Symmetry Reduction of Two-Dimensional Damped Kuramoto Sivashinsky Equation

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1 Commun. Theor. Phys. 56 ( Vol. 56 No. 2 August Symmetry Reduction of Two-Dimensional Damped Kuramoto Sivashinsky Equation Mehdi Nadjafikhah and Fatemeh Ahangari School of Mathematics Iran University of Science and Technology Narmak Tehran Iran (Received October Abstract In this paper the problem of determining the largest possible set of symmetries for an important nonlinear dynamical system: the two-dimensional damped Kuramoto Sivashinsky ((2D DKS equation is studied. By applying the basic Lie symmetry method for the (2D DKS equation the classical Lie point symmetry operators are obtained. Also the optimal system of one-dimensional subalgebras of the equation is constructed. The Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. The nonclassical symmetries of the (2D DKS equation are also investigated. PACS numbers: Jr p Key words: two-dimensional damped Kuramoto Sivashinsky equation symmetry optimal system similarity solutions 1 Introduction The appearance of spatially and temporally coherent cellular states is a common feature of many driven nonequilibrium systems such as directional solidification electroconvection in liquid crystals and parametrically excited surface waves or Faraday waves (see [1] and references therein. The generic feature of these systems is that under certain circumstances which are typically large driving forces the periodic or regular structures become dynamically unstable and appear to be disordered or chaotic in both space and time. While it is unlikely that all the nonequilibrium systems mentioned above can be described by a single generic model the Damped Kuramoto Sivashinsky (DKS equation can be regarded as a crude model of directional solidification that encompasses the generic behavior which is: at small driving forces a primary instability leads to stationary periodic or lamellar patterns while at large driving forces secondary instabilities generate spatiotemporal chaos. This equation displays many of the secondary instabilities seen in directional solidification and provides a simple mathematical model for studying the selection of nonequilibrium states and the transition to spatiotemporal chaos from a coherent cellular state. [1 2] The two-dimensional damped Kuramoto Sivashinsky ((2D DKS model represents a nonlinear dynamical system which is defined in a two-dimensional space {x y} the dependent variable h = h(x y t satisfies a fourth order partial derivative equation which is of the form as follows: h ( t = 2 h 4 h + h 2 αh = x. (1 y To the limited extent that this model describes directional solidification the field h(x y t represents the position of the liquid-solid interface and the damping factor α is related to the driving force. As the distance from the primary instability increases (i.e. increasing α c α where α c = 1/4 is where the primary instability occurs eventually the system is driven chaotic. [1] The DKS equation applied for the surface morphology evolution during the erosion by ion beam sputtering under normal incidence. [3] Some important results concerning the DKS model are already known. One of the major observations is characterization of secondary instabilities that can destroy the organized