On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

1 Introduction

Among the nonlinear evolution equations (NLEEs) [1–5], Korteweg–de Vries (KdV)-type equations can be used to describe the shallow water waves of long wavelength and small amplitude, stratified internal waves and ion-acoustic waves in plasmas [6, 7]. As a (2+1)-dimensional KdV-type equation, the Boiti–Leon–Manna–Pempinelli (BLMP) equation can be considered as a model for the incompressible fluid [7–9], which is

(1)vξτ + vζζζξ − 3vζvζξ − 3vζζvξ = 0, (1)

where v=v(ζ, ξ, τ) is an analytic function of the scaled spatial coordinates (ζ, ξ) and temporal coordinate τ. Equation (1) can also be used to describe the (2+1)-dimensional interaction of the Riemann wave propagating along the ξ-axis with a long wave propagating along the ζ-axis [10]. For ζ=ξ, (1) can be reduced to the KdV equation [6, 7]. Integrabilities of (1) in the Painlevé and Lax senses have been discussed [11]. Besides, for (1), multi-periodic solutions [12], variable-separation solutions [13, 14], and soliton-like solutions [15] have been obtained.

Conversely, integrable versions of the NLEEs usually have the soliton solutions [16–20]. A soliton is a solitary wave which preserves its velocity and shape after the interaction [7], i.e., the soliton can be considered as a quasiparticle [21, 22]. Hirota bilinear method has been used to construct the soliton solutions of the NLEEs [23–26]. Moreover, Bell polynomials approach has been applied to attain the bilinear form and Bäcklund transformations (BTs) [27].

Higher-dimensional NLEEs are scientifically interesting: For example, some (3+1)-dimensional KdV-type equations can describe the dust–ion–acoustic waves in cosmic nonmagnetised dusty plasmas such as those in the supernova shells and Saturn’s F-ring [28], and some (3+1)-dimensional NLEEs, with certain parameters, can reduce to the (2+1)-dimensional and (1+1)-dimensional NLEEs [29–35] (examples will be displayed following). A (3+1)-dimensional BLMP equation is [10, 36]

(2)uyt + uzt + uxxxy + uxxxz − 3uxuxy − 3uxuxz − 3uxxuy − 3uxxuz = 0, (2)

where u is an analytic function depending on the scaled spatial coordinates (x, y, z) and temporal coordinate t. By virtue of the exp-function method [10] and bilinear method via the logarithm transformation [36], the soliton solutions of (2) have been discussed [10, 36]. In the fluid and plasma dynamics, special cases of (2) are seen as follows:

1. When y=z, u(x, y, t)=ϱ(η, t)=ϱ [x + χ(y), t] with as χ an analytic function of y, (2) degenerates into the KdV equation [37],

   (3)ϱηt + ϱηηηη − 6ϱηϱηη = 0 , (3)

   for certain phenomena in fluids and plasmas [6, 7], where ϱ is an analytic function depending on the scaled spatial coordinate η and temporal coordinate t. Equation (3) is considered as the model of the shallow water waves for long wavelength and small amplitude [6, 7].

2. When y=z=ξ, t=τ, x=ζ, (2) degenerates into (1) for the incompressible fluid [7–9]. Further, through the variable transformation v˜ = vξ, (1) becomes [9]

   (4a)v˜τ + v˜ζζζ − 3(v˜v˜˜)ζ = 0, (4a)

   (4b)v˜˜ = vζ, (4b)

   where v˜ and v˜˜ are the components of the incompressible-fluid velocity [9]. Equation (4) works as a model for an incompressible fluid [9].

3. Equation (4) can be regarded as a generalization to some (2+1)-dimensional cases presented in [9, 38, 39].

Lax pairs, BTs, multi-solition solutions with two kinds of dispersion relations and bilinear form via a general logarithm transformation for (2) are our tasks in this article, which have not been obtained before. In Section 2, by virtue of the Bell polynomials, bilinear form of (2) are obtained. In Section 3, multi-soliton solutions with two kinds of nonlinear dispersion relations and another kind of analytic solutions are found. In Section 4, Lax pairs and BTs are obtained. In Section 5, interaction of the solitons are discussed. Finally, Section 6 discusses the conclusions.

