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Latin American applied research

versión impresa ISSN 0327-0793

Lat. Am. appl. res. v.34 n.1 Bahía Blanca ene./mar. 2004

 

Exact travelling wave solutions to the generalized Kuramoto-Sivashinsky equation

Changpin Li1, Guanrong Chen2 and Suchuan Zhao3

1 Department of Mathematics, Shanghai University, Shanghai 200436, P. R. China
leecp@online.sh.cn

2 Department of Electronic Engineering, City University of Hong Kong, Hong Kong, P. R. China
gchen@ee.cityu.edu.hk

3 Department of Physics, Shanghai University, Shanghai 200436, P. R. China

Abstract — By using a special transformation, the new exact travelling wave solutions to the generalized Kuramoto- Sivashinsky equation are obtained.

Keywords — Travelling Wave Solutions. Solitary Wave Solutions. Kuramoto-Sivashinsky Equation.

I. INTRODUCTION

In this paper, we consider the generalized Kuramoto-Sivashinsky equation (Yang, 1994):

ut + βuαux + γuτuxx + δuxxxx = 0, (1)

where α, β, γ, δ, τ ∈ R and αβγδ ≠ 0.

When α = β = 1 and τ = 0, (1) reduces to the original Kuramoto-Sivashinsky (K-S) equation. The K-S equation was derived by Kuramoto (1978) for the study of phase turbulence in the Belousov-Zhabotinsky reaction. An extension of this equation to two or more spatial dimensions was then given by Sivashinsky (1977, 1980) in the study of the propagation of a frame front for the case of mild combustion. The K-S equation represents one class of pattern formation equation (Yang, 1994; Temam, 1988), and it also serves as a good model of bifurcation and chaos (Abdel-Gawad & Abdusalam, 2001; Li and Chen 2001, 2002).

As far as the travelling wave solutions are concerned, one can always use the transform

u(x,t) = u(ξ), ξ = x - ct, (2)

where c is the wave velocity. The travelling wave solutions of (1) satisfy the following ordinary differential equation:

-cu' + βuαu' + γuτu'' + δu'''' = 0 . (3)

In (Yang, 1994), using the ansatz (Bernoulli equation)

u' = au + bun, (4)

where a, b, nR, ab < 0 and n ≠ 1, the exact travelling wave solution to (1) for a = 3τ = 9 was obtained. In this presentation, we further introduce the following ansatz:

u(ξ) = vh(ξ), v' = av + bvn (5)

where abh ≠ 0, n ≠ l and ab < 0, and obtain a new exact solution for the equation. From (5), one first gets

(6)

in which c0 is an arbitrary constant. If h/(n - 1) > 0, (6) is the solitary wave solution connecting the two stationary states u = 0 and (Lu, et al., 1993). So, the relative orbit is a heteroclinic orbit.

Repeating some differential calculations, one can obtain the following formulas:

v'' = (a + nbvn-1)v', (7)
v'''' = [a3 + a2bn(n2 + n + l)vn-1 + 3ab2n2(2n - l)v2n-2
+ b3n(2n - l)(3n - 2)v3n - 3]v'.
(8)
u' = hvh-lv', (9)
u'' = [h2avh-1 + hb(n + h - l)vn+h-2]v', (10)
u'''' = {h4a3vh-l + ha2b(n + h- l)[h2 + (n + h - 1)·
(n+2h - l)]vn+h-2 + 3hab2(n + h - l)2(2n + h-2)v2n+h-3
+ hb3(n + h - l)(2n + h - 2)(3n + h - 3)v3n+h-4} v'.
(11)

Then, by substituting the first formula of (5) and (9)-(11) into (3), one has

{(-ch + δh4a3)vh-1 + βhvαh+h-1 + γh2avτh+h-1
hb(n + h - 1)vn+(τ+1)h-2 + δha2b(n + h - l)·
[3h2 + 3h(n - l) + (n - l)2]vn+h-2 + 3δhab2(n + h - l)2·
(2n + h - 2)v2n+h-3 + δhb3(n + h - l)(2n + h - 2)·
(3n + h - 3)v3n+h-4} v' = 0.
(12)

By furthermore comparing the same orders of v, one can determine values of the parameters a, b, n and h. However, to consider all possible cases is rather complicated. In order to keep the presentation short, only the following interesting cases with h = 1, n = 0 and n = 2 are considered here.

II. CASE h = l

If h = 1, then u(ξ) = v(ξ) and u' = au + bun, so that (12) is reduced to

a3 - c) + βuα + γauτ + γbnuτ+n-1 +
δa2bn(n2 + n + l)un-1 + 3δab2n2(2n - l)u2n-2 +
δb3n(2n - l)(3n - 2)u3n-3 = 0.
(13)

By comparing the same orders of u, one finds the following situations.

that is,

In (14), a is a parameter. One should choose a such that ab < 0. The same should be done for the similar cases below.

that is,

(15)

If , then (13) can be translated into

The following results are immediate.

Therefore, one can easily find that

(16)

so,

(17)

so,

Besides and , one has the following case.

(3) τ = n - 1, α = 3n - 3

For this case,

δa3 - c = 0, γa + δa2bn(n2 + n + 1) = 0,

γbn + 3δab2n2(2n - l) = 0, β + δb3n(2n - l)(3n - 2) = 0

From the second and the third equations, it follows that n = 4. So, if and only if n = 4, α = 3τ = 9, there exist real number solutions for a, b and c, as

(19)

This result is the same as that obtained in Yang (1994).

