P
RAMANA °c Indian Academy of Sciences Vol. 67, No. 3—journal of September 2006
physics pp. 449–456
On exact solutions of the Bogoyavlenskii equation
YAN-ZE PENG and MING SHEN
Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
E-mail: yanzepeng@163.com
MS received 20 April 2006; revised 3 July 2006; accepted 3 August 2006
Abstract. Exact solutions for the Bogoyavlenskii equation are studied by the travelling wave method and the singular manifold method. It is found that the linear superposition of the shock wave solution and the complex solitary wave solution for the physical field is still a solution of the equation of interest, except for a phase-shift. The dromion-like structures with elastic and nonelastic interactions are found.
Keywords. Bogoyavlenskii equation; travelling wave method; singular manifold method.
PACS Nos 05.45.Yv; 02.30.Ik; 02.30.Jr
1. Introduction
This paper is devoted to a study of exact solutions for the Bogoyavlenskii equation [1]
4ut+uxxy−4u2uy−4uxv= 0,
uuy =vx. (1)
In ref. [1], the Lax pair and a nonisospectral condition for the spectral parameter is presented. Equation (1) was again derived by Kudryashov and Pickerling [2] as a member of a (2+1) Schwarzian breaking soliton hierarchy, and rational solutions of it were obtained. The equation also appeared in ref. [3] as one of the equations associated to nonisospectral scattering problems. The Painlev´e property of eq. (1) is recently checked by Est´evezet al[4].
Equation (1), as the modified version of a breaking soliton equation, 4uxt + 8uxuxy+ 4uyuxx+uxxxy = 0, describes the (2+1)-dimensional interaction of a Riemann wave propagating along they-axis with a long wave along thex-axis [1].
It is well-known that the solution and its dynamics of the equation can make re- searchers deeply understand the described physical process. The paper is organized as follows: In§2, the travelling wave method is used to obtain the travelling wave solutions of the polynomial form. In §3, a general explicit solution with two ar- bitrary functions is obtained by means of the singular structure analysis, and the
dynamical properties as well as the localized structures are studied. Conclusion and discussion are given in the last section.
2. The travelling wave method for eq. (1)
In the theory of nonlinear waves, one of the most important aspects is the study of travelling wave solutions which are solutions of constant form moving with a fixed velocity [5–7]. The travelling wave solution for eq. (1) is of the form
u(x, y, t) =u(ξ), v(x, y, t) =v(ξ), ξ=kx+ly−ωt. (2) Substituting eq. (2) into eq. (1) and integrating once the second equation of eq.
(1), we get
−4ωu0+k2lu000−4lu2u0−4ku0v= 0, v= l
2ku2+c1
k, (3)
where prime denotes the derivative with respect toξ,c1being the constant of inte- gration. The substitution of the second equation of eq. (3) into the first equation, after integrating once the resultant, yields
k2lu00−4(ω+c1)u−2lu3=c2, (4) with c2 being the integration constant. A mapping method and its extensions have been developed to obtain a series of exact solutions for nonlinear ordinary differential equations [5,6]. However, many of them are singular. Here, we introduce directly an ansatz approach to get nonsingular exact solutions for eq. (4). Now, we assume that eq. (4) has the solution of the form
u=A0+A1sn(ξ|m), (5)
whereA0andA1 are the constants to be determined, sn(ξ|m) is the Jacobi elliptic sine function [8] and m (0 < m < 1) the modulus of it. One can easily see that ansatz (5) is a natural extension of the tanh function method [7], since m → 1, sn(ξ|m)→tanhξ. Substituting eq. (5) into eq. (4), we obtain
A0= 0, c2= 0, A1=±
√2 2 kp
m(m+ 1), ω= 1
4(−k2l−k2lm−4c1). (6)
Thus, a periodic travelling wave solution of eq. (1) is obtained as follows:
u=±
√2 2 kp
m(m+ 1)sn(kx+ly−ωt|m), v=1
4klm(m+ 1)sn2(kx+ly−ωt|m) +c1
k, (7)
withω given by the last equation of eq. (6).