structure ultimately driving the system to spatiotemporal chaos. In [24] a onedimensional analysis of these secondary instabilities was presented. The phenomenological and statistical studies in two-dimensional case can be found in [15]. In [6] a complete numerical method for studying the DKS equation is presented. The symmetry group method plays a fundamental role in the analysis of differential equations. The theory of Lie symmetry groups of differential equations was first developed by Sophus Lie [7] at the end of the nineteenth century which was called classical Lie method. Nowadays application of Lie transformations group theory for constructing the solutions of nonlinear partial differential equations (PDEs can be regarded as one of the most active fields of research in the theory of nonlinear PDEs and applications. The fact that symmetry reductions for many PDEs are unobtainable by applying the classical symmetry method motivated the creation of several generalizations of the classical Lie group method for symmetry reductions. The nonclassical symmetry method of reduction was devised originally by Bluman and Cole in 1969 [8] to find new exact solutions of the heat equation. The description of the method is presented in [9 10]. Many authors have used m nadjafikhah@iust.ac.ir fa ahangari@iust.ac.ir c 2011 Chinese Physical Society and IOP Publishing Ltd

2 212 Communications in Theoretical Physics Vol. 56 the nonclassical method to solve PDEs. In [11] Clarkson and Mansfield have proposed an algorithm for calculating the determining equations associated to the nonclassical method. A new procedure for finding nonclassical symmetries has been proposed by Bîlǎ and Niesen in [12]. Classical and nonclassical symmetries of nonlinear PDEs may be applied to reduce the number of independent variables of the PDEs. Particularly the PDEs can be reduced to ODES. The ODEs may also have symmetries which enable us to reduce the order of the equation and we can integrate to obtain exact solutions. In this paper we will try to analyze the problem of the symmetries of the (2D DKS equation. The structure of the present paper is as follows. In Sec. 2 a number of necessary preliminaries about Lie symmetry method is presented. In Sec. 3 using the basic Lie symmetry method the most general classical Lie point symmetry group of the (2D DKS equation is determined and some results yield from the structure of the Lie algebra of symmetries are given. Section 4 is devoted to obtaining the one-parameter subgroups and the most general group-invariant solutions of the (2D DKS equation. In Sec. 5 the optimal system of one-dimensional subalgebras is constructed. The Lie invariants and similarity reduced equations corresponding to the infinitesimal symmetries of (1 are obtained in Sec. 6. In Sec. 7 the nonclassical symmetries of the (2D DKS equation is investigated. 