Multi-soliton Collisions and Bäcklund Transformations for the (2+1)-dimensional Modified Nizhnik–Novikov–Vesselov Equations

1 Introduction

Nonlinear evolution equations (NLEEs) have been used to describe some nonlinear wave phenomena in such fields as hydrodynamics, solid state physics and plasma physics [1–5]. Analytic techniques for studying the NLEEs have been presented, e.g. the Bell polynomials [6], inverse scattering method [7], Hirota method [8, 9], Bäcklund transformation (BT) [5, 10], Darboux transformation [11–16], Painlevé analysis [17], Wronskian technique [18, 19] and algebra-geometric method [20].

For some integrable NLEEs, solitary-wave collisions are completely elastic, which means that a solitary wave keeps its amplitude, velocity and wave shape unchanged after the collision [21]. However, there exists the concern about the inelastic collisions for certain integrable NLEEs, such as the Burgers equation in fluid dynamics [22, 23]. Fusion and fission exist in the inelastic collisions [23–25], which means that several solitary waves can fuse to one [23, 24], whereas conversely, one solitary wave can split into several solitary waves [23].

The Korteweg–de Vries (KdV) equation,

(1)Φτ + 6ΦΦξ + Φξξξ = 0, (1)

where Φ is the function of the variables ξ and τ, can describe the shallow water waves, stratified internal waves, ion-acoustic waves, plasma physics, and lattice dynamics [21, 26, 27], and the (2+1)-dimensional Nizhnik–Novikov–Vesselov equations, which are the isotropic Lax-integrable KdV extension, appear as [28–31]

(2a)ut + uxxx + uyyy + 3(uv)x + 3(uw)y = 0, (2a)

(2b)ux = vy,  uy = wx, (2b)

where u, v and w are the functions of two scaled space coordinates x, y, and time t [28]. Equations (2) have been claimed to depend strongly on the spatial coordinates x and y in a symmetric manner [28] and investigated via the binary Bell polynomials with an auxiliary variable [29]. The inverse problem of (2) has been formulated via the inverse scattering [30]. Bäcklund transformation and Wronskian solitons of (2) have been derived [31].

In this paper, we investigate the (2+1)-dimensional modified Nizhnik–Novikov–Vesselov equations [32–41],

(3a)ut + uxxx + uyyy + 3uxvxx + 3uyvyy − ux3 − uy3 = 0, (3a)

(3b)vxy = uxuy, (3b)

which are a member of the modified Nizhnik–Novikov–Vesselov hierarchy associated with the generalised Lamé system [32], and pass the three-soliton test and bilinear Painlevé test [33]. Soliton solutions for (3) have been obtained with the variable separation approach and bilinear method [34]. Quasi-periodic wave solutions of (3) have been constructed via the Hirota bilinear method and Riemann theta function [35]. The 1-dromion solution of (3) has been derived by means of the Hirota bilinear method [36]. Variable separation solutions of (3) with certain arbitrary functions have been obtained [37]. Binary Darboux transformation of (3) has been constructed [38]. For (3), collisions of the localised coherent solitons have been illustrated analytically and graphically [39], and geometrical significance has been addressed [40]. Multi-valued localised excitations of (3) have been derived [41].

However, inelastic collisions among the solitons of (3) have not been investigated. Through the binary Bell polynomials, bilinear forms, and soliton solutions for (3) will be obtained in Section 2, which are different from those in [34]. In Section 3, the BTs in the binary Bell polynomial form will be presented. In Section 4, propagation and collisions of the solitons will be investigated analytically and graphically. Finally, conclusions will be listed in Section 5.

2 Bilinear Forms and Soliton Solutions for (3)

In this section, according to Appendix A, we firstly transform (3) into the binary Bell polynomial form. Setting

(4)u = αp, v = βq, (4)

with p and q as the functions of x, y and t, while α and β as the complex constants, and substituting (4) into (3), we have

(5a)αpt + αp3x + αp3y + 3αβpxqxx + 3αβpyqyy − α3px3 − α3px3 = 0, (5a)

(5b)βqxt − α2pxpy = 0. (5b)

Through (30) in Appendix A, (3) can be transformed into the following form:

(6a)𝒴t(p, q) + 𝒴3x(p, q) + 𝒴3y(p, q) = 0, (6a)

(6b)𝒴xy(p, q) = 0, (6b)

where α=i and β=1.