III. CASE n = 0

For n = 0, (12) is reduced to

a3h3-c)vh-1 + βvαh+h-1 + γahvτh+h-1 + γb(h - 1)·
vτh+h-2 + δa2b(h - l)(3h2 - 3h + l)vh-2+3δab2(h - l)2·
(h - 2)vh-3 + δb3(h - l)(h - 2)(h - 3)vh-4 = 0.
(20)

After considering the coefficients of some orders of v, one has the following cases, with h = 1, h = 2 and h = 3, respectively.

(1) h = 1.

There exists one and only one sub-case with α = τ ≠ 0 for h = 1 (Note: α ≠ 0), as follows.

(1a) α = τ ≠ 0

(21)

(2) h = 2

For h = 2, one also has a sub-case.

For this sub-case, one has

a2b + γb + β = 0, 8δa3 - c + 2γa = 0.

Thus,

(22)

(3) h = 3

For h = 3, (20) can be changed to

(27δa3 - c)v2 + βv3α+2 + 3γav3τ+2 + 2γbv3τ+1
+ 38δa2bv + 12δab2 = 0,
(23)

and only three cases exist, as follows.

Here,

27δa3 - c = 0, 3γa + 38δa2b = 0, β + 2γb + 12βab2 = 0.

Therefore,

(24)

It is clear that

27δa3 - c = 0, β + 3γa + 38δa2b = 0,2γb + 12δab2 = 0.

Hence,

(25)

By the same reasoning, one has

27δa3 - c + 3γa = 0, 2γb + 38δa2b = 0, β + 12δab2 = 0,

so a, b, c are as below:

(26)

For h ≠ 1, h ≠ 2, and h ≠ 3, a comparison between the corresponding terms in (20) gives only one case, as follows.

It follows that

δa3h3 - c = 0 , (δa2b(h - 1) (3h2 - 3h + 1) + γah = 0 ,

ab2(h - l)2(h - 2) + γb(h - 1) = 0 ,

δb3(h - l)(h - 2)(h - 3) + β = 0 .

The second and the third equations of the above system give . So, if and only if , α = 3τ = 9, the above system has real number solutions for a, b and c, as

(27)

IV. CASE n = 2

Substituting n = 2 into (12) gives

(-c + δh3a3)vh-l + (δa2b(h + l)(3h2 + 3h + 1)vh
+ 3δab2(h + l)2(h + 2)vh+1 + δb3(h + l)(h + 2)(h + 3)vh+2
havτh+h-1 + γb(h + l)vτh+h + βvαh+h-1 = 0.
(28)

(1) h = -1

For h = - 1, there exist only one sub-case.

(la) γ = α ≠ 0

(29)

(2) h = -2

For h = - 2, there exist two sub-cases.

By the same reason, one has

-c - 8δa3 - 2γa = 0 , -7δa2b - γb + β = 0,

so,

(30)

Similarly, one has

-c - 8δa3 = 0, -7δa2b - 2γa = 0 , -γb + β = 0,

so,

(31)

(3) h = -3

For h = -3, there exist three sub-cases.

For this sub-case,

c + 27δa3 + 3γa = 0, 38δa2b + 2γb = 0, 12γab2 - β = 0.

The solutions are

(32)

Similarly,

c + 27δa3 = 0, 38δa2b + 3γa - β = 0, 12δab2 + 2γb = 0.

The solutions are

(33)

The following system is determined by the same reasoning:

c + 27δa3 = 0, 38δa2b + 3γa = 0, 12δab2 + 2γb - β = 0,

so,

(34)

Besides h = - 1, h = - 2 and h = -3, there exists only one case, with , α = 3γ.

(4) , α = 3γ.

It follows that

-c + δh3a3 = 0, δa2b(h + l)(3h2 + 3h + l) + γha = 0 ,

ab2(h + l)2(h + 2) + γb(h + 1) = 0 ,

δb3(h + l)(h + 2)(h + 3) + β = 0.

The second and third equations in the above system give . So, if and only if and α = 3τ = 9, parameters a, b and c are given by

(35)

ACKNOWLEDGEMENTS
This research was supported by the NSF of China (Grant No. 19971057) and the Hong Kong CERG (Grant No. 9040579).

REFERENCES
1. Abdel-Gawad, H. I. & Abdusalam, H. A., "Approximate solutions of the Kuramoto-Sivashinsky equation for periodic boundary value problems and chaos", Chaos, Solitons and Fractals, 12, 2039-2050 (2001).         [ Links ]
2. Kuramoto, Y., "Diffusion-induced chaos in reaction system", Prog. Theor. Phys. Suppl., 64, 346-367 (1978).         [ Links ]
3. Li, C. P. & Chen, G., "Bifurcation analysis of the Kuramoto-Sivashinsky equation in one spatial dimension", Int. J. Bifurcation and Chaos, 11 (9), 2493-2499 (2001).         [ Links ]
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7. Sivashinsky, G. I., "On flame propagation under conditions of stoichiometry", SIAM J. Appl. Math., 39, 67-82 (1980).         [ Links ]
8. Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York (1988).         [ Links ]
9. Yang, Z. J., "Travelling wave solutions to nonlinear evolution and wave eqations", J. Phys. A: Math. Gen., 27, 2837-2855 (1994).
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Received: April 30, 2002.
Accepted for publication: July 25, 2002.
Recommended by Subject Editor Jorge L. Moiola.