2 Method of Lie Symmetries In this section we recall the general procedure for determining symmetries for an arbitrary system of partial differential equations. [13 14] To begin let us consider a general system of partial differential equation containing p independent and q dependent variables is given as follows: µ (x u (n = 0 µ = 1... r (2 where u (n represents all the derivatives of u of all orders from 0 to n. We consider a one-parameter Lie group of transformations acting on the independent and dependent variables of the system (2: x i = x i + εξ i (x u + O(ε 2 ū α = u α + εη α (x u + O(ε 2 i = 1...p α = 1...q (3 where ξ i η α are the infinitesimals of the transformations for the independent and dependent variables respectively and ε is the transformation parameter. We consider the general vector field X as the infinitesimal generator associated with the above group of transformations: X = p ξ i (x u q x i + α=1 η α (x u u α. (4 A symmetry of a differential equation is a transformation which maps solutions of the equation to other solution. The invariance of the system (2 under the infinitesimal transformation leads to the invariance conditions (Theorem 6.5 of [14]: X (n [ µ (x u (n ] = 0 whenever µ (x u (n = 0 µ = 1...r (5 where X n is called the n th order prolongation of the infinitesimal generator given by q X (n = X + ηj α (x u(n u α (6 J α=1 J where J = (i 1... i k 1 i k p 1 k n and the sum is over all J s of order 0 < #J n. If #J = k the coefficient ηj α of / uα J will depend only on k-th and lower order derivatives of u and p p ηα(x J u (n = D J (η α ξ i u α i + ξ i u α Ji (7 where u α i : = uα / x i and u α Ji : = uα J / xi. 3 Classical Symmetries of (2D DKS Equation In this section we will perform Lie group method for Eq. (1. Firstly let us consider a one-parameter Lie group of infinitesimal transformation: x = x + εξ 1 (x y t h + O(ε 2 ȳ = y + εξ 2 (x y t h + O(ε 2 t = t + εξ 3 (x y t h + O(ε 2 h = h + εη(x y t h + O(ε 2 with a small parameter ε 1. The symmetry generator associated with the above group of transformations can be written as: X = ξ 1 (x y t h x + ξ2 (x y t h y + ξ3 (x y t h t + η(x y t h h. (8 The fourth prolongation of X is the vector field X (4 = X + η x + η y + η t + η 2x h x h y h t with coefficients h 2x + η xy + η xt + η 2y + η yt h xy h xt h 2y h yt + η 2t + η 3x + h 2t h 3x + η 4x + η 4y + (9 h 4x h 4y η J = D J (η 3 ξ i h α i + 3 ξ i h Ji (10 where J = (i 1... i k 1 i k 3 1 k 4 and the sum is over all J s of order 0 < #J 4. By Theorem (6.5 in [14] the invariance condition for the (2D DKS is given by the relation: X (4 [h t + h 2x + h 2y + h 4x + 2h (2x(2y + h 4y (h 2 x + h 2 y

3 No. 2 Communications in Theoretical Physics αh] = 0. (11 The invariance condition (11 is equivalent with the following equation: η t + η 2x + η 2y + η 4x + 2η (2x(2y + η 4y 2(η x h x + η y h y + αη = 0. (12 Substituting (10 into invariance condition (12 we are left with a polynomial equation involving the various derivatives of h(x y t whose coefficients are certain derivatives of ξ 1 ξ 2 ξ 3 and η. Since ξ 1 ξ 2 ξ 3 and η depend only on x y t h we can equate the individual coefficients to zero leading to the complete set of determining equations: ξ 2 tx = 0 ξ 2 y = 0 ξ 2 tt + αξ 2 t = 0 ξ 2 h = 0 ξ 3 x = 0 ξ 3 y = 0 ξ 1 y + ξ 2 x = 0 ξ 3 t = 0 ξ 3 h = 0 ξ1 h = 0 ξ2 xx = 0 η x ξ1 t = 0 η h = 0 η t + ηα = 0 η y ξ2 t = 0 ξ1 tt + αξ1 t = 0 ξ1 x = 0. By solving this system of PDEs we find that: Theorem 1 The Lie group of point symmetries of (2D DKS (1 has a Lie algebra generated by the vector fields X = ξ 1 x + ξ2 y + ξ3 t + η h where ξ 1 (x y t h = c 3 y + c 5 + c 6 e αt ξ 2 (x y t h = c 3 x + c 1 + c 2 e αt ξ 3 (x y t h = c 4 η(x y t h = 1 2( c2 αy + c 6 αx + 2c 7 e αt and c i i = are arbitrary constants. Corolary 2 