arXiv:1111.5178v1 [math.AP] 22 Nov 2011 Lie symmetries and conservation laws of the Hirota-Ramani equation Mehdi Nadjafikhah∗ Vahid Shirvani-Sh.† Abstract In this paper, Lie symmetry method is performed for the HirotaRamani (H-R) equation. We will find The symmetry group and optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group invariant solutions, symmetry reduction and nonclassical symmeries are investigated. Finally the conservation laws of the H-R equation are presented. Keywords. Lie symmetry, Invariant solutions, Nonclassical symmetries, Conservation laws, Hirota-Ramani equation. 1 Introduction In the present paper, we study the following equation ut − ux2 t + aux (1 − ut ) = 0, (1) where a 6= 0 and u(x, t) is the amplitude of relevant wave mode. This equation was introduced by Hirota and Ramani in [10]. Jie Ji obtained some travelling soliton solutions of this equation by using Exp-function method [12]. This equation is completely integrable by the inverse scattering method. Eq. (1) is studied in [10, 12, 9] where new kind of solutions were obtained. HirotaRamani equation is widely used in various branches of physics, such as plasma physics, fluid physics, quantum field theory. It also describes a variety of wave phenomena in plasma and solid state [10]. The theory of Lie symmetry groups of differential equations was developed by Sophus Lie [14], 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. Such Lie ∗ School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran. e-mail: m nadjafikhah@iust.ac.ir † Department of Mathematics, Islamic Azad University, Karaj Branch, Karaj 31485-313, Iran. e-mail: v.shirvani@kiau.ac.ir 1 groups are invertible point transformations of both the dependent and independent variables of the differential equations. The symmetry group methods provide an ultimate arsenal for analysis of differential equations and is of great importance to understand and to construct solutions of differential equations. Several applications of Lie groups in the theory of differential equations were discussed in the literature, the most important ones are: reduction of order of ordinary differential equations, construction of invariant solutions, mapping solutions to other solutions and the detection of linearizing transformations for many other applications of Lie symmetries see [17, 5, 4]. 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 [6], to find new exact solutions of the heat equation. The description of the method is presented in [7, 13]. Many authors have used the nonclassical method to solve PDEs. In [8] 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 Bı̂lǎ and Niesen in [1]. Many PDEs in the applied sciences and engineering are continuity equations which express conservation of mass, momentum, energy, or electric charge. Such equations occur in, e.g., fluid mechanics, particle and quantum physics, plasma physics, elasticity, gas dynamics, electromagnetism, magneto-hydro-dynamics, nonlinear optics, etc. In the study of PDEs, conservation laws are important for investigating integrability and linearization mappings and for establishing existence and uniqueness of solutions. They are also used in the analysis of stability and global behavior of solutions [2, 3, 19, 20]. This work is organized as follows. In section 2 we recall some results needed to construct Lie point symmetries of a given system of differential equations. In section 3, we give the general form of a infinitesimal generator admitted by Eq. (1) and find transformed solutions. In Section 4, we construct the optimal system of one-dimensional subalgebras. Lie invariants, similarity reduced equations and differential invariants corresponding to the infinitesimal symmetries of Eq. (1) are obtained in section 5 and 6. Section 7, is devoted to the nonclassical symmetries of the H-R model, symmetries generated when a supplementary condition, the invariance surface condition, is imposed. Finally in last section, the conservation laws of the Eq. (1) are obtained. 2 Method of Lie Symmetries In this section, we recall the general procedure for determining symmetries for any system of partial differential equations see [17, 15, 5, 4]. To begin, let us consider the general case of a nonlinear system E of partial differential equations 2 of order n in p independent and q dependent variables is given as a system of equations ∆ν (x, u(n) ) = 0, ν = 1, · · · , l, (2) involving x = (x1 , · · · , xp ), u = (u1 , · · · , uq ) and the derivatives of u with respect to x up to n, where u(n) represents all the derivatives of u of all orders from 0 to n. We consider a one-parameter Lie group of infinitesimal transformations acting on the independent and dependent variables of the system (2) x̃i = xi + sξ i (x, u) + O(s2 ), i = 1 · · · , p, ũj = uj + sϕj (x, u) + O(s2 ), j = 1 · · · , q, (3) where s is the parameter of the transformation and ξ i , η j are the infinitesimals of the transformations for the independent and dependent variables, respectively. The infinitesimal generator v associated with the above group of transformations can be written as v= p X ξ i (x, u)∂xi + q X ϕα (x, u)∂uα . (4) α=1 i=1 A symmetry of a differential equation is a transformation which maps solutions of the equation to other solutions. The invariance of the system (2) under the infinitesimal transformations leads to the invariance conditions (Theorem 2.36 of [17])   Pr(n) v ∆ν (x, u(n) ) = 0, ν = 1, · · · , l, whenever ∆ν (x, u(n) ) = 0, (5) where Pr(n) is called the nth order prolongation of the infinitesimal generator given by Pr(n) v = v + q X X ϕJα (x, u(n) )∂uαJ , (6) α=1 J where J = (j1 , · · · , jk ), 1 ≤ jk ≤ p, 1 ≤ k ≤ n and the sum is over all J’s of α order 0 < #J ≤ n. If #J = k, the coefficient ϕα J of ∂uJ will only depend on k-th and lower order derivatives of u, and ϕJα (x, u(n) ) = DJ (ϕα − p X ξ i uα i)+ i=1 p X ξ i uα J,i , (7) i=1 α i α α i where uα i := ∂u /∂x and uJ,i := ∂uJ /∂x . One of the most important properties of these infinitesimal symmetries is that they form a Lie algebra under the usual Lie bracket. 