2 Bell Polynomials and Bilinear Equation

In this section, we give the bilinear form of (2) by virtue of the multi-dimensional binary Bell polynomials. Let g=g(x, y, z, t) be a C^∞ function, and the multi-dimensional Bell polynomials are written as [40–44]

(5)ℬn1x,n2y,n3z,n4t(g) = e−g∂xn1∂yn2∂zn3∂tn4eg , (5)

where n_i′s (i=1, 2, 3, 4) are all the positive integers. Based on that, the multi-dimensional binary Bell polynomials, namely the 𝒴-polynomials, can be defined as [41–44]

(6)Yn1x,n2y,n3z,n4t(v, w)=ℬn1x,n2y,n3z,n4t(g)|gr1x,r2y,r3z,r4t = {vr1x,r2y,r3z,r4t,wr1x,r2y,r3z,r4t,r1 + r2 + r3 + r4 is oddr1 + r2 + r3 + r4is even, (6)

where gr1x,r2y,r3z,r4t = e−g∂xr1∂yr2∂zr3∂tr4eg and r_i=0, …, n_i (i=1, 2, 3, 4). For example,

(7a)Yx(v, w) = vx, (7a)

(7b)Yx,t(v, w) = wx,t + vxvt , (7b)

(7c)Y3x(v, w) = v3x + 3vxw2x + vx3, ⋮. (7c)

The link between the 𝙔-polynomials and Hirota bilinear operator Dxn1Dyn2Dzn3Dtn4F⋅G can be given by the following relations [41–44]

(8)Yn1x,n2y,n3z,n4t(v = lnFG, w = lnFG) = (FG)−1Dxn1Dyn2Dzn3Dtn4F⋅G, (8)

where n₁ + ··· + n₄ ≥ 1, F and G are the functions of x, y, z and t, and Dxn1Dyn2Dzn3Dtn4 is the Hirota bilinear derivative operator [26] defined by

(9)Dxn1Dyn2Dzn3Dtn4a(x, y, z, t)⋅a˜(x′, y′, z′, t′)≡(∂∂x − ∂∂x′)n1 (∂∂y − ∂∂y′)n2 (∂∂z − ∂∂z′)n3 (∂∂t − ∂∂t′)n4 a(x, y, z, t) a˜(x′, y′, z′, t′)|x′ = x, y′ = y, z′ = z,  t′ = t, (9)

with a(x, y, z, t) as the function of x, y, z, and t, a˜(x′, y′, z′, t′) as the function of formal variables x′, y′, z′, and t′.

In the particular case G=F, (8) becomes the 𝒫-polynomial [41–44]

(10)𝒫n1x,n2y,n3z,n4t(q˜) = Yn1x,n2y,n3z,n4t(0, lnF = q˜2) . (10)

The 𝒫-polynomials can be characterized by an equally-recognisable even-part-partitional structures [41–44],

(11)𝒫xt(q˜) = q˜xt,xxxy(q˜) = q˜xxxy + 3q˜xxq˜xy,⋯. (11)

Now we will bilinearise (2) with the aid of the Bell polynomials. Similar to the process in [45], under the scale transformations,

(12)x→λx′ , y→λ−1y′, z→λ−1z′ , t→λ−1t′, u→λ−1u′, (12)

Equation (2) is invariant. Introducing the variable q, we set

(13)u = cqx + Φ(y) + Ψ(z), (13)

where Φ and Ψ are both the arbitrary functions, c is a constant to be determined. Substituting (13) into (2) and integrating the resulting equation yield

(14)qzt + qyt − 3ϕ(y)qxx − 3ψ(z)qxx + qxxxy + qxxxz   + 3qxzqxx + 3qxyqxx = 0, (14)

where ϕ(y) = ∂∂yΦ(y), ψ(z) = ∂∂zΨ(z), and c=–1. Substituting (11) into (14), we can transform (2) into the form of the 𝒫-polynomials and bilinear form,

(15)𝒫zt + 𝒫yt + 𝒫xxxy + 𝒫xxxz − 3ϕ(y)𝒫xx − 3ψ(z)𝒫xx = 0. (15)

Meanwhile, (2) can be transformed into

(16)[DzDt + DyDt + Dx3Dy + Dx3Dz − 3ϕ(y)Dx2 − 3ψ(z)Dx2]f⋅f = 0, (16)

where f is a real function of x, y, z, and t. The dependent variable transformation of this procedure is

(17)u = −2fxf + Φ(y) + Ψ(z). (17)

We note that (17) is a generalization of what is obtained in [27], dealing with (1).

