Gramian solutions and soliton interactions for a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in a plasma or fluid

Abstract

Plasmas and fluids are of current interest, supporting a variety of wave phenomena. Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation for the nonlinear phenomena in a plasma or fluid. Based on the existing bilinear form, N-soliton solutions in the Gramian are derived, where N = 1, 2, 3…. With N = 3, three-soliton solutions are constructed. Fission and fusion for the three solitons are presented. Effects of the variable coefficients, i.e. h(t), l(t), q(t), n(t) and m(t), on the soliton fission and fusion are revealed: soliton velocity is related to h(t), l(t), q(t), n(t) and m(t), while the soliton amplitude cannot be affected by them, where t is the scaled temporal coordinate, h(t), l(t) and q(t) give the perturbed effects, and m(t) and n(t), respectively, stand for the disturbed wave velocities along two transverse spatial coordinates. We show the three parallel solitons with the same direction.

1. Introduction

Plasmas and fluids are of current interest, supporting a variety of wave phenomena [1,2]. Plasmas are believed to be possibly the most abundant form of visible matter in the Universe [2]. Nonlinear evolution equations (NLEEs) have been used to describe the nonlinear phenomena in such fields as fluid mechanics, plasma physics, nonlinear optics and solid-state physics [3–14]. For certain integrable NLEEs, soliton interaction has been regarded to be completely elastic for the solitons' amplitudes and wave shapes do not change after the interaction [15–17]. However, for some other integrable NLEEs, soliton interaction has been claimed not to be completely elastic since the solitons' amplitudes or shapes are changed after the interaction [18]: for instance, one soliton may split into two or more solitons at a special time, while two or more solitons may merge into one soliton [15]. Those phenomena have often been called the soliton fission and fusion, respectively [15].

Among the NLEEs, Kadomtsev–Petviashvili (KP)-category equations have been used to model the nonlinear phenomena in such fields as fluid mechanics and plasma physics [19–24], while the variable-coefficient NLEEs have been seen to model more complex physical phenomena than their constant-coefficient counterparts do [8,25,26]. Researchers [27–31] have worked on the following generalized (3 + 1)-dimensional KP equation with variable coefficients in a plasma or fluid:

$${[u_t+f(t)u{u}_x+g(t)u_{xxx}+h(t)u_x+q(t)u_y+l(t)u_z]}_x+m(t)u_{yy}+n(t)u_{zz}=0,$$ 1.1

where u(x, y, z, t) represents the amplitude of the shallow-water wave and/or surface wave in a fluid or electrostatic wave potential in a plasma [27,28], x, y and z are the scaled spatial coordinates, t is the scaled temporal coordinate, the subscripts denote the partial derivatives, f(t) and g(t), the non-zero differentiable functions, represent the nonlinearity and dispersion, respectively, the differentiable functions h(t), q(t) and l(t) give the perturbed effects, while the differentiable functions m(t) and n(t) stand for the disturbed wave velocities along the transverse spatial coordinates y and z, respectively [29]. One- and two-soliton solutions for equation (1.1) have been obtained [27]. Painlevé analysis and Bäcklund transformation for equation (1.1) have been given [28]. Breathers and rouge-wave solutions for equation (1.1) via the auto-Bäcklund transformation have been discussed [29]. Interaction between the two solitons for equation (1.1) has been analysed [30]. Lump solutions for equation (1.1) have been derived [31]. With the transformation

$$u=2{(\ln F)}_{xx},$$ 1.2

equation (1.1) has been shown to possess its bilinear form [28,29]:

$$[D_{xt}+g(t)D_x^4+h(t)D_x^2+q(t)D_{xy}+l(t)D_{xz}+m(t)D_y^2+n(t)D_z^2]F\cdot F=0,$$ 1.3

under the constraint

$$f(t)=6g(t),$$ 1.4

where F = F(x, y, z, t) is the real differentiable function, and D_x, D_y, D_z and D_t are the bilinear operators.¹

Special cases of equation (1.1) have been seen in physical sciences as follows:

• With h(t) = q(t) = l(t) = m(t) = n(t) = 0, equation (1.1) has been claimed to reduce to the (1 + 1)-dimensional variable-coefficient Korteweg–de Vries equation [34];

• With h(t) = q(t) = l(t) = n(t) = 0, equation (1.1) has been seen to reduce to the (2 + 1)-dimensional variable-coefficient generalized KP equation [35];

• With q(t) = l(t) = n(t) = 0, equation (1.1) has been seen to reduce to the (2 + 1)-dimensional variable-coefficient equation which can describe the internal waves propagating along the interface of two fluid layers [36].

