.19) correspond to rotational flow. Additionally, (4.20) corresponds to inrotational flow.

    Without loss of generality, choosing ν=η=κ=1 in Eq (4.1), the viscous MHD equations can be obtained as

    {u1t(u1xx+u1yy+u1zz)+(u1u1x+u2u1y+u3u1z)+(b2b2x+b3b3xb2b1yb3b1z)+px=0,u2t(u2xx+u2yy+u2zz)+(u1u2x+u2u2y+u3u2z)+(b1b1y+b3b3yb1b2xb3b2z)+py=0,u3t(u3xx+u3yy+u3zz)+(u1u3x+u2u3y+u3u3z)+(b1b1z+b2b2zb1b3xb2b3y)+pz=0,b1t(b1xx+b1yy+b1zz)(b1u1x+b2u1y+b3u1z)+(u1b1x+u2b1y+u3b1z)+rx=0,b2t(b2xx+b2yy+b2zz)(b1u2x+b2u2y+b3u2z)+(u1b2x+u2b2y+u3b2z)+ry=0,b3t(b3xx+b3yy+b3zz)(b1u3x+b2u3y+b3u3z)+(u1b3x+u2b3y+u3b3z)+rz=0,u1x+u2y+u3z=0, b1x+b2y+b3z=0. (4.21)

    Solving (4.3)–(4.9) with ν=η=κ=1, the coefficient functions of vector field V_ can be obtained as

    ζ1=2C1t+C2, ζ2=C1xC12yC13z+f1(t)+C3,ζ3=C12x+C1yC23z+f2(t)+C4, ζ4=C13x+C23y+C1z+f3(t)+C5,ϕ1=C1u1C12u2C13u3+f1(t), ϕ2=C12u1C1u2C23u3+f2(t),ϕ3=C13u1+C23u2C1u3+f3(t), φ1=C1b1C12b2C13b3,φ2=C12b1C1b2C23b3, φ3=C13b1+C23b2C1b3,ψ1=2C1pxf1(t)yf2(t)zf3(t)+α(t), ψ2=2C1r+β(t). (4.22)

    Case 1. C1=C2=C3=C4=C5=C12=C13=C23=f1(t)=f2(t)=0, V_=α(t)p+β(t)r.

    For invariants

    ˉζ0=t, ˉζ1=x, ˉζ2=y, ˉζ3=z,F1(ˉζ0,ˉζ2)=u1, F2(ˉζ0,ˉζ1)=u2, F3(ˉζ0,ˉζ3)=u3,G1(ˉζ0,ˉζ2)=b1, G2(ˉζ0,ˉζ1)=b2, G3(ˉζ0,ˉζ3)=b3. (4.23)

    Substituting invariants (4.23) into (4.21), and solving the reduced equations,

    u1=g1etcosy, u2=g2etcosx, u3=c7,b1=g3etcosy, b2=g4etcosx, b3=c8,p=[cos(2y)g23g4+2g34sin2x+4g3(g22g24)sinxsiny]e2t4g4+m(t), r=n(t), (4.24)

    is a sin/cos-type solution for MHD equations (4.21), where g1g4g2g3=0. c7 and c8 are arbitrary constants. m(t) and n(t) are arbitrary functions related to t only.

    Case 2. C2=1, Ci=C12=C13=C23=fj(t)=α(t)=β(t)=0 (i=1,3,4,5,j=1,2,3), V_=t.

    The corresponding invariants are

    ˉζ1=x, ˉζ2=y, ˉζ3=z, Q(ˉζ1,ˉζ2,ˉζ3)=p, R(ˉζ1,ˉζ2,ˉζ3)=r,F1(ˉζ1,ˉζ2,ˉζ3)=u1, F2(ˉζ1,ˉζ2,ˉζ3)=u2, F3(ˉζ1,ˉζ2,ˉζ3)=u3,G1(ˉζ1,ˉζ2,ˉζ3)=b1, G2(ˉζ1,ˉζ2,ˉζ3)=b2, G3(ˉζ1,ˉζ2,ˉζ3)=b3. (4.25)

