Multi-Soliton Solutions and Interaction for a Generalized Variable- Coefficient Calogero–Bogoyavlenskii–Schiff Equation

(1)

Multi-Soliton Solutions and Interaction for a Generalized Variable- Coefficient Calogero–Bogoyavlenskii–Schiff Equation

Long Xue, Yi-Tian Gao, Da-Wei Zuo, Yu-Hao Sun, and Xin Yu

Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

Reprint requests to Y.-T. G.; E-mail:gaoyt@public.bta.net.cn

Z. Naturforsch.69a, 239 – 248 (2014) / DOI: 10.5560/ZNA.2014-0018

Received December 13, 2013 / revised March 9, 2014 / published online May 21, 2014

In this paper, a generalized variable-coefficient Calogero–Bogoyavlenskii–Schiff equation is investigated. Based on the Bell polynomials and an auxiliary variable, bilinear forms of such an equation are obtained. One-, two-, and three-soliton solutions are given through the Hirota method and symbolic computation.N-soliton solutions are also constructed. Multi-soliton interaction and propagation are investigated and illustrated: (i) properties of the multi-soliton interaction on different planes in space depend on the forms of the only variable coefficient; (ii) positions of the solitons change when the wave numbers have the reverse signs.

Key words:Generalized Variable-Coefficient Calogero–Bogoyavlenskii–Schiff Equation; Bell Polynomials; Bilinear Form;N-Soliton Solutions.

PACS numbers:05. 45. Yv; 47. 35. Fg; 02. 30. Jr

1. Introduction

Nonlinear evolution equations (NLEEs) can be used to describe some nonlinear phenomena in fluids, plasmas, and nonlinear optics [1–4]. Among the NLEEs, the Calogero–Bogoyavlenskii–Schiff (CBS) equation [5–10],

u_t+uu_z+1

2u_x∂_x⁻¹u_z+1

4u_xxz=0, (1) describes the (2+1)-dimensional interaction of a Rie- mann wave propagating along the z-axis with a long wave propagating along thex-axis, whereuis a function of the scaled space coordinatesxandzwith time coordinatet, and∂_x⁻¹g=^Rgdx.

With the potential functionη(x,z,t) =∂_x⁻¹u(x,z,t) introduced, (1) can be written in the potential form as [8]

ηxt+ηxηxz+1

2ηxxηz+1

4ηxxxz=0. (2) Equation (2) has been transformed into the trilin- ear forms and N-soliton solutions in the Wronskian form have been constructed [5]. For (2), Lax pair [6],

symmetry reductions and some solutions [7], multi- front solutions [8], and travelling-wave solutions [9]

have been obtained. For a generalized version of (2), soliton-like and periodic-like solutions have been obtained [10].

In this paper, we will inquire into the generalized variable-coefficient version of (2) [11], i. e.,

η_xt+α(t)η_xη_xz+β(t)η_xxη_z+γ(t)η_xxxz=0, (3) whereα(t),β(t), andγ(t) are the analytic functions oft. Painlevé analysis of (3) has been performed, and Lie-point symmetries and similarity of (3) have been discussed [11]. In fluids and plasmas, special cases of (3) have been seen as follows:

(i) Whenα(t) +β(t) =6,γ(t) =1, by virtue of a dimensional reduction∂z=∂x and a potential function transformation h(x,t) =η_x(x,t), (3) can be degener- ated into the Korteweg–de Vries (KdV) equation for h=h(x,t)[12–14],

h_t+6hh_x+h_xxx=0, (4)

which can be used to describe the shallow-water waves, stratified internal waves in fluids, and ion- acoustic waves in plasmas [12–14].

(2)

(ii) Whenα(t) =1,β(t) =¹₂,γ(t) =¹₄, (3) becomes (2) [5–10].

(iii) Whenα(t) =−4,β(t) =−2,γ(t) =1, (3) becomes the breaking soltion equation [15–21],

ηxt−4η_xηxz−2η_xxηz+ηxxxz=0, (5) which can also be used to describe the (2+1)- dimensional interaction of a Riemann wave propagating along the z-axis with a long wave propagating along thex-axis.

