EXACT TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED KURAMOTO-SIVASHINSKY EQUATION

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1 Journal of Mathematial Sienes: Advanes and Appliations Volume 3, 05, Pages -3 EXACT TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED KURAMOTO-SIVASHINSKY EQUATION JIAN YANG, XIAOJUAN LU and SHENGQIANG TANG Shool of Mathematis and Computing Siene Guilin University of Eletroni Tehnology Guilin, Guangxi, P. R. China Abstrat By using transformation u = a + au + a3u, the method of sine-osine and the method of dynamial bifuration theory of the differentiable dynamis, we study the generalized Kuramoto-Sivashinsky equation. It is shown that the generalized Kuramoto-Sivashinsky equation gives solitary wave solution, solitary patterns wave solution, and periodi wave solution. Under different parametri onditions, various suffiient onditions to guarantee the existene of the above solutions are given. All exat expliit parametri representations of the above waves are determined. 00 Mathematis Subjet Classifiation: 35C08, 37B55. Keywords and phrases: solitary wave, periodi wave, solitary patterns wave, method of sineosine, method of dynamial bifuration theory of the differentiable dynamis, generalized Kuramoto-Sivashinsky equation. This researh was supported by NNSF of China (Nos and 603) and Natural Siene Foundation of Guangxi (No. 03GXNSFAA0900). Reeived November 8, Sientifi Advanes Publishers

2 JIAN YANG et al.. Introdution In reent years, muh attention has been paid on the study of nonlinear wave equations (NLWE) in low dimensions. But there is little work on the high dimensional ones. It is well known that most of high dimensional NLWEs fail the onventional integrability tests, so the natural and important problem is that are there exat solutions with good properties for the high dimensional NLWEs? To solve this problem, people have made some attempts. In 990, Aspe and Depassier [] onsidered the following evolution equation of surfae waves in a onveting fluid: σr ut + λ uux + λuxxx + uxx + λ3uxxxx + λ4 ( uux ) = 0, 5 x (.) where σ is the Prandtl number and is a small parameter suh that the exess of the Rayleigh number above its ritial value is given by R. The oeffiients λi, i = 4, are funtions of the parameters of the problem. Subsripts denote derivatives with respet to the time t and horizontal oordinate x. This equation without the effet of instability, that is, with R 0, has been found reently to be the generi equation = that desribes the evolution of marginally diffusively stable wave trains []. A similar equation but with λ 4 = 0 and with instability and diffusion of the same order as dispersion arises in the study of fluid flow along an inlined plane. Numerial studies of this equation have shown that in the dispersion-dominated regime, for periodi boundary onditions, regular arrangements of soliton like pulses appear. The overall evolution of the system for a suffiiently long periodiity interval is apparently governed by the interation of these pulses. to σr For λ,, and = λ3 = λ4 = λ = σ =, 5 Equation (.) beomes u uu + u + σu + u + ( uu ) = 0. (.) t + x xx xxx xxxx x x

3 EXACT TRAVELLING WAVE SOLUTIONS 3 Equation (.) is alled generalized Kuramoto-Sivashinsky (KS) equations [3] and by using the tanh-funtion method, find expliit solitary travelling wave solutions of Equation (.) In this paper, by using transformation u ( ξ) = a + au( ξ) + a3u ( ξ), ξ = x t, (.3) the method of sine-osine and the method of dynamial bifuration theory of the differentiable dynamis to Equation (.), the existene of solitary wave solution, solitary patterns wave solution, and unountably infinite many smooth periodi wave solutions is obtained. Under different parametri onditions, various suffiient onditions to guarantee the existene of the above solutions are given. All exat expliit parametri representations of the above waves are determined. The two methods will be desribed briefly, where details an be found in [4-6] and the referenes therein. Let u( x, t) = u( ξ), ξ = x t, where is the wave speed. Then (.) beomes to ( 4 u + uu + u + σu + u ) + ( uu ) = 0. (.4) Integrating (.4) one and setting integration onstants as g, we have u + u + u + σu + u + ( uu ) =. g (.5) Substitute (.3) into Equation (.5), we have σa g + ( aσ ) u + ( a3σ) u + ( + a ) u + ( + a3 ) uu = 0. (.6) g We know the Equation (.6) was established, for a =, a =, a 3 =, σ =, and =. Therefore, we only onsider Equation (.3).