To our knowledge, Gramian solutions and interaction among the three solitons for equation (1.1) have not been proposed. In §2, with symbolic computation [32,33,37–47], we give the N-soliton solutions in the Gramian for equation (1.1), where N = 1, 2, 3…. In §3, interactions among the three solitons and effects of the variable coefficients in equation (1.1), i.e. h(t), l(t), q(t), n(t) and m(t), on the soliton fission and fusion are discussed. Section 4 will be our conclusions.

2. N-soliton solutions in the Gramian for equation (eqn1.1)

Through bilinear form (1.3), we will derive the N-soliton solutions in the Gramian for equation (1.1).

Firstly, we construct the Gramian solutions for bilinear form (1.3). Hereby, we introduce the following Gramian:

$$F=\det {(a_{ij})}_{1\le i<j\le N},a_{ij}=c_{ij}+\int {\rho }_i{\varrho }_j\hspace{0.17em}\mathrm{d}x,$$ 2.1

where i and j are the positive integers, c_ij is a constant, and ρ_is and ϱ_js are the real functions of x, y, z and t which satisfy the following differential equations:

$$\begin{array}{ll}\hspace{0.17em}& {\rho }_{i,y}=\alpha {\rho }_{i,xx},{\varrho }_{j,y}=-\alpha {\rho }_{j,xx},\hfill \\ \hspace{0.17em}& {\rho }_{i,z}=\beta {\rho }_{i,xx},{\varrho }_{j,y}=-\beta {\rho }_{j,xx},\hfill \\ \hspace{0.17em}& {\rho }_{i,t}={\gamma }_1(t){\rho }_{i,x}+{\gamma }_2(t){\rho }_{i,xx}+{\gamma }_3(t){\rho }_{i,xxx}\hfill \\ \mathrm{and}\hfill & {\varrho }_{i,t}={\gamma }_1(t){\varrho }_{j,x}-{\gamma }_2(t){\varrho }_{j,xx}+{\gamma }_3(t){\varrho }_{j,xxx},\hfill \end{array}$$

where α and β are the real constants, and γ₁(t), γ₂(t) and γ₃(t) are the functions to be determined.

Next, we prove that expression (2.1) satisfies bilinear form (1.3).

We rewrite the Gramian F, i.e. expression (2.1), as a Pfaffian²:

$$F=(1,2,\dots ,N,N^{\diamond },\dots ,2^{\diamond },1^{\diamond }),$$ 2.2

where the elements of the Pfaffian are defined as (i, j^⋄) = a_ij, (i, j) = (i^⋄, j^⋄) = 0, and the superscript ⋄ as a mark to distinguish between the real constants k and k^⋄.

To compute the derivatives of a_ij and F, we introduce the Pfaffians defined as [32,33]

$$(d_n,j^{\diamond })=\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{x}^n}{\varrho }_j,(d_n^{\diamond },i)=\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{x}^n}{\varrho }_i,(d_m,d_n^{\diamond })=(d_n^{\diamond },j^{\diamond })=(d_n,i)=0,$$ 2.3

where m and n are both the positive integers.