However, to our knowledge, for (3),N-soliton so- lutions in the bilinear form have not been obtained, while features of the multi-soliton propagation and interaction have not been discussed, either. In this paper, we will inquire into (3) under the constraints presented in [11], i. e.,

α(t) =2β(t), γ(t) =Cβ(t), (6) where C is a nonzero constant. In Section2, a dependent-variable transformation will be proposed, (3) will be transformed into its bilinear forms with the Bell ploynomials and an auxiliary independent variable, and one-, two-, and three-soliton solutions of (3) will be derived by virtue of the Hirota method and symbolic computation. N-soliton solutions will also be constructed. In Section3, multi-soliton interaction [22,23] and propagation will be investigated and illustrated graphically. Section4will be devoted to our conclusions.

2. Bilinear Forms andN-Soliton Solutions for vc-CBS Equation

Let g be a C^∞ function of x and g_rx ≡∂_x^rg(r= 1,2, . . .), and the multi-dimensional Bell polynomials are defined as [24–31]

Y_nx(g)≡Y_n(g_1x,g_2x, . . .,g_nx)

=e^−g∂_xⁿe^g, n=1,2, . . . . (7) If we set g = g(x), (7) degrades into the one- dimensional Bell polynomials, where

Y₁(g) =g_x, Y₂(g) =g_2x+g²_x,

Y₃(g) =g_3x+3g_xg_2x+g³_x, ...

(8)

On the other hand, the binary Bell polynomials, i. e., theY-polynomials can be defined as [24–30]

Y_nx(p,q)

=Y_n(g_1x,g_2x, . . . ,g_nx) grx=







prx,ifris odd, qrx,ifris even,

(9)

where p andq are both differentiable functions of x.

For examples, Y_1x(p,q) =p_x, Y_2x(p,q) =q_2x+p²_x,

Y_3x(p,q) =p_3x+3p_xq_2x+p³_x.

(10) Relationship between theY-polynomials and HirotaD- operators can be given by the following formula [26]:

Y_nx

p=ln(F/G),q=ln(FG)

= (FG)⁻¹Dⁿ_xF·G, (11) where the HirotaD-operator is defined by [32–34]

D^m_xDⁿ_ta·b≡ ∂

∂x− ∂

∂x⁰ m

∂

∂t− ∂

∂t⁰ n

a(x,t)b(x⁰,t⁰) _x0=x,t⁰=t

,

(12)

witha(x,t)as the function ofxandt,b(x⁰,t⁰)as the function of formal variablesx⁰ andt⁰. In the case of F=G, (11) becomes theP-polynomials [25–27], i. e., P_nx(q=2 lnG) =Y_nx(p=0,q=2 lnG). (13) The P-polynomials can be characterized by the equally-recognizable even-part-partitional structures:

P_x,t(q) =q_x,t, P_4x(q) =q_4x+3q²_2x, P3x,z(q) =q3x,z+3q_2xqx,z, ...

(14)

Now we will bilinearize (3) with the aid of the Bell polynomials. If we set

η=τq_x, (15)

whereτis a parameter to be determined, through (15) and (6), (3) turns into

q_xt+Cβ(t)q_3x,z+τ β(t)q_xxq_xz +τ β(t)

Z

q_xxq_xxzdx=0. (16)

(3)

By virtue ofτ=2C,q3x,z=∂_x⁻¹∂zq4x, and (14), (16) can be written in theP-polynomials as

P_x,t(q) +2

3Cβ(t)P_3x,z(q) +1

3Cβ(t)∂_x⁻¹∂zP_4x(q) =0.