4 4 JIAN YANG et al. The paper is organized as follows. In Setion, the method of dynamial bifuration theory of the differentiable dynamis and the sine-osine method are briefly disussed. In Setion 3, represents exat analytial solutions of Equation (.3) by using the sine-osine method. In Setion 4, we disuss bifurations of phase portraits of Equation (.3). In Setion 5, all expliit parametri representations of travelling wave solutions are given. In Setion 6, the existene of smooth solitary wave solutions and unountable infinite many non-smooth periodi wave solutions of (.) is disussed. In the last setion, we onlude the paper and give some disussions.. Analysis of the Two Methods The sine-osine method and the method of dynamial bifuration theory of the differentiable dynamis have been applied for a wide variety of nonlinear problems. The main features of the two methods will be reviewed briefly. For both methods, we first use the wave variable PDE in two independent variables ξ = x t to arry a P ( u, u, u, u, u, u,, u, ) = 0, (.) t x tt xt xx xxx into an ODE Q ( u, u, u, u, ) = 0. (.) Equation (.) is then integrated as long as all terms ontain derivatives, where integration onstants are onsidered zeros... The sine-osine method admits the use of the solution in the form β π u ( ξ) = λ os ( µξ), µξ <, (.3) 0, otherwise, or in the form β λ sin ( ) ( µξ), µξ < π, u ξ = (.4) 0, otherwise,

5 EXACT TRAVELLING WAVE SOLUTIONS 5 where λ, µ, and β are parameters that will be determined. We therefore use and for (.4), we use β u ( ξ) = λ os ( µξ), ( ) os β u ξ = λ ( µξ ), (.5) β β ( u ) = µ β λ os ( µξ) + µ λ β( β ) os ( µξ), β u ( ξ) = λ sin ( µξ), ( ) sin β u ξ = λ ( µξ ), (.6) β β ( u ) = µ β λ sin ( µξ) + µ λ β( β ) sin ( µξ). We substitute (.3) or (.4) into the redued ordinary differential equation obtained above in (.), balane the terms of the osin funtions when (.3) is used, or balane the terms of the sine funtions when (.4) is used, and solving the resulting system of algebrai equations by using the omputerized symboli alulations to obtain all possible values of the parameters λ, µ, and β... The method of dynamial bifuration theory of the differentiable dynamis. The standard method of dynamial bifuration theory of the differentiable dynamis introdued in [4] and the referenes therein. 3. Using the Sine-Cosine Method Substituting (.3) into (.3) yields λ β µ β β β λ β os ( µξ) + λ µ β( β ) os ( µξ) = λ os ( µξ) + os ( µξ). (3.) (3.) is satisfied only if the following system of algebrai equations holds:

6 6 JIAN YANG et al. λ β 0, β = β, λ = λ β µ, λ µ β( β ) =. (3.) Solving the system (3.) gives β =, µ =, λ = 3. (3.3) The results (3.3) an be easily obtained if we also use the sine method (.4). Combining (3.3) with (.3) and (.4), the following periodi solution: 3 u ( x, y, t) = 3 se ( x t), 6 > 0, (3.4) 3 u ( x, y, t) = 3 s ( x t), 6 > 0 (3.5) are readily obtained. However, for < 0, we obtain the following solitary wave solutions and solitary patterns wave solutions: 3 u 3 ( x, y, t) = 3 seh ( x t), 6 < 0, (3.6) 3 u 4 ( x, y, t) = 3 sh ( x t), 6 < 0. (3.7) 4. Bifurations of Phase Portraits of (4.) Here we are onsidering a physial model where only bounded travelling waves are meaningful. So we only pay attention to the bounded solutions of (.4). The bifuration theory of dynamial systems (see [7-9]) plays an important role in our study. g For a =, a =, a3 =, σ =, and =, Equation (.3) beomes to g u ( ξ) = u( ξ) + u ( ξ). (4.)

7 EXACT TRAVELLING WAVE SOLUTIONS 7 Equation (4.) is equivalent to the two-dimensional systems as follows: du dξ = y, dy dξ = g u( ξ) + u ( ξ), (4.) with the first integral and g 3 y = u + u + h, (4.3) 3 g 3 H ( u, y) = y + u + u u = h. (4.4) 3 In this setion, we study all possible periodi annuluses defined by the vetor fields of (4.) when the parameters, g are varied. g Denote that = +, whih imply the relations in the (, g) -parameter plane L : g =. Thus, we have A ± (i) For = + g 0, > ( ( ± ), 0). there exist equilibrium points of (4.): (ii) For = + g 0, < there have no equilibrium points of (4.). Let M ( u e, y e ) be the oeffiient matrix of the linearized system of (4.) at an equilibrium point ( u y ). Then, we have e, e J( ui, 0) = det( M ( ui, 0) ) = ui. By the theory of planar dynamial systems, we know that for an equilibrium point of a planar integrable system, if J < 0, then the equilibrium point is a saddle point; if J > 0 and Trae ( M ( u i, y i )) = 0,

8 8 JIAN YANG et al. then it is a enter point; if J > 0 and Trae ( M ( u, y )) 4J( u, y ) > 0, then it is a node; if J = 0 and the index of the equilibrium point is 0, then it is a usp, otherwise, it is a high order equilibrium point. For the funtion defined by (4.4), we denote that i i i i h i ui = H( ui, 0) = [( 3g) ui g ], i =,. 3 We next use the above statements to onsider the bifurations of the phase portraits of (4.). In the (, g) -parameter plane, the urves L and = 0 partition it into 4 shown in Figure. Figure. The bifuration set of (4.) in (, g) -parameter plane.

9 EXACT TRAVELLING WAVE SOLUTIONS 9 Figure show the phase portraits of (4.) for > 0. (-) (, g) ( I) (-) (, g) ( II)

10 0 JIAN YANG et al. (-3) (, g) ( III) (-4) (, g) ( IV) Figure. The phase portraits of (4.) for > 0.