In terms of the expressions (2.3), the derivatives of a_ij are given by

$$\begin{array}{ccc}\hfill a_{ij,x}& =\hfill & {\rho }_i{\varrho }_j=(d_0,d_0^{\diamond },i,j^{\diamond }),\hfill \\ \hfill a_{ij,y}& =\hfill & {\int }^x({\rho }_y\varrho +\rho {\varrho }_y)\hspace{0.17em}\mathrm{d}x=\alpha {\int }^x({\rho }_{ixx}\varrho -\rho {\varrho }_{xx})\hspace{0.17em}\mathrm{d}x=\alpha [(d_0,d_1^{\diamond },i,j^{\diamond })-(d_1,d_0^{\diamond },i,j)],\hfill \\ \hfill a_{ij,z}& =\hfill & {\int }^x({\rho }_z\varrho +\rho {\varrho }_z)\hspace{0.17em}\mathrm{d}x=\beta {\int }^x({\rho }_{ixx}\varrho -\rho {\varrho }_{xx})\hspace{0.17em}\mathrm{d}x=\beta [(d_0,d_1^{\diamond },i,j^{\diamond })-(d_1,d_0^{\diamond },i,j)]\hfill \\ \mathrm{and}{a}_{ij,t}\hfill & =\hfill & {\int }^x({\rho }_t\varrho +\rho {\varrho }_t)\hspace{0.17em}\mathrm{d}x\hfill \\ \hspace{0.17em}& =\hfill & {\int }^x[{\gamma }_1(t)({\rho }_{ix}\varrho +\rho {\varrho }_x)+{\gamma }_2(t)({\rho }_{ixx}\varrho -\rho {\varrho }_{xx})\hspace{0.17em}\mathrm{d}x+{\gamma }_3(t)({\rho }_{ixxx}\varrho +\rho {\varrho }_{jxxx})]\hspace{0.17em}\mathrm{d}x\hfill \\ \hspace{0.17em}& =\hfill & {\gamma }_1(t)(d_0,d_0^{\diamond },i,j^{\diamond })+{\gamma }_2(t)[(d_0,d_1^{\diamond },i,j^{\diamond })-(d_1,d_0^{\diamond },i,j)]+{\gamma }_3(t)[(d_2,d_0^{\diamond },i,j^{\diamond })\hfill \\ \hspace{0.17em}& \hspace{0.17em}& -(d_1,d_1^{\diamond },i,j^{\diamond })+(d_0,d_2^{\diamond },i,j)].\hfill \end{array}\}$$ 2.4

Taking advantage of the expressions (2.4) and denoting (1, 2, …, N, N^⋄, …, 2^⋄, 1^⋄) = (•), we can calculate the derivatives for F,

$$\begin{array}{cc}\hspace{0.17em}\hfill & F_x=(d_0,d_0^{\diamond },\bullet ),F_{xx}=(d_1,d_0^{\diamond },\bullet )+(d_0,d_1^{\diamond },\bullet ),\hfill \\ \hspace{0.17em}\hfill & F_{xxx}=(d_2,d_0^{\diamond },\bullet )+2(d_1,d_1^{\diamond },\bullet )+(d_0,d_2^{\diamond },\bullet ),\hfill \\ \hspace{0.17em}\hfill & F_{xxxx}=(d_3,d_0^{\diamond },\bullet )+3(d_2,d_1^{\diamond },\bullet )+2(d_0,d_0^{\diamond },d_1,d_1^{\diamond },\bullet )+3(d_1,d_2^{\diamond },\bullet )+(d_0,d_3^{\diamond },\bullet ),\hfill \\ \hspace{0.17em}\hfill & F_y=[(d_0,d_1^{\diamond },\bullet )-(d_1,d_0^{\diamond },\bullet )],\hfill \\ \hspace{0.17em}\hfill & F_{yx}=\alpha [(d_1,d_1^{\diamond },\bullet )+(d_0,d_2^{\diamond },\bullet )-(d_2,d_0^{\diamond },\bullet )-(d_1,d_1^{\diamond },\bullet )],\hfill \\ \hspace{0.17em}\hfill & F_{yy}={\alpha }^2[(d_3,d_0^{\diamond },\bullet )-(d_2,d_1^{\diamond },\bullet )-(d_1,d_2^{\diamond },\bullet )+(d_0,d_3^{\diamond },\bullet )+2(d_0,d_0^{\diamond },d_1,d_1^{\diamond },\bullet )],\hfill \\ \hspace{0.17em}\hfill & F_z=\beta [(d_0,d_1^{\diamond },\bullet )-(d_1,d_0^{\diamond },\bullet )],\hfill \\ \hspace{0.17em}\hfill & F_{zx}=\beta [(d_1,d_1^{\diamond },\bullet )+(d_0,d_2^{\diamond },\bullet )-(d_2,d_0^{\diamond },\bullet )-(d_1,d_1^{\diamond },\bullet )],\hfill \\ \hspace{0.17em}\hfill & F_{zz}={\beta }^2[(d_3,d_0^{\diamond },\bullet )-(d_2,d_1^{\diamond },\bullet )-(d_1,d_2^{\diamond },\bullet )+(d_0,d_3^{\diamond },\bullet )+2(d_0,d_0^{\diamond },d_1,d_1^{\diamond },\bullet )],\hfill \\ \hspace{0.17em}\hfill & F_t={\gamma }_1(t)(d_0,d_0^{\diamond },\bullet )+{\gamma }_2(t)[(d_0,d_1^{\diamond },\bullet )-(d_1,d_0^{\diamond },\bullet )]\hfill \\ \hspace{0.17em}\hfill & +{\gamma }_3(t)[(d_2,d_0^{\diamond },\bullet )-(d_1,d_1^{\diamond },\bullet )+(d_0,d_2^{\diamond },\bullet )]\hfill \\ \mathrm{and}\hfill & F_{xt}={\gamma }_1(t)[(d_1,d_0^{\diamond },\bullet )+(d_0,d_1^{\diamond },\bullet )]+{\gamma }_2(t)[(d_0,d_2^{\diamond },\bullet )-(d_2,d_0^{\diamond },\bullet )]\hfill \\ \hspace{0.17em}\hfill & +{\gamma }_3(t)[(d_3,d_0^{\diamond },\bullet )+(d_0,d_3^{\diamond },\bullet )-(d_0,d_0^{\diamond },d_1,d_1^{\diamond },\bullet )].\hfill \end{array}\}$$ 2.5