Asm→1, eq. (7) is reduced to u=±ktanh[kx+ly+1
2(k2l+ 2c1)t], v= 1
2kltanh2[kx+ly+1
2(k2l+ 2c1)t] +c1
k. (8)
The physical field u is a shock wave and the potential v is a solitary wave. It is obvious that the wave velocity depends on the amplitude. If we replace sn- function by the Jacobi cosine function cn(ξ|m) in ansatz (5), we obtain another periodic wave solution of eq. (1)
u=±
√2 2 kp
m(m+ 1)icn(kx+ly−ωt|m), v=−1
4klm(m+ 1)cn2(kx+ly−ωt|m) +c1
k,
(9)
withω=1
4(−k2l+k2lm+k2lm2−4c1),c2= 0 andi2=−1.
Asm→1, it follows from eq. (9) that u=±iksech[kx+ly+1
4(−k2l+ 4c1)t], v=−1
2klsech2[kx+ly+1
4(−k2l+ 4c1)t] +c1
k. (10)
The physical fielduis a complex solitary wave solution while the potential fieldv is still a solitary wave. In the same way, the wave velocity has a relation to the amplitude. As is well-known, the principle of linear superposition for nonlinear equations does not, in general, hold. However, it is interesting to see that the linear superposition of eqs (8) and (10) for the physical fielduis also the solution of eq.
(4), except for a phase-shift. In other words, we have the complex solitary wave solution to the Bogoyavlenskii equation
u=±k
2tanhξ±ik 2sechξ, v= 1
8kl+c1
k −1
4klsech2ξ±i1
4kltanhξsechξ,
(11)
withξ=kx+ly+1
8(k2l+ 8c1)t.
3. The singular manifold method for eq. (1)
The travelling wave solution is only a special type of solution for the nonlinear PDE. However, eq. (1) may possess many other interesting solution structures. The singular manifold method [9–12] is a powerful tool for obtaining exact solutions of nonlinear PDEs. Recently, some authors used it successfully to construct coherent structures for nonlinear PDEs [13,14]. According to the singular structure analysis for nonlinear PDEs [9–12], we truncate the Painlev´e expansion of eq. (1) at the constant level term
u=ϕ−1u0+u1,
v=ϕ−2v0+ϕ−1v1+v2, (12)
where ϕ≡ϕ(x, y, t) is the singular manifold function. Substituting eq. (12) into eq. (1) and equating the coefficients of like powers ofϕto zero, we obtain
u0=ϕx, v0=1
2ϕxϕy, v1=−1
2ϕxy, (13)
whereϕsatisfies the system of equations 2u1ϕx+ϕxx= 0,
4ϕtϕx−4u21ϕxϕy−4v2ϕ2x +4u1yϕ2x+ 2u1xϕxϕy+ 8u1ϕxϕxy
+ϕxxϕxy+ 3ϕxϕxxy+ϕxxxϕy= 0,
−8u1u1yϕx+ 2u1xϕxy+ 4ϕxt
−4u21ϕxy−4v2ϕxx+ϕxxxy = 0, (14) withu1 andv2 satisfying the original equation (1). Note that u0=−ϕx has been omitted since no new meaningful results can be obtained for this case through similar analysis. Thus, eqs (12), (13) and (14) constitute an auto-B¨acklund trans- formation for the Bogoyavlenskii equation. In order to obtain exact solutions from this B¨acklund transformation, the seed solutionu1, v2 is usually taken to be zero.
However, we take the seed solution of eq. (1) as
u1= 1, v2=v2(y, t), (15)
withv2(y, t) being an arbitrary function of indicated variables. It is directly verified that eq. (14) with eq. (15) has the solution
ϕ=f(y) +g(y, t)e−2x, v2=−gt(y, t)
2g(y, t), (16)
wheref(y) andg(y, t) are two arbitrary functions of indicated variables. Thus, we obtain a general exact solution of eq. (1)
u=− 2ge−2x f+ge−2x + 1, v= f gy−fyg
(fex+ge−x)2 − gt
2g, (17)
including two arbitrary functionsf =f(y) andg=g(y, t). Thanks to the arbitrari- ness of functionsf and g in eq. (17), our result provides a rich solution structure and allows to study many interesting phenomena. However, we point out that eq.