11 EXACT TRAVELLING WAVE SOLUTIONS 5. Exat Expliit Parametri Representations of Travelling wave Solutions of (.) g 5.. Suppose that (, g) ( I), h = H( A+, 0) = u+ ( + u+ u+ ). 3 In this ase, we have the phase portrait of (4.) shown in Figure (-). We see from (4.) that the arh urve onneting A ( ( + ), 0) has the algebrai equation + y = ( u u+ ) ( u u ), 3 k (5.) where u+ = ( + ), uk = u+ 3. Thus, by using the first equation of (4.) and (5.), we obtain the parametri representation of this arh as follows: u ( ) ( ) + uk u5 x, t = uk + u+ uk tanh ( x t). (5.) 3 Solution (5.) gives rise to a smooth solitary wave solution of valley type of (.). 5.. Suppose that ( g) ( I), h = H( A, 0), h ( h, ). In this, h ase, we have the phase portrait of (4.) shown in Figure (-). We see from (4.) that arh urve around as the enter A ( u, 0) has the algebrai equation y = ( u um )( u ul )( u um ), (5.3) 3 where u < < u < u. Thus, by using the first equation of (4.) and m 0 l M (5.3), we obtain the parametri representation of this arh as follows: u ( x, t) = 6 u ( ) ( ) M um ul um ul um um um ul um sn ( ), x + y t 3 um um, u ( ) M um ul um um um ul um sn ( ), x + y t 3 um um (5.4)

12 JIAN YANG et al. where sn ( x, k ) is the Jaobi ellipti funtions with the modulo k [0]. Solution (5.4) gives rise to a smooth periodi wave solutions of (.) Suppose that ( g) ( II), h = H( A, 0). In this ase, we have, + the phase portrait of (4.) shown in Figure (-). We see from (4.) that the arh urve onneting A ( ( + ), 0) has the algebrai equation + y = ( u u ) ( un u ), 3 + (5.5) where u = ( + ), un = 3 u. Thus, by using the first equation of + + (4.) and (5.5), we obtain the parametri representation of this arh as follows: un u+ u7 ( x, t) = un ( un u+ ) tanh ( x t). (5.6) 3 Solution (5.6) gives rise to a smooth solitary wave solution of peak type of (.) Suppose that ( g) ( II), h = H( A, 0), h ( h, ). In this, h ase, we have the phase portrait of (4.) shown in Figure (-). We see from (4.) that arh urve around as the enter A ( u, 0) has the algebrai equation y = ( u φm )( u φl )( u φm ), (5.7) 3 where φ < φ < < φ. Thus, by using the first equation of (4.) and m l 0 M (5.7), we obtain the parametri representation of this arh as follows: u ( x, t) = 8 φl ( φm φ ) ( ) M φm φl φm φm φm φl φm sn ( x + y t), 3 φm φm φm φ φ φ φ φ ( φ φ ) M m l m m l m sn ( ), x + y t 3 φm φm. (5.8)

13 EXACT TRAVELLING WAVE SOLUTIONS 3 Solution (5.8) gives rise to a smooth periodi wave solutions of (.). 6. Disussion In this paper, we used the sine-osine method and the method of dynamial bifuration theory of the differentiable dynamis to study the generalized Kuramoto-Sivashinsky equation. The methods provided solitary wave solutions, solitary patterns wave solutions, and periodi wave solutions. Moreover, the obtained results in this work learly demonstrate the reliability of the methods that were used. Referenes [] H. Aspe and M. C. Depassier, Evolution equation of surfae waves in a onveting fluid, Phys. Rev. A 4 (990), [] A. J. Bernoff, Slowly varying fully nonlinear wave trains in the Ginzburg-Landau equation, Physia D 30 (988), [3] E. J. Parkes and B. R. Duffy, An automated tanh-funtion method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun. 98 (996), [4] A. M. Wazwaz, A sine-osine method for handling nonlinear wave equations, Math. Comput. Model. 40 (004), [5] S. Q. Tang, Y. X. Xiao and Z. J. Wang, Travelling wave solutions for a lass of nonlinear fourth order variant of a generalized Camassa-Holm equation, Appl. Math. Comput. 0 (009), [6] J. B. Li, Singular Nonlinear Travelling Wave Equations: Bifurations and Exat Solutions, Siene Press, Beijing, 03. [7] S. N. Chow and J. K. Hale, Method of Bifuration Theory, Springer-Verlag, New York, 98. [8] J. Gukenheimer and P. J. Holmes, Nonlinear Osillations, Dynamial Systems and Bifurations of Vetor Fields, Springer-Verlag, New York, 983. [9] J. B. Li and H. H. Dai, On the Study of Singular Nonlinear Traveling Wave Equations: Dynamial System Approah, Siene Press, Beijing, 007. [0] P. F. Byrd and M. D. Friedman, Ellipti Integrals for Engineers and Sientists, Springer-Verlag, New York, 97. g