Infinitesimal generators of every one parameter Lie group of point symmetries of the (2D DKS are: X 1 = x X 2 = y X 3 = t X 4 = e αt h X 5 = y x x y X 7 = e αt y + ( α 2 X 6 = e αt x + ( α 2 y e αt h. xe αt h The commutator table of symmetry generators of the (2D DKS is given in Table 1 where the entry in the i th row and j th column is defined as [X i X j ] = X i X j X j X i i j = Table 1 Commutation relations satisfied by infinitesimal generators. X 1 X 2 X 3 X 4 X 5 X 6 X 7 X X 2 ( α 2 X 4 0 X X 1 0 ( α 2 X 4 X αx 4 0 αx 6 αx 7 X αx X 5 X 2 X X 7 X 6 X 6 ( α 2 X 4 0 αx 6 0 X X 7 0 ( α 2 X 4 αx 7 0 X Group Invariant Solutions The equation (1 can be regarded as a submanifold of the jet space J 4 (R 3 R. So due to section (3.1 of [13] we can find the most general group of invariant solutions of equation (1. To obtain the group transformation which is generated by the infinitesimal generators X i = ξi 1 x+ξi 2 y +ξi 3 t+η i h for i = we need to solve the seven systems of first order ordinary differential equations d x(s = ξi 1 ds ( x(s ȳ(s t(s h(s x(0 = x dȳ(s = ξi 2 ( x(s ȳ(s t(s ds h(s ȳ(0 = x (13 d t(s ds = ξ 3 i ( x(s ȳ(s t(s h(s t(0 = t (14 d h(s = η i ( x(s ȳ(s t(s ds h(s h(0 = h i = exponentiating the infinitesimal symmetries of equation (1 we get the one parameter groups g k (s generated by X k for k = Theorem 3 The one-parameter groups g i (s : M M generated by the X i i = are given in the following table: g 1 (s : (x y t h (x + s y t h g 3 (s : (x y t h (x y t + s h g 2 (s : (x y t h (x y + s t h g 4 (s : (x y t h (x y t h + e αt s

4 214 Communications in Theoretical Physics Vol. 56 g 5 (s : (x y t h (xcos(s + y sin(s y cos(s xsin(s t h ( ( α g 6 (s : (x y t h x + s e αt y t e αt( s e αt + sx + h ( ( α g 7 (s : (x y t h x y + s e αt t e αt( s e αt + sy + h where entries give the transformed point exp(sx i (x y t h = ( x ȳ t h. Recall that in general to each one parameter subgroups of the full symmetry group of a system there will correspond a family of solutions called invariant solutions. Consequently we can state the following theorem: Theorem 4 If h = f(x y t is a solution of equation (1 so are the functions g 1 (s f(x y t = f(x + s y t g 2 (s f(x y t = f(x y + s t g 3 (s f(x y t = f(x y t + s g 4 (s f(x y t = f(x y t e αt s g 5 (s f(x y t = f(xcos(s + y sin(s y cos(s xsin(s t g 6 (s f(x y t = f(x + s e αt y t + 1 4( αs 2 e 2αt 2αsxe αt 2αs 2 e 2αt g 7 (s f(x y t = f((x y + s e αt t 1 4( αs 2 e 2αt + 2αsy e αt + 2αs 2 e 2αt. (15 5 Optimal System of One-Dimensional Subalgebras of (2D DKS Equation There is clearly an infinite number of one-dimensional subalgebras of the (2D DKS Lie algebra g each of which may be used to construct a special solution or class of solutions. So it is impossible to use all the one-dimensional subalgebras of the (2D DKS to construct invariant solutions. However a well-known standard procedure [15] allows us to classify all the one-dimensional subalgebras into subsets of conjugate subalgebras. This involves constructing the adjoint representation group which introduces a conjugate relation in the set of all one-dimensional subalgebras. In fact for one-dimensional subalgebras the classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation. If we take only one representative from each family of equivalent subalgebras an optimal set of subalgebras is