3 Multi-Soliton Solutions and Another Kind of Solution

For the multi-soliton solutions, we expand f into the power series of a small parameter ϵ as

(18)f = 1  + ϵf1 + ϵ2f2 + ⋯, (18)

where f_j′s(j=1, 2, ···) are all the real functions of x, y, z, and t.

Substituting (18) into (16) and collecting the coefficients of the same power of ϵ, we obtain

(19)ϵ0:[DzDt + DyDt + Dx3Dy + Dx3Dz − 3ϕ(y)Dx2  − 3ψ(z)Dx2](1⋅1) = 0 , (19)

(20)ϵ1:[DzDt + DyDt + Dx3Dy + Dx3Dz −3ϕ(y)Dx2 − 3ψ(z)Dx2](1⋅f1 + f1⋅1) = 0, (20)

(21)ϵ2:[DzDt + DyDt + Dx3Dy + Dx3Dz  −3ϕ(y)Dx2 − 3ψ(z)Dx2](1⋅f2 + f2⋅1 + f1⋅f1) = 0,    ⋮ . (21)

In order to obtain the one-soliton solutions for (2), we assume that

(22)f1 = eη1, (22)

where

(23)η1 = k1x + Φ(y)k1 + l1y + Ψ(z)k1 + m1z + w1t + δ1 (23)

whereas k₁, l₁, m₁, w₁, and δ₁ are all real constants.

Substituting (22) into (20), we derive out the relationship between l₁, m₁, k₁, ϕ(y), ψ(z), and w₁ as

Case I

(24a)ϕ(y) = −ψ(z) = C, (24a)

(24b)w1 = −k13, (24b)

Case II

(25a)l1 = −m1, (25a)

(25b)w1 = 2k13, (25b)

where C is a constant. Expressions (24b) and (25b) are both called the nonlinear dispersion relations [26]. Consequently, we can take f_j=0(j=2,3, ···), and ϵ=1. Then we get the one-soliton solutions for (2):

(26)u = −2[ln(1 + eη1)]x + Φ(y) + Ψ(z). (26)

Similarly, in order to derive the two-soliton solutions, we set

(27)f = 1 +  eη1 + eη2 + A12eη1 + η2 , (27)

where ηi = kix + Φ(y)ki + liy + Ψ(z)ki + miz + wit + δi(i = 1, 2), and l_i′s, m_i′s, k_i′s, w_i′s, δ_i′s (i=1, 2) and A₁₂ are all real constants. Substituting (27) into (20) and (21), we have

Case I

(28a)ϕ(y) = −ψ(z), wi = −ki3(i = 1, 2), (28a)

(28b)A12 = (k1 − k2)(l1 − l2 + m1 − m2)(k1 + k2)(l1 + l2 + m1 + m2) . (28b)

Case II

(29a)li = −mi, wi = 2ki3 (i = 1, 2) , (29a)

(29b)A12 = (k1 − k2)2(k12 + k1k2 + k22)k14 + k13k2 + k1k23 + k24 . (29b)

Thus, we can obtain the two-soliton solutions

(30)u = −2[ln(1 +  eη1 + eη2 + A12eη1 + η2)]x + Φ(y) + Ψ(z). (30)

This process can be continued for us to derive the three-soliton solutions, which can be obtained from

(31)f = 1 +  eη1 + eη2 +  eη3 + A12eη1 + η2 + A13eη1 + η3  + A23eη2 + η3 + A12A23A13eη1 + η2 + η3 , (31)

where ηi = kix + Φ(y)ki + liy + Ψ(z)ki + miz + wit + δi(i = 1, 2, 3),l_i′s, w_i′s, m_i′s, k_i′s, δ_i′s (i=1, 2, 3) are all real constants, and the others satisfy