If we set

(7)p = lnGF,  q = ln(FG), (7)

where F and G are both the functions of x, y and t, the bilinear forms for (3) can be obtained as

(8a)(Dt + Dx3 + Dy3)G ⋅ F = 0, (8a)

(8b)DxDyG ⋅ F = 0. (8b)

Based on (8), we construct the soliton solutions for (3). Expanding G and F into a power series of a small parameter ε as follows [42]:

(9a)F(x, y, t) = 1 + εF1(x, y, t) + ε2F2(x, y, t) + ε3F3(x, y, t)+⋯, (9a)

(9b)G(x, y, t) = 1 + εG1(x, y, t) + ε2G2(x, y, t) + ε3G3(x, y, t) + ⋯, (9b)

where F_l’s and G_l’s (l=1, 2, …) are the functions of x, y, and t.

Substituting (9) into (3), and collecting the coefficients of the same power of ε, we have

(10a)ε0:(Dt + Dx3 + Dy3)1 ⋅ 1 = 0, (10a)

(10b)DxDy1 ⋅ 1 = 0, (10b)

(10c)ε1:(Dt + Dx3 + Dy3)(1 ⋅ F1 + G1 ⋅ 1) = 0, (10c)

(10d)DxDy(1 ⋅ F1 + G1 ⋅ 1) = 0, (10d)

(10e)ε2:(Dt + Dx3 + Dy3)(1 ⋅ F2 + G2 ⋅ 1 + G2 ⋅ F2) = 0, (10e)

(10f)DxDy(1 ⋅ F2 + G2 ⋅ 1 + G2 ⋅ F2) = 0⋮ (10f)

Multi-soliton solutions for (3) can be constructed under certain assumptions. To obtain the one-soliton solutions for (3), we truncate (9) to F₁(x, y, t) and G₁(x, y, z), respectively. Setting

(11)F1(x, y, t) = eθ, G1(x, y, t) = − eθ′, (11)

where θ = πi2 + ax + by + ct + η and θ′ = πi2 + ax + by + ct + δ, with δ, η, a, b and c as all the constants, and substituting (11) into (10), we have

(12)δ = η,  c = − a3 − b3. (12)

Without loss of generality, when we set ε=1, the one-soliton solutions and monotonic function for (3) can be written as

(13a)u = 2 arctan [eax + by − (a3 + b3)t + η], (13a)

(13b)v = ln{[1 − ieax + by − (a3 + b3)t + η][1 + ieax + by − (a3 + b3)t + η]}. (13b)

Similarly, truncating (9) to F₂(x, y, t) and G₂(x, y, z), respectively, setting

(14a)G1(x, y, t) = − eθ′1 − eθ′2,  G2(x, y, t) = eθ′3, (14a)

(14b)F1(x, y, t) = eθ1 + eθ2, F2(x, y, t) = eθ3, (14b)

and substituting (14) into (10), we obtain the two-soliton solutions and monotonic function as

(15a)u = 2 arctan[ea1x + b1y − (a13 + b13)t + η1 + ea2x + b2y − (a23 + b23)t + η21 − ea1x + b1y − (a13 + b13)t + η1 + a2x + b2y − (a23 + b23)t + η2 + η3], (15a)

(15b)v = ln{[1 + iea1x + b1y − (a13 + b13)t + η1 + iea2x + b2y − (a23 + b23)t + η2 − ea1x + b1y − (a13 + b13)t + η1 + a2x + b2y − (a23 + b23)t + η2 + η3][1 − iea1x + b1y − (a13 + b13)t + η1 − iea2x + b2y − (a23 + b23)t + η2− ea1x + b1y − (a13 + b13)t + η1 + a2x + b2y − (a23 + b23)t + η2 + η3]}. (15b)

with

(16a)θ1 = πi2 + a1x + b1y − (a13 + b13)t + η1, θ2 = πi2 + a2x + b2y − (a23 + b23)t + η2, (16a)

(16b)θ3 = πi + a1x + b1y − (a13 + b13)t + η1 + a2x + b2y − (a23 + b23)t + η2 + η3, (16b)

(16c)θ′1 = πi2 + a1x + b1y − (a13 + b13)t + δ1, θ′2 = πi2 + a2x + b2y − (a23 + b23)t + δ2, (16c)

(16d)θ′3 = πi + a1x + b1y − (a13 + b13)t + δ1 + a2x + b2y − (a23 + b23)t + δ2 + δ3, (16d)

(16e)δ1 = η1, δ2 = η2, δ3 = η3, eη3 = a1b1 − a2b1 − a1b2 + a2b2(a1 + a2)(b1 + b2), (16e)

where η₁, η₂, η₃, a₁, a₂, b₁ and b₂ are all the constants.