(17)

Introducing an auxiliary independent variable ζ, we can transform (17) into a pair of equations in the form ofP-polynomials:

P_4x(q)−sP_x,ζ(q) =0, (18a) P_x,t(q) +2

3Cβ(t)P_3x,z(q) +1

3C sβ(t)P_z,ζ(q) =0,

(18b)

wheresis a nonzero constant. Withq=2 lnf, the bilinear forms of (3) can be obtained as

(D⁴_x−sD_xD_ζ)f·f =0, (19a) h

D_xD_t+2

3Cβ(t)D³_xD_z +1

3Csβ(t)D_zD_ζi

f·f =0.

(19b)

The relation betweenηand f is

η=4C(lnf)_x. (20)

We next expand f into the power series of small pa- rameterεas

f =1+εf₁+ε²f₂+ε³f₃+. . . , (21)

where f_j (j=1,2, . . .) are the functions of x, z, ζ, andt. Substituting (21) into (19) and collecting the coefficients of the same powers ofε, we have

ε⁰: D⁴_x−sD_xD_ζ

(1·1) =0, (22a)

DxDt+2

3Cβ(t)D³_xDz+1

3C sβ(t)D_zD_ζ

(22b)

×(1·1) =0, ε¹: D⁴_x−sDxD_ζ

(1·f1+f1·1) =0, (22c)

D_xD_t+2

3Cβ(t)D³_xD_z+1

3C sβ(t)D_zD_ζ

(22d)

×(1·f₁+f₁·1) =0, ε²: D⁴_x−sD_xD_ζ

(1·f₂+f₁·f₁+f₂·1) =0, (22e)

D_xD_t+2

3Cβ(t)D³_xD_z+1

3C sβ(t)D_zD_ζ

(22f)

×(1·f₂+f₁·f₁+f₂·1) =0, ε³: D⁴_x−sD_xD_ζ

(22g)

×(1·f₃+f₁·f₂+f₂·f₁+f₃·1) =0,

D_xD_t+2

3Cβ(t)D³_xD_z+1

3C sβ(t)D_zD_ζ

(22h)

×(1·f₃+f₁·f₂+f₂·f₁+f₃·1) =0, ...

In order to obtain the one-soliton solutions of (3), we assume that

f₁=e^ξ¹, ξ₁=k₁x+m₁ζ+n₁z+ω₁(t) +ξ₁⁰, (23) wherek₁,m₁,n₁, andξ₁⁰are the constants, andω₁(t) is a function oft.

According to the properties of D-operators [34], (22a) and (22b) are satisfied. Substituting (23) into (22c) and (22d), the relations ofk₁,m₁,n₁, andω1(t) can be obtained as

m₁=k³₁

s , (24a)

ω₁(t) =−C k²₁n₁ Z

β(t)dt. (24b)

Consequently, we can takef_j=0(j=2,3, . . .)andε= 1, so (21) is truncated as

f =1+e^ξ¹. (25)

Thus, the one-soliton solutions for (3) are η=4C

ln(1+e^ξ¹)

x. (26)

From Figures1and2, we can see: (i) Figure1a and 1b show the periodic solutions on both thext- andzt- planes, while Figure1c shows the kink-shape soliton on thexz-plane whenβ(t) = sin^t₂. (ii) Figure2a,2b, and2c show the kink-shape soliton on thext-,zt-, and xz-planes whenβ(t) =−¹₂.

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

f₁=e^ξ¹+e^ξ², (27a)

ξj=k_jx+m_jζ+n_jz+ω_j(t) +ξ⁰_j (j=1,2), (27b) wherek_j,m_j,n_j, andξ⁰_j (j=1,2)are the constants, andω_j(t) (j=1,2)are the functions oft. Substituting

(4)

Fig. 1 (colour online). One-soliton solutions via (26) withζ=0,k1=1.4,n1=2,C=1,β(t) =sin₂^t,ξ₁⁰=0: (a) ofz=1;

(b) ofx=1; (c) oft=0.

Fig. 2 (colour online). One-soliton solutions via (26) withζ=0,k1=1.4,n1=2,C=1,β(t) =−¹₂,ξ₁⁰=0: (a) ofz=1;

(b) ofx=1; (c) oft=0.