Case I

(32a)ϕ(y) = −ψ(z), wi = −ki3(i = 1, 2, 3) , (32a)

(32b)A13 = (k1 − k3)(l1 − l3 + m1 − m3)(k1 + k3)(l1 + l3 + m1 + m3) , (32b)

(32c)A23 = (k2 − k3)(l2 − l3 + m2 − m3)(k2 + k3)(l2 + l3 + m2 + m3) . (32c)

Case II

(33a)li = −mi, wi = 2ki3(i = 1, 2, 3), (33a)

(33b)A13 = (k1 − k3)2(k12 + k1k3 + k32)k14 + k13k3 + k1k33 + k34 , (33b)

(33c)A23 = (k2 − k3)2(k22 + k2k3 + k32)k24 + k23k3 + k2k33 + k34 . (33c)

If we continue such a process, we might derive the four-, five-soliton solutions and so on.

Besides, when ϕ(y)=–ψ(z), the solutions for (16) can be assumed to be in the form [46, 47]

(34)f(x,y,z,t) = Q0eγ1 + Q1cos(γ2) + Q2e−γ1, (34)

where γ1 = k˜1x + Φ(y)k˜1 + l˜1y + m˜1z + Ψ(z)k˜1 + w˜1t, γ2 = k˜2x + Φ(y)k˜2 + l˜2y + m˜2z + Ψ(z)k˜2 − w˜2t,k˜i′s, l˜i′s, w˜i′s, m˜i′s(i = 1, 2) and Q_i′s(i=0, 1, 2) are all constants. We note that Φ(y) and Ψ(z) in (34) are the constants and they can be absorbed in γ₁ and γ₂. Taking (34) into (16) and collecting the coefficients of different terms, we can find that the coefficients of (34) should satisfy the following relations:

Case 1

(35a)li = −mi (i = 1, 2), (35a)

(35b)w˜1 = −(k˜1)3 + 3k˜1(k˜2)2, (35b)

(35c)w˜2 = −(k˜2)3 + 3(k˜1)2k˜2. (35c)

Case II

(36a)li = −mi(i = 1, 2), (36a)

(36b)w˜1 = −(k˜1)5 − 11(k˜1)3(k˜2)2(k˜1)2 + (k˜2)2, (36b)

(36c)w˜2 = −−6k˜2(k˜1)4 + 7(k˜1)2(k˜2)3 + (k˜2)5(k˜1)2 + (k˜2)2. (36c)

Thus, we get a kind of analytic solution u with different nonlinear dispersion relations in the following form:

(37)u = −2[ln(Q0eγ1 + Q1cos(γ2) + Q2e−γ1)]x + c1(y − z) + c2, (37)

where c_i′s (i=1, 2) are the constants.

4 Lax Pairs and BTs

Lax pairs play a role in the integrable properties of the NLEEs [35]. Bell polynomials have been used [27] to get the Lax pairs of (1), based on which we hereby construct the Lax pairs of (2) as

(38a)L(Θ) = Θt + Θxxx − 3uxΘx, (38a)

(38b)M(Θ) = Θxy − uyΘ − uzΘ, (38b)

where Θ is a function of x, y, z, and t. Our direct calculation indicates that the condition L[M(Θ)]=M[L(Θ)] leads to (2). By the way, Lax pairs (38) can also be obtained via the Bell polynomials.

Conversely, a BT is related to the integrability for a NLEE, which can also be used for people to construct the soliton solutions [27]. Thus, we search for the bilinear BTs of (2).

Let p and p′ be two different solutions of (14), and we have

(39)(p′ − p)zt + (p′ − p)yt − 3ϕ(y)(p′ − p)xx − 3ψ(z)(p′ − p)xx + (p′ − p)xxxy+(p′ − p)xxxz + 3(p′xzp′xx − pxzpxx) + 3(p′xyp′xx  − pxypxx) = 0. (39)

With the mixing variable transformations as

(40)ν = p′ − p2, ω = p′ + p2 , (40)