created. The corresponding set of invariant solutions is then the minimal list from which we can get all other invariant solutions of one-dimensional subalgebras simply via transformations. Each X i i = of the basis symmetries generates an adjoint representation (or interior automorphism Ad(exp(εX i defined by the Lie series Ad(exp(ε X i X j = X j ε [X i X j ] + ε2 2 [X i [X i X j ]] (16 where [X i X j ] is the commutator for the Lie algebra ε is a parameter and i j = ([13] page 199. In Table 2 we give all the adjoint representations of the (2D DKS Lie group with the (i j the entry indicating Ad(exp(εX i X j. We can expect to simplify a given arbitrary element X = a 1 X 1 + a 2 X 2 + a 3 X 3 + a 4 X 4 + a 5 X 5 + a 6 X 6 + a 7 X 7 (17 of the (2D DKS Lie algebra g. Note that the elements of g can be represented by vectors a = (a 1... a 7 R 7 since each of them can be written in the form (17 for some constants a 1...a 7. Hence the adjoint action can be regarded as (in fact is a group of linear transformations of the vectors (a 1...a 7. Table 2 Adjoint representation generated by the basis symmetries of the (2D DKS Lie algebra. Ad X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 1 X 1 X 2 X 3 X 4 X 5 + εx 2 X 6 ( εα 2 X 4 X 7 X 2 X 1 X 2 X 3 X 4 X 5 εx 1 X 6 X 7 ( εα 2 X 4 X 3 X 1 X 2 X 3 e εα X 4 X 5 e εα X 6 e εα X 7 X 4 X 1 X 2 X 3 εαx 4 X 4 X 5 X 6 X 7 X 5 cos(εx 1 sin(εx 2 cos(εx 2 + sin(εx 1 X 3 X 4 X 5 cos(εx 6 sin(εx 7 cos(εx 7 + sin(εx 6 X 6 X 1 + ( εα 2 X 4 X 2 X 3 εαx 6 X 6 X 5 + εx 7 X 6 X 7 X 7 X 1 X 2 + ( εα 2 X 4 X 3 εαx 7 X 4 X 5 εx 6 X 6 X 7

5 No. 2 Communications in Theoretical Physics 215 Therefore we can state the following theorem: Theorem 5 A one-dimensional optimal system of the (2D DKS Lie algebra g is given by (i : X 4 + ax 5 (ii : ax 3 + bx 5 (iii : X 1 + ax 3 (iv : X 2 + ax 3 (v : ax 1 + bx 2 + X 6 (vi : ax 1 + bx 2 + X 7 where a b c R and a 0. Proof Fi s : g g defined by X Ad(exp(s i X i.x is a linear map for i = The matrix Mi s of Fi s i = with respect to basis {X 1... X 7 } is M1 s = 0 s sκ M2 s = s sκ M3 s = e sα e sα e sα sα M4 s = C S S C M5 s = C S S C Sκ sα 0 M6 s = s sκ sα M7 s = s respectively where S = sin s C = coss and κ = α/2. Let X = 7 a ix i then F s7 F s6 6 F s1 1 : X ( cos(s 5 a 1 sin(s 5 a 2 + (κs 6 cos(s 5 κs 7 sin(s 5 a 4 X1 + ( sin(s 5 a 1 + cos(s 5 a 2 + (κs 7 cos(s 5 + κs 6 sin(s 5 a 4 X2 + ( a 2 s 4 αa 4 s 6 αa 6 s 7 αa 7 X3 + e s3α a 4 X 4 + (( s 2 cos(s 5 + s 1 sin(s 5 a 1 + ( s 2 sin(s 5 + s 1 cos(s 5 a 2 + ( (s 2 sin(s 5 + s 1 cos(s 5 κs 7 + ( s 2 cos(s 5 + s 1 sin(s 5 κs 6 a4 + a 5 s 7 a 6 + s 6 a 7 X5 + ( κs 1 e s3α a 4 + e s3α cos(s 5 a 6 e s3α sin(s 5 a 7 X6 + ( κs 2 e s3α a 4 + e s3α sin(s 5 a 6 + e s3α cos(s 5 a 7 X7. (18 Now we can simplify X as follows: If a 4 0 then we can make the coefficients of X 1 X 2 X 3 X 6 and X 7 vanish by F s1 1 F s2 2 F s4 4 F s6 6 and F s7 7 ; By setting s 1 = a 6 /κa 4 s 2 = a 7 /κa 4 s 4 = a 3 /αa 4 s 6 = a 1 /κa 4 and s 7 = a 2 /κa 4 respectively. Scaling X if necessary we can assume that a 4 = 1. So X is reduced to the case (i. If a 4 = 0 and a 6 0 then we can make the coefficients of X 3 X 5 and X 7 vanish by F s6 6 F s5 s7 5 and F7 ; By setting s 6 = a 3 /αa 6 s 5 = arctan(a 7 /a 6 and s 7 = a 5 /a 6. Scaling X if necessary we can assume that a 6 = 1. So X is reduced to the case (v.