(27) into (22), we obtain m_j=k³_j

s (j=1,2), (28a)

ωj(t) =−C k²_jn_j Z

β(t)dt (j=1,2), (28b) f2=e^ξ¹^+ξ²^+A¹², (28c) e^A¹²=(k₁−k2)²

(k₁+k2)², (28d)

f_j=0 (j=3,4, . . .). (28e) Therefore, the two-soliton solutions are

f =1+e^ξ¹+e^ξ²+e^ξ¹^+ξ²^+A¹². (29) In order to derive the three-soliton solutions, we set

f1=e^ξ¹+e^ξ²+e^ξ³, (30a) f₂=e^ξ¹^+ξ²^+A¹²+e^ξ¹^+ξ³^+A¹³+e^ξ²^+ξ³^+A²³, (30b)

ξj=k_jx+m_jζ+n_jz+ω_j(t) +ξ⁰_j

(30c) (j=1,2,3),

e^A^jl =(k_j−k_l)²

(k_j+k_l)² (j,l=1,2,3 and j<l), (30d) wherekj,mj,nj, andξ⁰_j (j=1,2,3)are the constants, andωj(t) (j=1,2,3)are the functions oft. Substitut- ing (30) into (22), we obtain

m_j=k³_j

s (j=1,2,3), (31a)

ω_j(t) =−C k²_jn_j Z

β(t)dt (j=1,2,3), (31b) f₃=e^ξ¹^+ξ²^+ξ³^+A¹²³, (31c) e^A¹²³=(k₁−k₂)²

(k₁+k₂)²

(k₁−k₃)² (k₁+k₃)²

(k₂−k₃)²

(k₂+k₃)², (31d)

(5)

fj=0 (j=4,5, . . .). (31e) Therefore, the three-soliton solutions are

f=1+e^ξ¹+e^ξ²+e^ξ³+e^ξ¹^+ξ²^+A¹²+e^ξ¹^+ξ³^+A¹³ +e^ξ²^+ξ³^+A²³+e^ξ¹^+ξ²^+ξ³^+A¹²³. (32) This process can be continued for us to derive theN- soliton solutions, which can be denoted by the following formula:

f =

∑

µ=0,1

exph ^N

∑

j=1

µ_jξ_j+

N 1≤j<l

∑

µ_jµ_lA_{j l}i

, (33)

with

m_j=k³_j

s , (34a)

ω_j(t) =−C k²_jn_j Z

β(t)dt, (34b)

Fig. 3 (colour online). Two-soliton interaction via (33) and (20) withN=2,k₁=1.4,k₂=−2,n₁=n₂=2,C=1,β(t) = sin₂^t,ξ₁⁰=ξ₂⁰=0: (a) ofz=1; (b) ofx=1; (c) oft=0; (d) profiles of (a) witht=0,π,2π; (e) profiles of (b) with t=0,π,2π.

ξj=kjx+mjζ+n_jz+ωj(t) +ξ⁰_j, (34c) e^A^jl =(k_j−k_l)²

(k_j+k_l)² (j,l=1,2, . . .,N), (34d) and the sum taken over all the possible combinations ofµ_j=0,1(j=1,2, . . .,N).

3. Multi-Soliton Interaction

In this section, we will analyze the multi-soliton interaction for (3).

Taking N=2,3 in (33), setting k_j, n_j as the real constants andξ⁰_j =0 (j=1,2, . . .,N), and then substituting them into (20), we have the two-soliton and three-soliton solutions. Without loss of generality, we can set the auxiliary variableζ =0.

Comparing Figure3 with4, we find that : (i) the two-soliton solutions have the periodic properties on

(6)

Fig. 4 (colour online). Two-soliton interaction via (33) and (20) withN=2,k1=1.4,k2=2,n1=n2=2,C=1,β(t) =sin^t₂, ξ₁⁰=ξ₂⁰=0: (a) ofz=1; (b) ofx=1; (c) oft=0; (d) profiles of (a) witht=0,π,2π; (e) profiles of (b) witht=0,π,2π.