3 Bilinear BTs

A BT can be used to construct new solutions from the known ones [10]. In this section, from (5), we derive the BTs in the binary Bell polynomial form and bilinear form.

Considering

(17a)P1 = [𝒴t(p′, q′) + 𝒴3x(p′, q′) + 𝒴3y(p′, q′)] − [𝒴t(p, q) + 𝒴3x(p, q) + 𝒴3y(p, q)], (17a)

(17b)P2 = [𝒴xy(p′, q′)] − [𝒴xy(p, q)], (17b)

where P₁ and P₂ are both the functions of p, q, p′ and q′, with p′ and q′ as the functions of x, y and t, and satisfy (5), p′ = lnGF and q′=ln(FG), we obtain the BTs as follows:

(18a)𝒴t(v1, w1) + 𝒴3x(v1, w1) + 3λμ𝒴x(v1, w1) + 𝒴3y(v1, w1) + 3sh𝒴t(v1, w1) = 0, (18a)

(18b)𝒴t(v2, w2) + 𝒴3x(v2, w2) + 3λμ𝒴x(v2, w2) + 𝒴3y(v2, w2) + 3sh𝒴t(v2, w2) = 0, (18b)

(18c)𝒴x(v3, w3) = λ exp(v1 − v2),  𝒴y(v3, w3) = μ exp(v2 − v1), (18c)

(18d)𝒴x(v4, w4) = s exp(v1 − v2),  𝒴y(v4, w4) = h exp(v2 − v1), (18d)

where

(19a)v1 = lnG′G,  v2 = lnF′F,  v3 = lnF′G,  v4 = lnG′F, (19a)

(19b)w1 = ln(G′G), w2 = ln(F′F), w3 = ln(F′G), w4 = ln(G′F), (19b)

F′ and G′ are both the functions of x, y and t, with λ, μ, s, h as the real constants.

Via (34), (18) can be transformed into the following form:

(20a)(Dt + Dx3 + 3λμDx + Dy3 + 3shDy)G′ ⋅ G = 0, (20a)

(20b)(Dt + Dx3 + 3λμDx + Dy3 + 3shDy)F′ ⋅ F = 0, (20b)

(20c)DxF′ ⋅ G = λFG′, DxG′ ⋅ F = μF′G, (20c)

(20d)DyF′ ⋅ G = sFG′, DyG′ ⋅ F = hF′G. (20d)

4 Discussion

In this section, based on (13) and (15), we discuss the propagation and collision of soliton solutions.

Figure 1 describes the shapes and motion of the one soliton u, with the monotonic function v also displayed. In Figure 1, we can observe that the one soliton maintains its shape during the propagation with its amplitude invariant.

Figure 1:

One soliton and the monotonic function via (13) with η=0, a=0.125, b=0.5: (a) t=−15; (b) t=0; (c) t=15.

As shown in Figure 2, the parameters a and b can respectively affect the propagation direction of the one soliton but have no influence on the amplitude, compared with Figure 1b. By the way, effects of a and b on the monotonic function are illustrated.

Figure 2:

The same as Figure 1b, except (a) a=−0.1; (b) b=0.35.

Figure 3 displays the collision between the two solitons, and we can see that the shapes of the two solitons have changed after the collision. Also, the monotonic function, v, is shown.

Figure 3:

Collision between the two solitons for u, and monotonic function v via (15) with η₁=0, η₂=0, a₁=0.1, a₂=0.2, b₁=0.4, b₂=−0.1: (a) t=−100; (b) t=0; (c) t=100.

Compared with Figure 3b, Figures 4 and 5 illustrate that the parameters a₁, a₂, b₁, and b₂ are able to respectively change the propagation directions of the two solitons, but cannot affect the amplitudes. Meanwhile, the influences of a₁, a₂, b₁, and b₂ on the monotonic function, v, are displayed.

Figure 4:

The same as Figure 3b except (a) a₁=−0.1; (b) a₂=0.35.