6 216 Communications in Theoretical Physics Vol. 56 If a 4 = a 6 = 0 and a 7 0 then we can make the coefficients of X 3 and X 5 vanish by F s6 6 and F s7 7 ; By setting s 6 = a 5 /a 7 and s 7 = a 3 /αa 7 respectively. Scaling X if necessary we can assume that a 7 = 1. So X is reduced to the case (vi. If a 4 = a 6 = a 7 = 0 and a 1 0 then we can make the coefficient of X 2 and X 5 vanish by F s5 5 and F s2 2 ; By setting s 5 = arctan(a 2 /a 1 and s 2 = a 5 /a 1. Scaling X if necessary we can assume that a 1 = 1. So X is reduced to the case (iii. If a 1 = a 4 = a 6 = a 7 = 0 and a 2 0 then we can make the coefficients of X 5 vanish by F s1 1 ; By setting s 1 = a 5 /a 2. Scaling X if necessary we can assume that a 2 = 1. So X is reduced to the case (iv. If a 1 = a 2 = a 4 = a 6 = a 7 = 0 then X is reduced to the case (ii. 6 Similarity Reduction of (2D DKS Equation The (2D DKS equation (1 is expressed in the coordinates (x y t h so we ought to search for this equation s form in specific coordinates in order to reduce it. Those coordinates will be constructed by looking for independent invariants (z w r corresponding to the infinitesimal symmetry generator. Hence by applying the chain rule the expression of the equation in the new coordinate leads to the reduced equation. We can now compute the invariants associated with the symmetry operators. They can be obtained by integrating the characteristic equations. For example for the operator H 5 := X 6 = e αt ( α x + xe αt 2 h this means: dx dy = e αt 0 = dt 0 = 2dh. (19 αxe αt The corresponding invariants are as follows: z = y w = t r = h 1 4 αx2. (20 Taking into account the last invariant we assume a similarity solution of the form: h = f(z w αx2 (21 and we substitute it into (1 to determine the form of the function f(z w: We obtain that f(z w has to be a solution of the following differential equation: f w + f zz + f 4z f 2 z + αf α = 0. (22 Having determined the infinitesimals the similarity solutions z j w j and h j are listed in Table 3. In Table 4 we list the reduced form of the (2D DKS equation corresponding to infinitesimal symmetries. 7 Nonclassical Symmetries of (2D DKS Equation In this section the so called nonclassical symmetry method [8] will be applied. The nonclassical symmetry method has become the focus of a lot of research and many applications to physically important partial differential equations as in [9 12]. For the nonclassical method we must add the invariance surface condition to the given equation and then apply the classical symmetry method. This can also be conveniently written as: X (4 1 1=0 2=0 = 0 (23 where X is defined in (8 and 1 and 2 are given as: 1 : = h t + h 2x + h 2y + h 4x + 2h (2x(2y + h 4y (h 2 x + h2 y αh 2 : = η ξ 1 h x ξ 2 h y ξ 3 h t. Without loss of generality we choose ξ 3 = 1. In this case using 2 we have: h t = η ξ 1 h x ξ 2 h y. (24 By reducing the initial system using the invariant surface condition (24 the equivalent form of (1 is obtained. Invariance of the equivalent equation under a Lie group of point transformations with infinitesimal generator (8 leads to the determining equations. Substituting ξ 3 = 1 into the determining equations we obtain the determining equations of the nonclassical symmetries of the original equation (1. Solving the system obtained by this procedure the only solutions we found were exactly the solution obtained through the classical symmetry approach (theorem 1. This means that no supplementary symmetries of nonclassical type are specific for the (2D DKS equation. 8 Conclusion In this paper by applying the criterion of invariance of the equation under the infinitesimal prolonged infinitesimal generators we find the most general Lie point symmetries group of the (2D (DKS equation. Also we have constructed the optimal system of one-dimensional subalgebras of (2D (DKS equation. The latter creates the preliminary classification of group invariant solutions. The Lie invariants and similarity reduced equations corresponding to infinitesimal symmetries are obtained. By applying the nonclassical symmetry method for the (2D DKS model we have concluded that the analyzed model does not admit supplementary nonclassical type symmetries. Using this procedure the classical Lie operators only are generated.