    Substituting invariants (4.25) into (4.21), and solving the reduced equations,

    u1=sech2(x+iy), u2=isech2(x+iy), u3=c9, p=m,b1=sech2(x+iy), b2=isech2(x+iy), b3=c10, r=n, (4.26)

    is a sech-type solution for MHD equations (4.21), where c9 and c10 are arbitrary constants. Using symmetry V_=tx+ty+tz+u1+u2+u3, solution (4.26) can further generate the following invariant solution,

    u1=sech2(xεt+i(yεt))+ε, u2=isech2(xεt+i(yεt))+ε,u3=c9+ε, b1=sech2(xεt+i(yεt)), b2=isech2(xεt+i(yεt)),b3=c10, p=m(t), r=n(t). (4.27)

    Remark 4.2. (1) The lump solution (4.20) for inviscid MHD equations (4.10) also satisfies the viscous MHD equations (4.21).

    (2) If b1=b2=b3=0 and r=0 in (4.27), then (4.27) reduces to exact solutions for (3 + 1)-dimensional Navier-Stokes equation.

    (3) Since ω=×U0 in (4.24) and ω=×U=0 in (4.27), it can be concluded that (4.24) corresponds to rotational flow. Moreover, (4.27) corresponds to inrotational flow.

    Using traveling wave transformation ζ=k2x+k3y+k4zk1t, Eq (4.1) are transformed into following ODEs as

    {k1u1ζν(k22u1ζζ+k23u1ζζ+k24u1ζζ)+k2u1u1ζ+k3u2u1ζ+k4u3u1ζ+κ(k2b2b2ζ+k2b3b3ζk3b2b1ζk4b3b1ζ)+k2pζ=0,k1u2ζν(k22u2ζζ+k23u2ζζ+k24u2ζζ)+k2u1u2ζ+k3u2u2ζ+k4u3u2ζ+κ(k3b1b1ζ+k3b3b3ζk2b1b2ζk4b3b2ζ)+k3pζ=0,k1u3ζν(k22u3ζζ+k23u3ζζ+k24u3ζζ)+k2u1u3ζ+k3u2u3ζ+k4u3u3ζ+κ(k4b1b1ζ+k4b2b2ζk2b1b3ζk3b2b3ζ)+k4pζ=0,k1b1ζη(k22b1ζζ+k23b1ζζ+k24b1ζζ)(k2b1u1ζ+k3b2u1ζ+k4b3u1ζ)+k2u1b1ζ+k3u2b1ζ+k4u3b1ζ+k2rζ=0,k1b2ζη(k22b2ζζ+k23b2ζζ+k24b2ζζ)(k2b1u2ζ+k3b2u2ζ+k4b3u2ζ)+k2u1b2ζ+k3u2b2ζ+k4u3b2ζ+k3rζ=0,k1b3ζη(k22b3ζζ+k23b3ζζ+k24b3ζζ)(k2b1u3ζ+k3b2u3ζ+k4b3u3ζ)+k2u1b3ζ+k3u2b3ζ+k4u3b3ζ+k4rζ=0,k2u1ζ+k3u2ζ+k4u3ζ=0, k2b1ζ+k3b2ζ+k4b3ζ=0. (4.28)

    Suppose that the solution of ODEs (4.28) can be expressed as a polynomial of ϕ(ζ) as follows.

    u1=N1i=0aiϕi(ζ), u2=N2i=0miϕi(ζ), u3=N3i=0diϕi(ζ), b1=N4i=0niϕi(ζ),b2=N5i=0siϕi(ζ), b3=N6i=0fiϕi(ζ), p=l(t)+N7i=0liϕi(ζ), r=q(t)+N8i=0qiϕi(ζ), (4.29)

    where ai,mi,di,ni,si,fi,li,qi are undetermined constants and aN1, mN2, dN3, nN4, sN5, fN60. l(t) and q(t) are arbitrary functions related to t only. ϕ(ζ) satisfies Eq (3.27). We choose N1=N2=N3=N4=N5=N6=N7=N8=2 with can balance the highest order of the derivative and nonlinear terms in ODEs.