Figure 5:

The same as Figure 3b except (a) b₁=0.2; (b) b₂=0.1.

Next, our purpose will be to analyse the coalescence of the two solitons, which means that during the collision the two solitions can be fused [43]. Setting

(21)G(x, y, t) = 1 − eθ′1 − eθ′2, F(x, y, t) = 1 + eθ1 + eθ2, (21)

and substituting (21) into (10), we obtain the two solitons and monotonic function with

(22)u = 2 arctan[eπi2 + a1x + b1y − (a13 + b13)t + η1 + eπi2 + a2x + b2y − (a23 + b23)t + η2], (22)

(23)v = ln{[1 − eπi2 + a1x + b1y − (a13 + b13)t + η1 − eπi2 + a2x + b2y − (a23 + b23)t + η2]× [1 + eπi2 + a1x + b1y − (a13 + b13)t + η1 + eπi2 + a2x + b2y − (a23 + b23)t + η2]}. (23)

Without loss of generality, we assume that the wave parameters a₁ and a₂ satisfy the relation a₂>a₁>0 and η₁=η₂=0. Figure 6 illustrates the two-soliton collision, whereas the two solitons propagate the same propagation direction. Away from the collision region, the two solitons move forward and merge, with the limits of boundary values

Figure 6:

Collision between the two solitons for u, and monotonic function v via (22) with a1 = 150, a2 = 25, b1 = − 16, b2 = − 16: (a) t=−30; (b) t=0; (c) t=30.

(24a)u → 2 arctan [eπi2 + a1x + b1y − (a13 + b13)t], when t → − ∞, y → − ∞, x ≅ a13 + b13a1t − a1b1y, (24a)

(24b)u → 2 arctan [eπi2 + a2x + b2y − (a23 + b23)t], when t → +∞, y → +∞, x ≅ a23 + b23a2t − a2b2y. (24b)

Setting the solitons in (24) as S₁ and S₂, respectively, we can find that far from the collision region, the amplitudes of S₁ and S₂ can be expressed as a₁ and a₂, respectively.

Moreover,

(25a)v → ln{[1−eπi2 + a1x + b1y − (a13 + b13)t][1 + eπi2 + a1x + b1y − (a13 + b13)t]},when  t → − ∞, y → − ∞, x ≅ a13 + b13a1t − a1b1y, (25a)

(25b)v → ln{[1 − eπi2 + a2x + b2y − (a23 + b23)t][1 + eπi2 + a2x + b2y − (a23 + b23)t]},when  t → +∞, y → +∞, x ≅ a23 + b23a2t − a2b2y. (25b)

From Figure 6, we can observe the coalescence of the two solitons propagating with the same propagation direction, the shapes of which have changed after the collision.

5 Conclusions

The KdV-type equations can describe the shallow water waves, stratified internal waves, ion-acoustic waves, plasma physics, and lattice dynamics, whereas the (2+1)-dimensional Nizhnik–Novikov–Vesselov equations are the isotropic extensions of the KdV-type equations. In this paper, we investigated the (2+1)-dimensional modified Nizhnik–Novikov–Vesselov equations, i.e. (3). By virtue of the binary Bell polynomials, we have derived bilinear forms (8) for (3), based on which (13) and (15) for (3) have been obtained. The BTs in the binary Bell polynomial form and bilinear form for (3) have been derived. In Figures 1 through 6, we have illustrated the propagation and collision features of the solitons u for (3): Figure 1 has shown the motion of the one soliton which maintains its shape and amplitude unchanged during the propagation. In order to discuss the influence of the parameters a and b in (13) on the one soliton, compared with Figure 1b, we have chosen a and b with different values in Figure 2 and found that a and b, respectively, can affect the propagation direction of the one soliton but have no influence on the amplitude.

Meanwhile, the effects of a and b on the monotonic function, v, have been shown. Figure 3 has displayed the collision between the two solitons for u, where the shapes of the two solitons have changed after the collision. Figures 4 and 5 have illustrated that the parameters a₁, a₂, b₁, and b₂ in (15), respectively, change the propagation directions of the two solitons but cannot affect the amplitudes, compared with Figure 3b. Moreover, the influences of a₁, a₂, b₁, and b₂ on the monotonic function, v, have been illustrated. Figure 6 has shown the coalescence of the two solitons with the same propagation direction from which we have found that the collision is inelastic.