7 No. 2 Communications in Theoretical Physics 217 Table 3 Lie invariants and similarity solutions. J H j z j w j r j h j 1 X 1 y t h f(z w 2 X 2 x t h f(z w 3 X 3 x y h f(z w 4 X 4 x 2 + y 2 t h f(z w 5 X 6 y t h 1 4 αx2 f(z w αx2 6 X 7 x t h 1 4 αy2 f(z w αy2 7 X 4 + X 5 x 2 + y 2 t h e αt (arctan( x y f(z w + e αt (arctan( x y 8 X 3 + X 5 x 2 + y 2 t arctan( x y h f(z w 9 X 1 + X 3 y t x h f(z w 10 X 2 + X 3 x t y h f(z w 11 X 1 + X 2 + X 6 t (y xe αt +y e αt +1 h αx2 4 e αt +4 f(z w + αx2 4 e αt X 1 + X 2 + X 7 t y (1 + e αt x h xαe 2αt ((x 2y e αt + x f(z w 1 4 xαe 2αt ((x 2y e αt + x J Table 4 Reduced equations corresponding to infinitesimal symmetries. Similarity reduced equations 1 f w + f zz + f 4z f 2 z + αf = 0 2 f w + f zz + f 4z f 2 z + αf = 0 3 f zz + f ww + f 4z + 2f zzww + f 4w (f 2 z + f 2 w + αf = 0 4 f w + 4zf zz + 4f z + 16z 2 f 4z + 64zf zzz + 32f zz 4zf z + αf = 0 5 f w + f zz + f 4z f 2 z + αf α = 0 6 f w + f zz + f 4z f 2 z + αf α = z 3 f 4z + 4z 2 f zz + 32zf zz + 64z 2 f 3z 4z 2 f z + 4zf + αzf + zf w e 2αw = z 4 f 4z + 64z 3 f 3z + f 4w + αz 2 f + 4(z 2 z 3 f z + 4(z 3 + 8z 2 f zz + 8z 2 f zzww + (z + 4f ww + (z 2 zf w = 0 9 f w + f ww + f zz + f 4w + 2f zzww + f 4z (f 2 w + f2 z + αf = 0 10 f w + f zz + f ww + f 4z + 2f zzww + f 4w (f 2 z + f 2 w + αf = 0 11 f z + α 2( e αz +1 + αf + 2 e2αz +2 e αz +1 (1+ e αz 2 f ww + αf + (2 e2αz +2e αz +1 2 (1+ e αz 4 f 4w + 2 e2αz +2 e αz +1 (1+ e αz 2 fw 2 = 0 12 f z + (e 2αz + 2e αz + 2f 4w + (e 2αz + 2e αz + 2f ww + αw(e αz + e 2αz f w (e 2αz + 2e αz + 2f 2 w 1 4 α2 w 2 e 2αz 1 2 α(e αz + e 2αz + αf = 0 References [1] M. Paniconi and K.R. Elder Phys. Rev. E 56 ( [2] K.R. Elder J.D. Gunton and N. Goldenfeld Phys. Rev. E 56 ( [3] S. Facsko T. Bobek A. Stahl H. Kurz and T. Deksosy Phys. Rev. B 69 ( [4] C. Misbah and A. Valance Phys. Rev. E 49 ( [5] I. Daumont K. Kassner C. Misbah and A. Valance Phys. Rev. E 55 ( [6] L. Cueto-Felgueroso and J. Peraire Journal of Computational Physics 227 ( [7] S. Lie Arch. for Math. 6 ( translation by N.H. Ibragimov. [8] G.W. Bluman and J.D. Cole Journal of Mathematics and Mechanics 18 ( [9] P.A. Clarkson Chaos Solitons and Fractals 5 ( [10] D. Levi and P. Winternitz Journal of Physics A 22 ( [11] P.A. Clarkson and E.L. Mansfield SIAM Journal on Applied Mathematics 55 ( [12] N. Bîlǎ and J. Niesen Journal of Symbolic Computation 38 ( [13] P.J. Olver Applications of Lie Groups to Differential Equations Springer New York (1986. [14] P.J. Olver Equivalence Invariants and Symmetry Cambridge University Press Cambridge (1995. [15] L.V. Ovsiannikov Group Analysis of Differential Equations Academic Press New York (1982.