Fig. 5 (colour online). Two-soliton interaction via (33) and (20) withN=2,k1=1.4,k2=−2,n1=n2=2,C=1,ξ₁⁰= ξ₂⁰=0,z=1: (a) ofβ(t) =sin^2t₃; (b) ofβ(t) =t; (c) ofβ(t) =₄^t; (d) ofβ(t) =−¹₂.

thext- andzt-planes, and have the kink-shape properties on thexz- plane, whenβ(t) =sin₂^t; (ii) the two- soliton positions change when the wave numbers are just opposite in sign. As shown in Figures3d and4d, when k1>0 and k2<0, the one soliton locates at η>0, and the other atη<0. Whilek₁>0 andk₂>0, both of the two solitons locate atη>0.

Comparing Figure3a with5, we find that: (i) in Fig- ures3a and5a, the two-soliton interaction has the periodic properties on thext-plane when the only variable coefficientβ(t) is a sine function of time, i. e., β(t) =sin₂^t in Figure3a andβ(t) =sin^2t₃ in Figure5a, and the period ofβ(t)has an affect on the period and range of the soliton interaction; (ii) in Figure5b and5c,

(7)

Fig. 6 (colour online). Two-soliton interaction via (33) and (20) withN=2,k1=1.4,k2=−2,n1=n2=2,C=1,ξ₁⁰= ξ₂⁰=0,x=1: (a) ofβ(t) =sin^2t₃; (b) ofβ(t) =t; (c) ofβ(t) =₄^t; (d) ofβ(t) =−¹₂.

Fig. 7 (colour online). Three-soliton interaction via (33) and (20) withN=3,k1=1.4,k2=−2,k3=1.2,n1=n2=n3=2, C=1,β(t) =sin^t₂,ξ₁⁰=ξ₂⁰=ξ₃⁰=0: (a) ofz=1; (b) ofx=1; (c) oft=0; (d) profiles of (a) witht=0,π,2π; (e) profiles of (b) witht=0,π,2π.

the two-soliton interaction has the quadric properties on thext-plane whenβ(t)is a linear function oft, i. e., β(t) =t in Figure5b andβ(t) = ^t₄ in Figure5c, and herebyβ(t)has an affect on the range of the soliton interaction; (iii) in Figure5d, the two-soliton interaction has the kink-shape properties on thext-plane when β(t)is a nonzero constant, i. e.,β(t) =−¹₂.

Comparing Figure3b with6, we find that the two- soliton interaction on thezt-plane has the similar properties with those on thext-plane under the same variable coefficientβ(t), i. e.: (i) in Figures3b and6a, the two-soliton interaction has the periodic properties on thezt-plane when the only variable coefficientβ(t)is a sine function oft, i. e.,β(t) =sin₂^t in Figure3b and

(8)

Fig. 8 (colour online). Three-soliton interaction via (33) and (20) withN=3,k₁=1.4,k₂=2,k₃=1.2,n₁=n₂=n₃=2, C=1,β(t) =sin^t₂,ξ₁⁰=ξ₂⁰=ξ₃⁰=0: (a) ofz=1; (b) ofx=1; (c) oft=0; (d) profiles of (a) witht=0,π,2π; (e) profiles of (b) witht=0,π,2π.

β(t) =sin^2t₃ in Figure6a, and the period ofβ(t)has an affect on the period and range of the soliton interaction; (ii) in Figure6b and6c, the two-soliton interaction has the quadric properties on thezt-plane when β(t)is a linear function of time, i. e.,β(t) =t in Fig- ure6b andβ(t) =₄^t in Figure6c, and herebyβ(t)has an affect on the range of the soliton interaction; (iii) in Figure6d, the two-soliton interaction has the kink- shape properties on thext-plane whenβ(t)is a nonzero constant, i. e.,β(t) =−¹₂.