    When ν=η=0 and κ=1 in ODEs (4.28), substituting (4.29) and (3.27) into (4.28), we collect the coefficients of ϕi(ζ) and set them to be zeros, we obtain

    a0=a0,a1=a1,a2=a2,d0=d0,d1=d1,d2=d2,f0=f0,f1=f1,f2=0,k1=k1,k2=k2,k3=k3,k4=k4,l1=(k22k23k24)f0f1k22+k23,l2=f21(k22+k23+k24)2(k22+k23),m0=a0k2k4d0+k1k3,m1=a1k2k4d1k3,m2=k2a2k4d2k3,n0=n0,n1=f1k2k4k22k23,n2=0,q1=0,q2=0,s0=k4f0k2n0k3,s1=f1k3k4k22k23,s2=0.

    Combined with the general solutions of (3.27) (c.f. [29]), it folllows from (4.29) that (4.30)–(4.38) are four kinds of solutions for (3 + 1)-dimensional inviscid MHD equations.

    Case 1. When ξ214ξ2ξ0>0 and ξ1ξ20 (or ξ0ξ20), the following tanh-type solution can be obtained.

    u1=4a0ξ222a1ξ1ξ2+a2ξ214ξ22+a2ξ1a1ξ22ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)+a2(ξ214ξ2ξ0)4ξ22tanh2(ξ214ξ2ξ02ζ), (4.30)
    u2=4ξ22(a0k2d0k4+k1)+2ξ1ξ2(k2a1+d1k4)ξ21(k2a2+d2k4)4k3ξ22+ξ2(a1k2+d1k4)ξ1(k2a2+d2k4)2k3ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)k2a2+d2k44k3ξ22(ξ214ξ2ξ0)tanh2(ξ214ξ2ξ02ζ), (4.31)
    u3=4d0ξ222d1ξ1ξ2+d2ξ214ξ22+d1ξ2+d2ξ12ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)+d2(ξ214ξ2ξ0)4ξ22tanh2(ξ214ξ2ξ02ζ), (4.32)
    b1=2n0ξ2(k22+k23)+f1k2k4ξ12ξ2(k22+k23)+f1k2k4ξ214ξ2ξ02ξ2(k22+k23)tanh(ξ214ξ2ξ02ζ), (4.33)
    b2=2ξ2(k22+k23)(k4f0+k2n0)+k23f4f1ξ12k3ξ2(k22+k23)+k3f4f1ξ214ξ2ξ02ξ2(k22+k23)tanh(ξ214ξ2ξ02ζ), (4.34)
    b3=2f0ξ2f1ξ12ξ2f1ξ214ξ2ξ02ξ2tanh(ξ214ξ2ξ02ζ), (4.35)

    where ζ=k2x+k3y+k4zk1t.

    Setting a0=6,a1=6,a2=2, d0=1,d1=3,d2=2,k1=2, k2=4,k3=1,k4=1,ξ0=1, ξ1=2,ξ2=3,y=1 and t=5 in (4.31), we obtain Figure 10 of kink solution u2 as follows.

    Figure 10.  (a) u2, (b) u2(z = 4, 5, 6).

    In particular, when ξ0=0 and ξ1ξ20, the following sinh-cosh-type solution can be obtained.

    u1=a0a1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+a2ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,u2=a0k2d0k4+k1k3+(a1k2+d1k4)ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]k3ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C](k2a2+d2k4)ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2k3ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,u3=d0d1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+d2ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,b1=n0+f1k2k4ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)](k22+k23)ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C],b2=k4f0k2n0k3+k3k4f1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)](k22+k23)ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]b3=f0f1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C] (4.36)

    Setting a0=2,a1=2,a2=127,d0=10,d1=12,d2=3,k1=14,k2=7,k3=2, k4=2,ξ0=0,ξ1=1,ξ2=12,y=1,C=1 and x=5 in u2 in (4.36), we obtain Figure 11 of kink-like solution u2 as follows.

    Figure 11.  (a) u2, (b) u2(z = 4, 5, 6).