Under the constraints

$$\begin{array}{rlr}\hspace{0.17em}& {\gamma }_1(t)=-h(t),\hfill & {\gamma }_2(t)=-q(t)\alpha -l(t)\beta \hfill \\ \mathrm{and}\hfill & {\gamma }_3(t)=-4g(t),\hfill & g(t)=\frac{1}{3}[m(t){\alpha }^2+n(t){\beta }^2],\hfill \end{array}\}$$ 2.6

substituting the expressions (2.5) into bilinear form (1.3) yields

$$\begin{array}{rl}\hspace{0.17em}& \{D_{xt}+g(t)D_x^4+h(t)D_x^2+q(t)D_{xy}+l(t)D_{xz}+m(t)D_y^2+n(t)D_z^2\}F\cdot F\hfill \\ \hspace{0.17em}& =24g(t)[(d_0,d_0^{\diamond },d_1,d_1^{\diamond },\bullet )+(d_0,d_1^{\diamond },\bullet )(d_1,d_0^{\diamond },\bullet )-(d_0,d_0^{\diamond },\bullet )(d_1,d_1^{\diamond },\bullet )]\hfill \\ \hspace{0.17em}& =0.\hfill \end{array}$$ 2.7

Therefore, we have verified that expression (2.1) satisfies bilinear form (1.3).

Finally, based on expressions (1.2) and (2.1), we give the N-soliton solutions in the Gramian for equation (1.1) as

$$u=2{\left[\ln \det (c_{ij}+\frac{1}{{\eta }_i+{\mu }_j}{e}^{{\xi }_i+{\varsigma }_j})\right]}_{xx},$$ 2.8

where ${\xi }_i={\eta }_{i}x+\alpha {\eta }_i^{2}y+\beta {\eta }_i^{2}z-\frac{1}{3}{\eta }_i\int [3h(t)+3{\eta }_i\beta l(t)+4{\eta }_i^2{\alpha }^{2}m(t)+4{\eta }_i^2{\beta }^{2}n(t)+3{\eta }_i\alpha q(t)]\hspace{0.17em}\mathrm{d}t$, ${\varsigma }_j={\mu }_{j}x-\alpha {\mu }_i^{2}y-\beta {\mu }_i^{2}z-\frac{1}{3}{\mu }_j\int [3h(t)-3{\mu }_j\beta l(t)+4{\mu }_j^2{\alpha }^{2}m(t)+4{\mu }_j^2{\beta }^{2}n(t)-3{\mu }_j\alpha q(t)]\hspace{0.17em}\mathrm{d}t$, and η_is and μ_js are the real constants.

3. Discussion

With N = 1 and 2 in solutions (2.8), one- and two-soliton solutions can be obtained, which are the same as those presented in [27–29]. Knowing that, we will go further. With N = 3 in solutions (2.8), three-soliton solutions for equation (1.1) can be expressed as