(1) possesses some special types of localized coherent structures for the following potential field:
-4 -2
0 2
4
x -4
-2 0
2 4
y -1
-0.5 0 0.5
1 u
-4 -2
0 2
4 x
Figure 1. A typical spatial structure of the physical fieldufor eq. (19).
w=uy=− 2(f gy−fyg)
(fex+ge−x)2, (18)
rather than the physical field u itself. In what follows, several special cases of interest are considered. First, taking f = 1, g = tanh(y−t) + tanh(2y+t) + 3, from eq. (17), one has
u=− 2(tanh(y−t) + tanh(2y+t) + 3)e−2x 1 + (tanh(y−t) + tanh(2y+t) + 3)e−2x + 1, v= sech2(y−t) + 2sech2(2y+t)
[ex+ (tanh(y−t) + tanh(2y+t) + 3)e−x]2 (19) + sech2(y−t)−sech2(2y+t)
2(tanh(y−t) + tanh(2y+t) + 3).
For eq. (19), u is a simple kink wave solution without interaction, whose typical spatial structure att= 0 is depicted in figure 1, andv is the combination solitary wave solution of a line with a peak and two half-lines, whose nonelastic interaction process is shown in figure 2. In this case, from eq. (18), we obtain the two dromion- like structure
w=− 2[sech2(y−t) + 2 sech2(2y+t)]
[ex+ (tanh(y−t) + tanh(2y+t) + 3)e−x]2, (20) whose evolution process is illustrated in figure 3. It is found that the interaction of two dromion-like structure is nonelastic because of the change in their amplitudes.
By the way, dromions are exponentially localized structures in two dimensions (decaying exponentially in all spatial directions) while the familiar solitary wave solution means only, in general, the localized structure. Whenf = 1, g= sech(y− t) + sech(2y+t) + 3, it follows from eq. (17) that
Figure 2. The evolution of solitary wave solutionvfor eq. (19).
Figure 3. The evolution of two dromion-like solution (20).
Figure 4. The evolution of four dromion-like solution (22).
u=− 2(sech(y−t) + sech(2y+t) + 3)e−2x 1 + (sech(y−t) + sech(2y+t) + 3)e−2x + 1, v=−sech(y−t)tanh(y−t) + 2 sech(2y+t)tanh(2y+t)
[ex+ (sech(y−t) + sech(2y+t) + 3)e−x]2
−sech(y−t)tanh(y−t)−sech(2y+t)tanh(2y+t)
2(sech(y−t) + sech(2y+t) + 3) , (21) whereuis still a simple kink wave solution without interaction, whose typical spatial structure is very similar to figure 1 and is omitted, whilevis a combination solitary wave solution of two lines, whose nonelastic interaction process is also similar to figure 2 and is omitted. From eq. (18), one has
w=2[sech(y−t)tanh(y−t) + 2 sech(2y+t)tanh(2y+t)]
[ex+ (sech(y−t) + sech(2y+t) + 3)e−x]2 , (22) a four dromion-like structure (two dromions and two anti-dromions). Figure 4 shows the interaction detail of the four dromion-like solution (22). It is easily seen that the interaction of four dromion-like structure is completely elastic.
4. Conclusion and discussion
Exact solutions and their properties for the Bogoyavlenskii equation have been studied by the travelling wave method and the singular manifold method. For the
physical fieldu, the linear superposition of the shock wave solution and the complex solitary wave solution is still a solution of the equation in question, except for a phase-shift. For the potential fieldw=uy, we have found not only a two dromion- like solution with nonelastic interaction, but also a four dromion-like solution with elastic interaction. However, we have not found a two dromion-like solution with elastic interaction by selecting appropriately arbitrary functions appearing in the solution formula (18), and this problem still remains to be studied in future.
Acknowledgements
The authors would like to acknowledge the anonymous referee for his valuable comments.
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