    Case 2. When ξ214ξ2ξ0<0 and ξ1ξ20 (or ξ0ξ20), the following tan-type solution can be obtained.

    u1=4a0ξ222a1ξ1ξ2+a2ξ214ξ22+a1ξ2a2ξ12ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)+a2(4ξ2ξ0ξ21)4ξ22tan2(4ξ2ξ0ξ212ζ),u2=4ξ22(a0k2d0k4+k1)+2ξ1ξ2(a1k2+d1k4)ξ21(k2a2+k4d2)4k3ξ22+ξ2(a1k2+d1k4)+ξ1(k2a2+k4d2)2k3ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)k2a2+k4d24k3ξ22(4ξ2ξ0ξ21)tan2(4ξ2ξ0ξ212ζ),u3=4d0ξ222d1ξ1ξ2+d2ξ214ξ22+d1ξ2d2ξ12ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)+d2(4ξ2ξ0ξ21)4ξ22tan2(4ξ2ξ0ξ212ζ),b1=2n0ξ2(k22+k23)+f1k2k4ξ12ξ2(k22+k23)f1k2k44ξ2ξ0ξ212ξ2(k22+k23)tan(4ξ2ξ0ξ212ζ),b2=2ξ2(k22+k23)(k4f0+k2n0)+k23k4f1ξ12k3ξ2(k22+k23)k3k4f14ξ2ξ0ξ212ξ2(k22+k23)tan(4ξ2ξ0ξ212ζ),b3=2f0ξ2f1ξ12ξ2+f14ξ2ξ0ξ212ξ2tan(4ξ2ξ0ξ212ζ). (4.37)

    Case 3. When ξ1=ξ0=0 and ξ20, the following rational solution can be obtained.

    u1=a0a1ξ2ζ+C+a2(ξ2ζ+C)2, u2=a0k2d0k4+k1k3+a1k2+d1k4k3(ξ2ζ+C)a2k2+d2k4k3(ξ2ζ+C)2,u3=d0d1ξ2ζ+C+d2(ξ2ζ+C)2, b1=n0+f1k2k4(k22+k23)(ξ2ζ+C),b2=k4f0k2n0k3+k3k4f1(k22+k23)(ξ2ζ+C), b3=f0f1ξ2ζ+C. (4.38)

    When ν=η=κ=1 in ODEs (4.28), substituting (4.29) and (3.27) into (4.28), we obtain

    a0=a0,a1=a1,a2=a2,d0=d0,d1=ia1k22+k23k2,d2=ia2k22+k23k2,f0=f0,f1=f1,f2=f2,k1=k1,k2=k2,k3=k3,k4=ik22+k23,m0=m0,m1=a1k3k2,m2=a2k3k2,n0=n0,n1=f1k2ik22+k23,n2=f2k2ik22+k23,s0=s0,s1=f1k3ik22+k23,
    s2=f2k3ik22+k23,l1=a1(a0k2id0k22+k23m0k3+k1)k2,l2=a2(i(a0k2+m0k3k1)k22+k23d0(k22+k23))ik2k22+k23,q1=(a1n0+f1d0)k2k3s0a1k2+(a0f1+a1f0)k22f1(m0k3+k1)k2+f0a1k23ik2k22+k23,q2=(a2n0+f2d0)k2k3s0a2k2+(a0f2+a2f0)k22f2(m0k3+k1)k2+f0a2k23ik2k22+k23.

    Combined with general solutions of (3.27) (c.f. [29]), it follows from (4.29) that (4.39)–(4.52) are four kinds of solutions for (3 + 1)-dimensional viscous MHD equations.

    Case 1. When ξ214ξ2ξ0>0 and ξ1ξ20 (or ξ0ξ20), the following tanh-type solution can be obtained.

    u1=4a0ξ222a1ξ1ξ2+a2ξ214ξ22+a2ξ1a1ξ22ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)+a2(ξ214ξ2ξ0)4ξ22tanh2(ξ214ξ2ξ02ζ),  (4.39)
    u2=4m0k2ξ222a1k3ξ1ξ2+a2k3ξ214k2ξ22+k3(a1ξ2+a2ξ1)2k2ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)+a2k3(ξ214ξ2ξ0)4k2ξ22tanh2(ξ214ξ2ξ02ζ),  (4.40)
    u3=d0+i(2a1ξ1ξ2+a2ξ21)k22+k234k2ξ22+ia2(ξ214ξ2ξ0)k22+k234k2ξ22tanh2(ξ214ξ2ξ02ζ)+i(a2ξ1a1ξ2)k22+k232k2ξ22ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ), (4.41)
    b1=n0+2f1k2ξ1ξ2+f2k2ξ21i4ξ22k22+k23+k2(f1ξ2+f2ξ1)i2ξ22k22+k23ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)+f2k2(ξ214ξ2ξ0)i4ξ22k22+k23tanh2(ξ214ξ2ξ02ζ),  (4.42)
    b2=s0+2f1k3ξ1ξ2+f2k3ξ21i4ξ22k22+k23+k3(f1ξ2+f2ξ1)i2ξ22k22+k23ξ214ξ2ξ0tanh(ξ214ξ2ξ02ζ)+f2k3(ξ214ξ2ξ0)i4ξ22k22+k23tanh2(ξ214ξ2ξ02ζ),  (4.43)
    b3=4f0ξ222f1ξ1ξ2+f2ξ214ξ22+f1ξ2+f2ξ12ξ22ξ124ξ2ξ0tanh(ξ214ξ2ξ02ζ)+f2(ξ214ξ2ξ0)4ξ22tanh2(ξ214ξ2ξ02ζ). (4.44)