Expression (39) can be rewritten as

(41)νzt + νyt − 3ϕ(y)νxx − 3ψ(z)νxx + νxxxy + νxxxz + 3ωxxνxz + 3νxxωxz + 3ωxxνxy + 3νxxωxy = I + II + III + IV + V = 0, (41)

where

I = −3ϕ(y)νxx + 3νxxωxy + 3νxνyνxx = 3νxx[Yxy(ν, ω) − ϕ(y)],II = −3ψ(z)νxx + 3νxxωxz + 3νxνzνxx   = 3νxx[Yxz(ν, ω) − ψ(z)],III = νyt + νxxxy + 3ωxxνxy + 3νxωxxy + 3νx2νxy   =(νt + νxxx + 3νxωxx + νx3)y = [Y(ν, ω)t + Y(ν, ω)xxx]y,IV = νzt + νxxxz + 3ωxxνxz + 3νxωxxz + 3νx2νxz=(νt + νxxx + 3νxωxx + νx3)z = [Y(ν, ω)t + Y(ν, ω)xxx]z.

V = −3νxνyνxx − 3νxνzνxx − 3νxωxxy − 3νx2νxy − 3νxωxxz − 3νx2νxz = −3νx(νyνxx + ωxxy + νxνxy) − 3νx(νzνxx + ωxxz + νxνxz) = −3νx(νxνy + ωxy)x − 3νx(νxνz + ωxz)x, = −3νx[Y(ν, ω)xy + Y(ν, ω)xz]x .

Via I=II=III=IV=0, people can obtain that V=0 and the coupled system of the 𝒴-polynomials is

(42a)Yt(ν, ω) + Yxxx(ν, ω) = λ, (42a)

(42b)Yxy(ν, ω) − ϕ(y) = 0, (42b)

(43c)Yxz(ν, ω) − ψ(z) = 0, (43c)

where λ is a constant.

With (8) and the variable transformations

(43)lnf = p2, lnf′ = p′2 , (43)

the bilinear BTs of (2) can be written as

(44a)[Dt + Dx3]f′⋅f = λ, (44a)

(44b)[DxDy − ϕ(y)]f′⋅f = 0, (44b)

(44c)[DxDz − ψ(z)]f′⋅f = 0. (44c)

5 Soliton Interaction

In this section, the soliton interaction is investigated.

Figures 1a and 2a show the kink-shape solitons on the x-t plane. Figure 1b has one kink-shape soliton, whereas Figure 2b has both one kink-shape soliton and one periodic backgrounds caused by Φ(y)=sin(2y). Figure 1c has one kink-shape soliton, whereas Figure 2c has both one kink-shape soliton and one periodic backgrounds caused by Ψ(z) = −12 cos (2z).Figures 3 and 4 show the two-soliton interaction on the x–t plane. The periodic backgrounds in Figure 4b and c are caused by Φ(y) = 12sin(2y) and Ψ(z) = −12cos(2z) accordingly. We can find that the three-soliton pictures in Figures 5 and 6 have the same properties as those in Figures 1–4. Figure 7 shows a kink-shape soliton with the periodic properties.

Figure 1:

One soliton in Case I of (26) with the parameters k₁=–1.8, m₁=1, l₁=1.2, Φ(y)=Ψ(z)=1, δ₁=0; and (a) of z=y=1, (b) of z=t=1, (c) of t=y=1.

Figure 2:

One soliton in Case II of (26) with the parameters l₁=–m₁=–1, k₁=–1.8, δ1 = 5, Φ(y) = sin(2y), Ψ(z) = −12 cos(2z); and (a) of y=z=1, (b) of z=t=1, (c) of y=t=1.

Figure 3:

Two solitons in Case I of (30) with the parameters Φ(y)=Ψ(z)=1, k₁=1.5, k₂=–1.1, m₁=1.8, m₂=1, l₁=1.4, l₂=1.2, δ₁=0, δ₂=–2.5; and (a) of y=z=1, (b) of z=t=1, (c) of y=t=1.

Figure 4:

Two solitons in Case II of (30) with the parameters Φ(y) = 12sin(2y),Ψ(z) = −12cos(2z),l₁=–m₁=–1, l₂=–m₂=–1.7, k₁=1.2, k₂=–1.8, δ₁=0, δ₂=2, and (a) of y=z=1, (b) of z=t=1, (c) of y=t=1.