As seen in Figures7 and8, the three-soliton solutions have the similar properties to the two-soliton solutions, that is: (i) the three-soliton solutions have the periodic properties on the xt- andzt-planes, and have the kink-shape properties on thexz-plane, whenβ(t) = sin^t₂; (ii) the three-soliton positions change when the wave numbers are just opposite in sign. As shown in Figures7d and8d, whenk₁>0,k₂<0, and k₃>0,

the two solitons locate atη>0, and the third atη<0.

Whilek₁>0,k₂>0, andk₃>0, all of the three solitons locate atη>0.

4. Conclusions

Among the NLEEs, the CBS equation, i. e., (2), can be used to describe the (2+1)-dimensional interaction of a Riemann wave propagating along thez-axis with a long wave propagating along the x-axis; the KdV equation, i. e., (4), describes certain phenomena in fluids and plasmas; and the breaking soliton equation, i. e., (5), describes the (2+1)-dimensional wave interaction. Generalized variable-coefficient equation, i. e., (3), covers (2), (4), and (5) with different forms of the variable coefficients. We have investigated (3) under the constraints of (6). The results can be concluded as follows:

(9)

(i) Via the Bell polynomials and an auxiliary independent variable ζ, Bilinear Forms (19) have been derived under Transformation (20). With Bi- linear Forms (19), One-Soliton Solutions (23), Two- Soliton Solutions (20) and (27) and Three-Soliton So- lutions (20) and (30) have been obtained via the Hirota method.N-Soliton Solutions (20) and (33) have been constructed.

(ii) From Figures1and2, it can be found that Fig- ures1a and 1b show the periodic solutions on both thext- andzt-planes, while Figure1c shows the kink- shape soliton on thexz-plane whenβ(t) =sin^t₂, and Figure2a,2b, and2c all show the kink-shape soliton on thext-,zt-, andxz-planes whenβ(t) =−¹₂.

(iii) From Figures3–8, it can be found that the properties of the multi-soliton interaction on the xt- andzt-planes depend on the forms of the only variable coefficientβ(t), and the wave numberskj(j=1, 2, 3) andβ(t)have an effect on the positions of the multi- soliton interaction, on thext-,zt-, andxz-planes, as follows:

Comparing Figure3a with5, and Figure3b with6, we have found that the two-soliton interaction has the periodic properties on thext andzt-planes whenβ(t) is a sine function of time, i. e., β(t) =sin₂^t in Fig- ure3a and3b, whileβ(t) =sin^2t₃ in Figures5a and6a;

the two-soliton interaction has the quadric properties on the xt- and zt-planes when β(t) is a linear function of time, i. e., β(t) =t in Figures5b and6b, while β(t) =₄^t in Figures5c and6c; the two-soliton interaction has the kink-shape properties on the xz- plane when β(t) is a nonzero constant, i. e., β(t) =

−¹₂ in Figures5d and6d; the period of β(t) has an affect on the period and range of the soliton interaction as shown in Figures3a and5a, Figures3b

and6a; β(t) has an affect on the positions the soliton interaction, as shown in Figures5b and5c,6b and 6c.

Comparing Figure3with4, we have found that the two-soliton interaction has the periodic properties on thext- andzt-planes and kink-shape soltion on thexz- plane whenβ(t) =sin₂^t. Positions of the two solitons change when the wave number k₂ changes its sign:

when k₁>0 and k₂<0, the one soliton locates at η>0, and the other atη<0, as shown in Figure3d and3e; whenk₁>0 andk₂>0, both of the solitons locate atη>0, as shown in Figure4d and4e.

Comparing Figure7 with8, we have found that the three-soliton solutions have the similar properties to those of the two-soliton solutions, i. e., the three- soliton interaction has the periodic properties on the xt- and zt-planes and kink-shape soltion on the xz- plane whenβ(t) =sin^t₂. Positions of the three solitons change when the wave numberk₂changes its sign:

whenk1>0,k2<0, andk3>0, two of the solitons locate atη >0, and the third at η<0, as shown in Figure7d and7e; whenk1>0,k2>0, andk3>0, all of the three solitons locate atη>0, as shown in Fig- ure8d and8e.