    Setting a1=1,a2=2,d0=1,k1=1,k2=1,k3=1,ξ0=2,ξ1=6,ξ2=3 and y=1 in (4.41), we obtain Figure 12 for solution u3 as follows.

    Figure 12.  (a) The evolution of interaction solution between anti-kink and solition wave via |u3|(t=1), (b) Overview of |u3|(t=1), (c) |u3|(t=1,2,3,z=1).

    In particular, when ξ0=0 and ξ1ξ20, the following sinh-cosh-type solution can be obtained.

    u1=a0a1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+a2ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,u2=m0a1k3ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]k2ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+a2k3ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2k2ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,u3=d0ia1ξ1k22+k23[sinh(ξ1ζ)+cosh(ξ1ζ)]k2ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+ia2ξ21k22+k23[sinh(ξ1ζ)+cosh(ξ1ζ)]2k2ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2, b1=n0f1k2ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]iξ2k22+k23[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+f2k2ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2iξ22k22+k23[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2,b2=s0f1k3ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]iξ2k22+k23[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+f2k3ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2iξ22k22+k23[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2b3=f0f1ξ1[sinh(ξ1ζ)+cosh(ξ1ζ)]ξ2[sinh(ξ1ζ)+cosh(ξ1ζ)+C]+f2ξ21[sinh(ξ1ζ)+cosh(ξ1ζ)]2ξ22[sinh(ξ1ζ)+cosh(ξ1ζ)+C]2. (4.45)

    Case 2. When ξ214ξ2ξ0<0 and ξ1ξ20 (or ξ0ξ20), the following tan-type solution can be obtained.

    u1=4a0ξ222a1ξ1ξ2+a2ξ214ξ22+a1ξ2a2ξ12ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)+a2(4ξ2ξ0ξ21)4ξ22tan2(4ξ2ξ0ξ212ζ), (4.46)
    u2=4m0k2ξ222a1k3ξ1ξ2+a2k3ξ214k2ξ22+k3(a1ξ2a2ξ1)2k2ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)+a2k3(4ξ2ξ0ξ21)4k2ξ22tan2(4ξ2ξ0ξ212ζ), (4.47)
    u3=d0+i(2a1ξ1ξ2+a2ξ21)k22+k234k2ξ22+ia2(4ξ2ξ0ξ21)k22+k234k2ξ22tan2(4ξ2ξ0ξ212ζ)+i(a1ξ2a2ξ1)k22+k232k2ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ), (4.48)
    b1=n0+k2(2f1ξ1ξ2+f2ξ21)i4ξ22k22+k23+k2(f1ξ2f2ξ1)i2ξ22k22+k234ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)+f2k2(4ξ2ξ0ξ21)i4ξ22k22+k23tan2(4ξ2ξ0ξ212ζ), (4.49)
    b2=s0+k3(2f1ξ1ξ2+f2ξ21)i4ξ22k22+k23+k3(f1ξ2f2ξ1)i2ξ22k22+k234ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)+k3f2(4ξ2ξ0ξ21)i4ξ22k22+k23tan2(4ξ2ξ0ξ212ζ), (4.50)
    b3=4f0ξ222f1ξ1ξ2+f2ξ214ξ22+f1ξ2f2ξ12ξ224ξ2ξ0ξ21tan(4ξ2ξ0ξ212ζ)+f2(4ξ2ξ0ξ21)4ξ22tan2(4ξ2ξ0ξ212ζ). (4.51)