$$u=2{\left[\ln \left(\left|\begin{array}{ccc}{c}_{11}+\frac{1}{{\eta }_1+{\mu }_1}\hspace{0.17em}{\mathrm{e}}^{{\xi }_1+{\varsigma }_1}\hfill & c_{12}+\frac{1}{{\eta }_1+{\mu }_2}\hspace{0.17em}{\mathrm{e}}^{{\xi }_1+{\varsigma }_2}\hfill & c_{13}+\frac{1}{{\eta }_1+{\mu }_3}\hspace{0.17em}{\mathrm{e}}^{{\xi }_1+{\varsigma }_3}\hfill \\ c_{21}+\frac{1}{{\eta }_2+{\mu }_1}\hspace{0.17em}{\mathrm{e}}^{{\xi }_2+{\varsigma }_1}\hfill & c_{22}+\frac{1}{{\eta }_2+{\mu }_2}\hspace{0.17em}{\mathrm{e}}^{{\xi }_2+{\varsigma }_2}\hfill & c_{23}+\frac{1}{{\eta }_2+{\mu }_3}\hspace{0.17em}{\mathrm{e}}^{{\xi }_2+{\varsigma }_3}\hfill \\ c_{31}+\frac{1}{{\eta }_3+{\mu }_1}\hspace{0.17em}{\mathrm{e}}^{{\xi }_3+{\varsigma }_1}\hfill & c_{32}+\frac{1}{{\eta }_3+{\mu }_2}\hspace{0.17em}{\mathrm{e}}^{{\xi }_3+{\varsigma }_2}\hfill & c_{33}+\frac{1}{{\eta }_3+{\mu }_3}\hspace{0.17em}{\mathrm{e}}^{{\xi }_3+{\varsigma }_3}\hfill \end{array}\right|\right)\right]}_{xx}.$$ 3.1

Fission for the three solitons via solutions (3.1) is presented in figure 1, where one large-amplitude soliton splits into three small-amplitude ones. On the y–z plane, we observe that one large-amplitude soliton splits into three parallel small-amplitude ones, as shown in figure 1c.

Figure 1.

Fission for the three solitons via solutions (3.1) with η₁ = 1, η₂ = 0.7, η₃ = 0.5, α = 1.2, c₁₁ = 1, c₁₂ = 0, c₁₃ = 0, c₂₁ = 0, c₂₂ = 0.6, c₂₃ = 0, c₃₁ = 0, c₃₂ = 0, c₃₃ = 0.8, β = 0.6, μ₁ = − 0.3, μ₂ = − 0.3, μ₃ = − 0.3, h(t) = − 1, n(t) = 1.1, l(t) = − 0.8, q(t) = − 1, m(t) = 0.8. (a) z = 0; (b) y = 0; (c) x = 0. (Online version in colour.)

Next, on the soliton fission, on the x–t plane as an example³, we will study the effects of the variable coefficients in equation (1.1), i.e. h(t), l(t), q(t), n(t) and m(t), as shown in figure 2. If we take h(t), l(t), q(t), n(t) and m(t) as the constants, the soliton fission occurs for the velocity- and amplitude-unvarying solitons, as shown in figure 2a. However, if we, respectively, take those variable coefficients as the t-dependent functions, we observe that the one velocity-varying but amplitude-unvarying soliton splits into three velocity-varying but amplitude-unvarying solitons, as shown in figure 2b–f . Therefore, we can conclude that the soliton velocity is related to h(t), l(t), q(t), n(t) and m(t), while the soliton amplitude cannot be affected by them.

Figure 2.

Effects of h(t), l(t), q(t), n(t) and m(t) on the fission for the three solitons via solutions (3.1) on the x–t plane with (a) η₁ = 1, η₂ = 0.7, η₃ = 0.5, α = 1.2, c₁₁ = 1, c₁₂ = 0, c₁₃ = 0, c₂₁ = 0, c₂₂ = 0.6, c₂₃ = 0, c₃₁ = 0, c₃₂ = 0, c₃₃ = 0.8, β = 0.6, μ₁ = − 0.3, μ₂ = − 0.3, μ₃ = − 0.3, h(t) = − 1, n(t) = 1.1, l(t) = − 0.8, q(t) = − 1, m(t) = 0.8, y = z = 0. (b–f ): The same as (a) except that (b) h(t) = − 0.2t; (c) n(t) = sin(t); (d) l(t) = cos(t); (e) q(t) = 3tanh(t); (f ) m(t) = 0.2t. (Online version in colour.)

Fusion for the three solitons via solutions (3.1) is presented in figure 3, where three small-amplitude solitons merge into one large-amplitude soliton. On the y–z plane, we observe that three parallel small-amplitude solitons merge into one large-amplitude soliton, as shown in figure 3c.

Figure 3.