Acknowledgements

We express our sincere thanks to all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Sci- ence Foundation of China under Grant No. 11272023, and by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) under Grant No. IPOC2013B008 .

[1] B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, and Y. T.

Gao, Eur. Phys. J. B47, 329 (2005).

[2] B. Tian, G. M. Wei, C. Y. Zhang, W. R. Shan, and Y. T.

Gao, Phys. Lett. A356, 8 (2006).

[3] Y. T. Gao and B. Tian, Europhys. Lett. 77, 15001 (2007).

[4] X. Yu, Y. T. Gao, Z. Y. Sun, and Y. Liu, Phys. Rev. E 83, 056601 (2011).

[5] S. J. Yu, K. Toda, and T. Fukuyama, J. Phys. A 31, 10181 (1998).

[6] K. Toda, S. J. Yu, and T. Fukuyama, Rep. Math. Phys.

44, 247 (1999).

[7] M. S. Bruzón, M. L. Gandarias, C. Muriel, S. Saez, and F. R. Romero, Theor. Math. Phys. 137, 1367 (2003).

[8] A. M. Wazwaz, Appl. Math. Comput.203, 592 (2008).

[9] A. M. Wazwaz, Appl. Math. Comput.196, 363 (2008).

[10] B. Li and Y. Chen, Czech. J. Phys.54, 517 (2004).

[11] A. Bansal and R. K. Gupta, Phys. Scr. 86, 035005 (2012).

[12] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, Society for Industrial and Ap- plied Mathematics, Philadelphia 1981.

(10)

[13] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, New York 1991.

[14] X. Yu, Y. T. Gao, Z. Y. Sun, and Y. Liu, Phys. Scr.81, 045402 (2010).

[15] O. I. Bogoyavlenskii, Math. USSR. Izv.34, 245 (1990).

[16] O. I. Bogoyavlenskii, Russ. Math. Surv.45, 1 (1990).

[17] B. Tian, K. Y. Zhao, and Y. T. Gao, Int. J. Eng.35, 1081 (1997).

[18] B. Li, Y. Chen, H. N. Xuan, and H. Q. Zhang, Chaos Solitons Fract.17, 885 (2003).

[19] Z. Y. Yan and H. Q. Zhang, Comput. Math. Appl.44, 1439 (2002).

[20] F. Calogero and A. Degasperis, Nuov. Cim. B32, 201 (1976).

[21] S. A. Elwakil, S. K. El-labany, M. A. Zahran, and R. Sabry, Z. Naturforsch.58a, 39 (2003).

[22] H. L. Zhen, B. Tian, Y. F. Wang, H. Zhong, and W. R.

Sun, Phys. Plasmas21, 012304 (2014).

[23] W. R. Sun, B. Tian, H. Zhong and H. L. Zhen, Stud.

Appl. Math.132, 65 (2014).

[24] E. T. Bell, Ann. Math.35, 258 (1934).

[25] F. Lambert, I. Loris, J. Springael, and R. Willox, J. Phys. A27, 5325 (1994).

[26] F. Lambert and J. Sprigael, Chaos Solitons Fract.12, 2821 (2001).

[27] F. Lambert and J. Sprigael, Acta. Appl. Math.102, 147 (2008).

[28] Y. Jiang, B. Tian, M. Li, and P. Wang, Nonlinear Dy- nam.74, 1053 (2013).

[29] Y. Jiang and B. Tian, Europhys. Lett. 102, 10010 (2013).

[30] H. L. Zhen, B. Tian, and W. R. Sun, Appl. Math. Lett.

27, 90 (2014).

[31] W. R. Sun, B. Tian, Y. Jiang, and H. L. Zhen, Ann.

Phys.343, 215 (2014).

[32] R. Hirota, Phys. Rev. Lett.27, 1192 (1971).

[33] J. Satsuma, J. Phys. Soc. Jpn.40, 286 (1976).

[34] R. Hirota, The Direct Method in Soliton Theory, Cambridge Univ. Press, Cambridge 2004.