    Case 3. When ξ1=ξ0=0 and ξ20, the following rational solution can be obtained.

    u1=a0a1ξ2ζ+C+a2(ξ2ζ+C)2, b1=n0f1k2ik22+k23(ξ2ζ+C)+f2k2ik22+k23(ξ2ζ+C)2,u3=d0ia1k22+k23k2(ξ2ζ+C)+ia2k22+k23k2(ξ2ζ+C)2, u2=m0a1k3k2(ξ2ζ+C)+a2k3k2(ξ2ζ+C)2, b2=s0f1k3ik22+k23(ξ2ζ+C)+f2k3ik22+k23(ξ2ζ+C)2 b3=f0f1ξ2ζ+C+f2(ξ2ζ+C)2. (4.52)

    Setting a1=2i,a2=i,d0=2,k1=3,k2=2,k3=2,ξ0=0,ξ1=0,ξ2=1,C=1,z=2 and x=15 for u3 in (4.52), we obtain Figure 13 for bright–dark soliton solution u3 as follows.

    Figure 13.  (a) |u3|, (b) |u3|(t = 1, 2, 3).

    We analyze the continuous dependence of solution for MHD equations (2.2) on initial data, or namely the stability of MHD equations (2.2) from a qualitative perspective.

    Lemma 5.1. [31] For q[2,), there exists C>0 such that for fH1(R2),

    fqLq(R2)Cf2L2(R2)fq2L2(R2). (5.1)

    Lemma 5.2. [32] For p[2,6], there exists C>0 such that for gH1(R3),

    gpLp(R3)Cg(6p)/2L2(R3)g(3p6)/2L2(R3). (5.2)

    Theorem 5.3. For n=2,3, if the initial data U0,B0(L2(Rn))n, then the solutions (U,B) for the (2 + 1)- and (3 + 1)-dimensional MHD equations (2.2) with periodic boundary condition at infinity depend on the initial data continuously in (L2(Rn))n.

    Proof. Let (U1,B1) and (U2,B2) be two solutions to MHD equations (2.2) with initial data U0,B0(L2(Rn))n. Set ˜U=U1U2, ˜B=B1B2, ˜p=p1p2, ˜r=r1r2, then (˜U,˜B) is the solution to the following system,

    {˜UtνΔ˜U+((˜U)U1+(U2)˜U)κ((˜B)B1+(B2)˜B)+˜p+12κ(|B1|2|B2|2)=0,(5.3)˜BtηΔ˜B+((˜U)B1+(U2)˜B)((˜B)U1+(B2)˜U)+˜r=0,(5.4)div ˜U=0, div ˜B=0.(5.5)

    Case 1. n=2.

    It follows from H¨older inequality and Lemma 5.1 that

    ((˜U)U1,˜U)U1L2˜U2L4C0U1L2˜UL2˜UL2, (5.6)
    ((˜B)U1,˜B)U1L2˜B2L4C0U1L2˜BL2˜BL2, (5.7)
    ((˜B)B1,˜U)B1L2˜BL4˜UL4C0B1L2˜B12L2˜B12L2˜U12L2˜U12L2, (5.8)
    ((˜U)B1,˜B)B1L2˜UL4˜BL4C0B1L2˜U12L2˜U12L2˜B12L2˜B12L2. (5.9)

    Take L2 inner product of (5.3) with ˜U and (5.4) with ˜B, respectively. Without loss of generality, choose ν=η=κ=1 in (5.3) and (5.4). Since

    ((U2)˜U,˜U)=0,  ((U2)˜B,˜B)=0,  ((B2)˜B,˜U)+((B2)˜U,˜B)=0, (5.10)

    and

    ((˜p+12κ(|B1|2|B2|2)),˜U)=0,  (˜r,˜B)=0, (5.11)

    using (5.6)–(5.11), we have

    ddt(˜U2L2+˜B2L2)+2˜U2L2+2˜B2L2CU1L2˜UL2˜UL2+CU1L2˜BL2˜BL2+2CB1L2˜B12L2˜B12L2˜U12L2˜U12L2. (5.12)

    where C=2C0. It follows from Young inequality and (5.12) that

    ddt(˜U2L2+˜B2L2)+2˜U2L2+2˜B2L2C22(U12L2+B12L2)(˜U2L2+˜B2L2)+(˜U2L2+˜B2L2). (5.13)

    Using Gr¨onwall's inequality, ˜U2L2+˜B2L2M(˜U2L2+˜B2L2)|t=t0. Then, solution (U,B) for (2 + 1)-dimensional MHD equations (2.2) with periodic boundary condition at infinity depends on the initial data continuously in (L2(R2))2.