Figure 5:

Three solitons in Case I of (31) and (17) with the parameters Φ(y)=Ψ(z)=1, l₁=1.2, l₂=1.6, l₃=1.7, m₁=1.3, m₂=1.4, m₃=1.15, k₁=1.07, k₂=1.9, k₃=1.7, δ₁=1, δ₂=1.6, δ₃=1.5; and (a) of y=z=1, (b) of z=t=1, (c) of y=t=1.

Figure 6:

Three solitons in Case II of (31) and (17) with the parameters Φ(y) = 12sin(2y), Ψ(z) = −12cos(2z),l₁=–m₁=–1, l₂=–m₂=–2.2, l₃=–m₃=–3.4, k₁=–1.9, k₂=1.2, k₃=1.7, δ₁=0, δ₂=3, δ₃=5; and (a) of y=z=1. (b) of z=t=1, (c) of y=t=1.

Figure 7:

Analytic solutions as given by (37) and (36) with the parameters l˜1 = −m˜1 = −1, l˜2 = −m˜2 = −1.2,Q0 = Q1 = Q2 = 1, k˜1 = −1,k˜2 = 1.5, c1 = c2 = 1, and z=y=1; where (a) is the surface of u and (b) is the wave profile of (a) with t=0.5, 0, –0.5.

6 Conclusions

Equation (1), the BLMP equation, can be considered as a model for the incompressible fluid and used for the (2+1)-dimensional interaction of the Riemann wave with a long wave, whereas people know that the higher-dimensional NLEEs are scientifically interesting. In this article, (2), a (3+1)-dimensional BLMP equation, has been investigated, with our results as follows:

1. Using the Bell polynomials, we have obtained Hirota Bilinear Form (16) of (2). By virtue of (16), two kinds of the multi-soliton solutions [i.e., (31) and (17)] and another kind of the analytic solutions [i.e., (37)] have been obtained, both with different nonlinear dispersion relations, as illustrated in Figures 1–7. For investigating the integrability of (2), we have obtained Lax Pairs (38) and BTs (44) for (2), with our integrability results as the multi-soliton solutions.

2. From Figures 1–6, it can be found that the functions Φ(y) and Ψ(z) have an effect on the solutions on both the x-y and x-z planes. The explanation is as follows:

   • As the one-soliton pictures, Figures 1a and 2a show the kink-shape solitons on the x-t plane; Figure 1b has one kink-shape soliton, whereas Figure 2b has both one kink-shape soliton and the periodic backgrounds caused by Φ(y)=sin(2y); Figure 1c has one kink-shape soliton, whereas Figure 2c has both one kink-shape soliton and the periodic backgrounds caused by Ψ(z) = −12cos(2z).

   • As the two-soliton pictures, Figures 3a and 4a show the interaction of two kink-shape solitons on the x-t plane; Figure 3b has two kink-shape solitons, whereas Figure 4b has both two kink-shape solitons and the periodic backgrounds caused by Φ(y) = 12sin(2y);Figure 3c has two kink-shape solitons, whereas Figure 4c has both two kink-shape solitons and the periodic backgrounds caused by Ψ(z) = −12cos(2z).

   • As the three-soliton pictures, Figures 5a and 6a show the interaction of the three kink-shape solitons on the x–t plane; Figure 5b has three kink-shape solitons, whereas Figure 6b has both three kink-shape solitons and the periodic backgrounds caused by Φ(y) = 12sin(2y);Figure 5c has three kink-shape solitons, whereas Figure 6c has both three kink-shape solitons and the periodic backgrounds caused by Ψ(z) = −12cos(2z).

   • We can see that the aforementioned phenomena in Figures 1–6 are caused by two kinds of nonlinear dispersion relations, i.e., (24) and (25) for the one soliton, (28) and (29) for the two solitons, and (32) and (33) for the three solitons.

3. By virtue of (16) and (34), another kind of analytic solutions [i.e., (37)] with two kinds of nonlinear dispersion relations [i.e., (35) and (36)] have been obtained, and one of those has been illustrated in Figure 7. We see the kink-shape solutions with the periodic property from Figure 7.