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## Latin American applied research

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*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 Li ^{1}, Guanrong Chen^{2} and Suchuan Zhao^{3}**

^{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

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):

*u _{t}* + β

*u*

^{α}

*u*+ γ

_{x}*u*

^{τ}

*u*+ δ

_{xx}*u*= 0, (1)

_{xxxx} 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'* + β*u*^{α}*u'* + γu^{τ}*u''* + δ*u''''* = 0 . (3)

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

*u'* = *au* + *bu ^{n}*, (4)

where *a*, *b*, *n* ∈ *R*, *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) = *v ^{h}*(ξ),

*v' = av + bv*(5)

^{n} 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 *c*_{0} 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 + nbv^{n-1})v', | (7) |

v'''' = [a^{3} + a^{2}bn(n^{2} + n + l)v^{n-1} + 3ab^{2}n^{2}(2n - l)v^{2n-2} + b^{3}n(2n - l)(3n - 2)v^{3n - 3}]v'. | (8) |

u' = hv^{h-l}v', | (9) |

u'' = [h2av^{h-1} + hb(n + h - l)v^{n+h-2}]v', | (10) |

u'''' = {h^{4}a^{3}v^{h-l} + ha^{2}b(n + h- l)[h^{2} + (n + h - 1)· ( n+2h - l)]v^{n+h-2} + 3hab^{2}(n + h - l)^{2}(2n + h-2)v^{2n+h-3} + hb^{3}(n + h - l)(2n + h - 2)(3n + h - 3)v^{3n+h-4}} v'. | (11) |

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

{(-ch + δh^{4}a^{3})v^{h-1} + βhv^{αh+h-1} + γh^{2}av^{τh+h-1} +γ hb(n + h - 1)v^{n+(τ+1)h-2} + δha^{2}b(n + h - l)· [3 h^{2} + 3h(n - l) + (n - l)^{2}]v^{n+h-2} + 3δhab^{2}(n + h - l)^{2}· (2 n + h - 2)v^{2n+h-3} + δhb^{3}(n + h - l)(2n + h - 2)· (3 n + h - 3)v^{3n+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* + *bu ^{n}*, so that (12) is reduced to

(δa^{3} - c) + βu^{α} + γau^{τ} + γbnu^{τ+n-1} + δ a^{2}bn(n^{2} + n + l)u^{n}^{-1} + 3δab^{2}n^{2}(2n - l)u^{2n-2} + δ b^{3}n(2n - l)(3n - 2)u^{3n-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 = 3*n* - 3

For this case,

δ*a*^{3} - *c* = 0, γ*a* + δ*a*^{2}*bn*(*n*^{2} + *n* + 1) = 0,

γ*bn* + 3δ*ab*^{2}*n*^{2}(2*n *- l) = 0, β + δ*b*^{3}*n*(2*n* - l)(3*n *- 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

(δa^{3}h^{3}-c)v^{h-1} + βv^{αh+h-1} + γahv^{τh+h-1} + γb(h - 1)· v^{τh+h-2} + δa^{2}b(h - l)(3h^{2} - 3h + l)v^{h-2}+3δab^{2}(h - l)^{2}· ( h - 2)v^{h-3} + δb^{3}(h - l)(h - 2)(h - 3)v^{h-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

7δ*a*^{2}*b* + γ*b* + β = 0, 8δ*a*^{3} - *c* + 2γ*a* = 0.

Thus,

(22) |

(3) *h* = 3

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

(27δa^{3} - c)v^{2} + βv^{3α+2} + 3γav^{3τ+2} + 2γbv^{3τ+1} + 38δ a^{2}bv + 12δab^{2} = 0, | (23) |

and only three cases exist, as follows.

Here,

27δ*a*^{3} - *c* = 0, 3γ*a* + 38δ*a*^{2}*b* = 0, β + 2γ*b* + 12β*ab*^{2} = 0.

Therefore,

(24) |

It is clear that

27δ*a*^{3} - *c* = 0, β + 3γ*a* + 38δ*a*^{2}*b* = 0,2γ*b* + 12δ*ab*^{2} = 0.

Hence,

(25) |

By the same reasoning, one has

27δ*a*^{3} - *c* + 3γ*a* = 0, 2γ*b* + 38δ*a*^{2}*b* = 0, β + 12δ*ab*^{2} = 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

δ*a*^{3}*h*^{3} - *c* = 0 , (δ*a*^{2}*b*(*h* - 1) (3*h*^{2} - 3*h* + 1) + γ*ah* = 0 ,

3δ*ab*^{2}(*h* - l)^{2}(*h* - 2) + γ*b*(*h* - 1) = 0 ,

δ*b*^{3}(*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 + δh^{3}a^{3})v^{h-l} + (δa^{2}b(h + l)(3h^{2} + 3h + 1)v ^{h}+ 3δ ab^{2}(h + l)^{2}(h + 2)v^{h+1} + δb^{3}(h + l)(h + 2)(h + 3)v^{h+2} +γ hav^{τh+h-1} + γb(h + l)v^{τh+h} + βv^{αh+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* - 8δ*a*^{3} - 2γ*a* = 0 , -7δ*a*^{2}*b* - γ*b* + β = 0,

so,

(30) |

Similarly, one has

-*c* - 8δ*a*^{3} = 0, -7δ*a*^{2}*b* - 2γ*a* = 0 , -γ*b* + β = 0,

so,

(31) |

(3) *h* = -3

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

For this sub-case,

* c *+ 27δ*a*^{3} + 3γ*a* = 0, 38δ*a*^{2}*b* + 2γ*b* = 0, 12γ*ab*^{2} - β = 0.

The solutions are

(32) |

Similarly,

*c* + 27δ*a*^{3} = 0, 38δ*a*^{2}*b* + 3γ*a* - β = 0, 12δ*ab*^{2} + 2γ*b* = 0.

The solutions are

(33) |

The following system is determined by the same reasoning:

*c* + 27δ*a*^{3} = 0, 38δ*a*^{2}*b* + 3γ*a* = 0, 12δ*ab*^{2} + 2γ*b* - 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* + δ*h*^{3}*a*^{3} = 0, δ*a*^{2}*b*(*h* + l)(3*h*^{2} + 3*h* + l) + γ*ha* = 0 ,

3δ*ab*^{2}(*h* + l)^{2}(*h* + 2) + γ*b*(*h* + 1) = 0 ,

δ*b*^{3}(*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 ]

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**Received: April 30, 2002. Accepted for publication: July 25, 2002. Recommended by Subject Editor Jorge L. Moiola.**