    Case 2. n=3.

    It follows from H¨older inequality and Lemma 5.2 that

    ((˜U)U1,˜U)U1L2˜UL3˜UL6C0U1L2˜U12L2˜U32L2, (5.14)
    ((˜B)U1,˜B)U1L2˜BL3˜BL6C0U1L2˜B12L2˜B32L2. (5.15)
    ((˜B)B1,˜U)B1L2˜BL3˜UL6C0B1L2˜B12L2˜B12L2˜UL2, (5.16)
    ((˜U)B1,˜B)B1L2˜UL3˜BL6C0B1L2˜U12L2˜U12L2˜BL2. (5.17)

    Using Young inequality with ε, without loss of generality, choosing ν=η=κ=1 in (5.3) and (5.4), there exists ε<23 such that

    ddt(˜U2L2+˜B2L2)+2˜U2L2+2˜B2L2C(ε)(U14L2+B14L2)(˜U2L2+˜B2L2)+3ε(˜U2L2+˜B2L2).

    Similarly, using Gr¨onwall's inequality, it can be obtained that solution (U,B) for (3 + 1)-dimensional MHD equations (2.2) with periodic boundary condition at infinity depends on the initial data continuously in (L2(R3))3.

    Next, we further analyze the stability of MHD equations (2.2) combining with the exact solutions obtained above from a quantitative perspective, which provide an accurate mathematical description for the stability of MHD systems. Denote ˉU=U+U, ˉB=B+B, where U,B are disturbances to the velocity and magnetic field, respectively. (U,B) and (ˉU,ˉB) are solutions before and after being affected by disturbances, respectively. Therefore U,B satisfy the following system

    {UtνΔU+((U)U+((U+U))U)κ((B)B+((B+B))B)+p+12κ(|B|2+|B+B|2)=0,(5.18)BtηΔB+((U)B+((U+U))B)((B)U+((B+B))U)+r=0,(5.19)divU=0, divB=0.(5.20)

    We select several obtained exact solutions of MHD system to study the impact of disturbances on stability of the system.

    Case 1. Harmonic disturbance.

    The initial disturbance is

    u1(t0,x,y)=A1cos(2πw1(xy(ˉv1ˉv2)t0)),  u2(t0,x,y)=A2cos(2πw2(xy(ˉv1ˉv2)t0)),

    where A1, A2 are amplitude of disturbance waves. We analyze the behavior of u1, u2 in (3.13) after being affected by disturbances u1(t,x,y), u2(t,x,y). Set Ai=0.1,wi=5 (i=1,2), the evolution of u1+u1 can be displayed intuitively as following Figure 14 (u2+u2 is similar).

    Figure 14.  (a) u1+u1, (b) u1+u1(t = 1, 3, 5).

    From Figure 14, it can be seen that with the evolution of time, the amplitude of U under the influence of the harmonic disturbance is limited. The solutions (U,B) for the (2 + 1)-dimensional MHD equations (2.2) depend on the initial data continuously in (L2(R2))2, which is also consistent with the conclusion of qualitative analysis.

    Case 2. Bell shaped solitary wave disturbance.

    The initial disturbance is

    u1(t0,x,y)=A1sech(2πw1(xy(ˉv1ˉv2)t0)),  u2(t0,x,y)=A2sech(2πw2(xy(ˉv1ˉv2)t0)),

    where A1, A2 are amplitude of disturbance waves. We analyze the behavior of u1, u2 in (3.14) after being affected by disturbances u1(t,x,y), u2(t,x,y). Set Ai=0.1,wi=5(i=1,2), the evolution of u1+u1 can be displayed intuitively as following Figure 15 (u2+u2 is similar).