Fusion for the three solitons via solutions (3.1) with η₁ = 1, η₂ = 0.7, η₃ = 0.5, α = 1.2, c₁₁ = 1, c₁₂ = 0, c₁₃ = 0, c₂₁ = 0, c₂₂ = 0.6, c₂₃ = 0, c₃₁ = 0, c₃₂ = 0, c₃₃ = 0.8, β = 0.6, μ₁ = − 0.3, μ₂ = − 0.3, μ₃ = − 0.3, h(t) = 0.5, n(t) = 1, l(t) = 2, q(t) = − 1, m(t) = − 0.5. (a) z = 0; (b) y = 0; (c) x = 0. (Online version in colour.)

Next, we will study the effects of the variable coefficients in equation (1.1), i.e. h(t), l(t), q(t), n(t) and m(t), on the soliton fusion. If we take h(t), l(t), q(t), n(t) and m(t) as the constants, we display the velocity- and amplitude-unvarying soliton fusion, as shown in figure 4a. However, if we, respectively, take those variable coefficients as the t-dependent functions, we present the velocity-varying but amplitude-unvarying soliton fusion, as shown in figure 4b–f . Therefore, we can draw the same conclusion as for the soliton fission.

Figure 4.

Effects of h(t), l(t), q(t), n(t) and m(t) on the fusion for the three solitons via solutions (3.1) on the x–t plane with (a) η₁ = 1, η₂ = 0.7, η₃ = 0.5, α = 1.2, c₁₁ = 1, c₁₂ = 0, c₁₃ = 0, c₂₁ = 0, c₂₂ = 0.6, c₂₃ = 0, c₃₁ = 0, c₃₂ = 0, c₃₃ = 0.8, β = 0.6, μ₁ = − 0.3, μ₂ = − 0.3, μ₃ = − 0.3, h(t) = 0.5, n(t) = 1, l(t) = 2, q(t) = − 1, m(t) = − 0.5, y = z = 0. (b–f ) The same as (a) except that (b) h(t) = − 0.2t; (c) n(t) = sin(t); (d) l(t) = 0.2t; (e) q(t) =  − 0.1t²; (f ) m(t) = 2tanh(t). (Online version in colour.)

Figure 5 shows the three parallel solitons via solutions (3.1) with the same direction. Since the soliton amplitude, velocity and wave shape do not change after the interaction, the interaction among the three parallel solitons is elastic.

Figure 5.

Three parallel solitons via solutions (3.1) with η₁ = 0.8, η₂ = 0.7, η₃ = 0.6, α = 1.2, c₁₁ = 1, c₁₂ = 0, c₁₃ = 0, c₂₁ = 0, c₂₂ = 0.6, c₂₃ = 0, c₃₁ = 0, c₃₂ = 0, c₃₃ = 0.8, β = 0.6, μ₁ = 0.3, μ₂ = 0.2, μ₃ = 0.1, h(t) = − 1, n(t) = 1.1, l(t) = − 0.8, q(t) = − 1, m(t) = 0.8. (a) z = 0; (b) y = 0; (c) x = 0. (Online version in colour.)

4. Conclusion

Plasmas and fluids are of current interest, supporting a variety of wave phenomena. Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Some nonlinear phenomena in a plasma or fluid have been modelled by a generalized (3 + 1)-dimensional variable-coefficient KP equation, i.e. equation (1.1). Via bilinear form (1.3), the N-soliton solutions in the Gramian for equation (1.1) have been constructed, i.e. solutions (2.8). With N = 3, three-soliton solutions (3.1) have been presented. Fission and fusion for the three solitons have been presented in figures 1 and 3, respectively. As shown in figures 2 and 4, effects of the variable coefficients in equation (1.1), i.e. h(t), l(t), q(t), n(t) and m(t), on the soliton fission and fusion have been revealed, respectively: soliton velocity is related to h(t), l(t), q(t), n(t) and m(t), but the soliton amplitude cannot be affected by them, where t is the scaled temporal coordinate, h(t), l(t) and q(t) give the perturbed effects, and m(t) and n(t), respectively, stand for the disturbed wave velocities along two transverse spatial coordinates. We have shown the three parallel solitons with the same direction in figure 5.

Data accessibility

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Competing interests

We declare we have no competing interests.

Funding

This work has been supported by the National Natural Science Foundation of China under grant nos. 11772017, 11272023, 11805020 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05) and by the Fundamental Research Funds for the Central Universities of China under grant no. 2011BUPTYB02.