    Figure 15.  u1+u1(t=1, 4, 7, respectively).

    From Figure 15, it can be seen that the amplitude of U under the influence of the Bell shaped solitary wave disturbance has increased but is limited. The velocity U under the influence of Bell shaped solitary wave disturbance is stable.

    In this paper, several novel classes of solutions and stability analysis are presented for MHD flows. When the magnetic field vanishes, some of the exact solutions can be reduced to solutions of Euler or Navier-Stokes equation. Through Lie symmetry analysis and the generalized Riccati equation expansion method, the MHD system achieves order reduction and dimensionality reduction, and the complex nonlinear and strongly coupled terms in fluid dynamics systems are handled technically. The Lie group of transformations and the similarity reductions of (2 + 1)- and (3 + 1)-dimensional inviscid and viscous MHD equations are studied. The exact solutions with rich forms are obtained, which can describe certain solition-like surface waves, such as periodic solution, single soliton solution, and lump solution. The mechanisms of rotational and irrotational fluids are analyzed. Furthermore, using the generalized Riccati equation expansion method, we obtain miscellaneous traveling wave solutions, including kink, kink-like, anti-kink-like, breather, and interaction solutions. In addition, the continuous dependence of solutions for MHD equations for initial values is studied from qualitative and quantitative perspectives.

    Compared with the related work, the novelty of this paper lies in that we consider the problem from multiple perspectives and obtain new exact solutions. For instance, Dorodnitsyn et al. [23] studied (1 + 1)-dimensional inviscid MHD flows in the mass Lagrangian coordinates, while we studied from the perspective of both inviscid and viscous of (2 + 1)- and (3 + 1)-dimensional MHD equations. Liu et al. [24] obtained analytical solutions of (2 + 1) -dimensional inviscid incompressible MHD equations by Lie symmetry analysis. Picard et al. [26] obtained some exact solutions of (3 + 1)-dimensional inviscid MHD equations by the symmetry reduction method. We used Lie symmetry analysis as well as generalized Riccati equation expansion methods to study both inviscid and viscous of (2 + 1)- and (3 + 1)-dimensional MHD equations. Moreover, based on the study, we obtain new exact solutions with richer forms. Xia et al. [25] used the Lie symmetry method to obtain some exact solutions of (2 + 1)-dimensional incompressible ideal MHD equations. Cheung et al. [33] obtained bounded soliton solutions of (2 + 1)-dimensional incompressible MHD equations. However, we obtain some new exact solutions for both inviscid and viscous of (2 + 1)- and (3 + 1)-dimensional MHD equations, such as lump solutions, kink solutions, kink-like solution, breather solutions, and interaction solution between anti-kink and solition. Ayub et al. [34] studied solitary wave solutions for two-dimensional viscous incompressible MHD flow regarding space evolution, while we studied from the perspective of both inviscid and viscous of (2 + 1)- and (3 + 1)-dimensional MHD flows, which consider both time and space evolution.

    The exact solutions we obtain can correspond to different physical behaviors for MHD flows. For instance, solitons can maintain their shape and thier amplitude is unchanged during propagation. This property is important for understanding and predicting some wave propagations in MHD flow. Soliton waves can be used to describe some wave phenomena in plasma, which has potential application value for achieving and maintaining the stability of fusion plasma. Periodic solutions can describe some periodic oscillation phenomena in MHD flow. Lump solution can correspond to waves that are localized in time and space, while the amplitude of peak and valley is several times higher than the surrounding background height. Breather solutions can explain MHD flow that exhibits periodicity in certain direction and locality in other directions. The kink and kink-like solutions can manifest as a rapid change or discontinuity in some fields at the macro level. Considering the physical significance and the importance of studying analytical solutions of MHD equations, compressible case and MHD systems with other factors such as time-dependent density and Coriolis force deserve to be further studied.

    Shengfang Yang worked on conceptualization, writing-original draft, formal analysis, software. Huanhe Dong worked on conceptualization, resources, validation, supervision. Mingshuo Liu worked on methodology, writing-review & editing, formal analysis, validation.

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

    This work was supported by the National Natural Science Foundation of China (Nos. 12105161, 12305003).

    The authors declare there is no conflict of interest.



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