3 3 Lie symmetries of the H-R equation We consider the one parameter Lie group of infinitesimal transformations on (x1 = x, x2 = t, u1 = u), x̃ = x + sξ(x, t, u) + O(s2 ), t̃ ũ = x + sη(x, t, u) + O(s2 ), = x + sϕ(x, t, u) + O(s2 ), (8) where s is the group parameter and ξ 1 = ξ, ξ 2 = η and ϕ1 = ϕ are the infinitesimals of the transformations for the independent and dependent variables, respectively. The associated vector field is of the form: v = ξ(x, t, u)∂x + η(x, t, u)∂t + ϕ(x, t, u)∂u . (9) and, by (6) its third prolongation is Pr(3) v 2 2 v + ϕx ∂ux + ϕt ∂ut + ϕx ∂ux2 + ϕxt ∂uxt + ϕt ∂ut2 = 3 2 2 3 +ϕx ∂ux3 + ϕx t ∂ux2 t + ϕxt ∂uxt2 + ϕt ∂ut3 . (10) where, for instance by (7) we have ϕx ϕt t3 ϕ = Dx (ϕ − ξ ux − η ut ) + ξ ux2 + η uxt , = Dt (ϕ − ξ ux − η ut ) + ξ uxt + η ut2 , .. . = Dt3 (ϕ (11) − ξ ux − η ut ) + ξ uxt3 + η ut4 , where Dx and Dt are the total derivatives with respect to x and t respectively. By (5) the vector field v generates a one parameter symmetry group of Eq. (1) if and only if Pr(3) v[ut − ux2 t + aux (1 − ut )] = 0, (12) whenever ut − ux2 t + aux (1 − ut ) = 0. The condition (12) is equivalent to (1 − aux )ϕt + a(1 − ut )ϕx − ϕx 2 t = 0, (13) whenever ut − ux2 t + aux (1 − ut ) = 0. Substituting (11) into (13), and equating the coefficients of the various monomials in partial derivatives with respect to x and various power of u, we can 4 find the determining equations for the symmetry group of the Eq. (1). Solving this equations, we get the following forms of the coefficient functions ξ= −x c1 + c3 , 3 η = c1 t + c2 , ϕ=( 2t u 2x + − )c1 + c4 . 3 3 3a (14) where c1 , c2 , c3 and c4 are arbitrary constant. Thus, the Lie algebra g of infinitesimal symmetry of the Eq. (1) is spanned bye the four vector fields v1 = ∂x , v2 = ∂t , v3 = 1 ∂u , a v4 = 3t∂t − x∂x + (2t + u − 2x )∂u . a (15) The commutation relations between these vector fields are given in the Table 1. The Lie algebra g is solvable, because if g(1) = hvi , [vi , vj ]i = [g, g], we have g(1) = hv1 , · · · v4 i, and g( 2) = [g(1) , g(1) ] = h−v1 − 2v3 , 3v2 + 2av3 , v3 i, so, we have a chain of ideals g(1) ⊃ g(2) ⊃ 0. Table 1: The commutator table [vi , vj ] v1 v2 v3 v4 v1 0 0 0 v1 + 2v3 v2 0 0 0 −3v2 − 2av3 v3 0 0 0 -v3 v4 −v1 − 2v3 3v2 + 2av3 v3 0 To obtain the group transformation which is generated by the infinitesimal generators vi for i = 1, 2, 3, 4 we need to solve the three systems of first order ordinary differential equations dx̃(s) ds dt̃(s) ds dũ(s) ds = ξi (x̃(s), t̃(s), ũ(s)), x̃(0) = x, = ηi (x̃(s), t̃(s), ũ(s)), t̃(0) = t, = ϕi (x̃(s), t̃(s), ũ(s)), ũ(0) = u. i = 1, · · · , 4 (16) Exponentiating the infinitesimal symmetries of Eq. (1), we get the one-parameter groups Gi (s) generated by vi for i = 1, · · · , 4 G1 : (t, x, u) G2 : (t, x, u) 7−→ (x + s, t, u), 7−→ (x, t + s, u), G3 : (t, x, u) 7−→ (x, t, u + s/a), G4 : (t, x, u) 7−→ (xe−s , te3s , te3s + Consequently, 5 (17) x x −s e + (u − t − )es ). a a Theorem 3.1 4 If u = f (x, t) is a solution of Eq. (1), so are the functions G1 (s) · f (x, t) = f (x − s, t), G2 (s) · f (x, t) = G3 (s) · f (x, t) = f (x, t − s), f (x, t) + s/a, G4 (s) · f (x, t) = es f (xes , te−3s ) + (18) x (1 − e2s ) + t(1 − e−2s ). a Optimal system of the H-R equation In general, to each s-parameter subgroup H of the full symmetry group G of a system of differential equations in p > s independent variables, there will correspond a family of group-invariant solutions. Since there are almost always an infinite number of such subgroups, it is not usually feasible to list all possible group-invariant solutions to the system. We need an effective, systematic means of classifying these solutions, leading to an ”optimal system” of group-invariant solutions from which every other such solution can be derived.[17] Definition 4.1 Let G be a Lie group with Lie algebra g. An optimal system of s−parameter subgroups is a list of conjugacy inequivalent s−parameter subalgebras with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s−parameter subalgebras forms an optimal system if every s−parameter subalgebra of g is equivalent to a unique member of the list under some element of the adjoint representation: h = Ad(g(h)).[17] Theorem 4.2 Let H and H be connected s-dimensional Lie subgroups of the Lie group G with corresponding Lie subalgebras h and h of the Lie algebra g of G. Then H=gHg −1 are conjugate subgroups if and only if h = Ad(g(h)) are conjugate subalgebras. [17] By theorem (4.2), the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. For onedimensional subalgebras, this classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation, since each onedimensional subalgebra is determined by nonzero vector in g. This problem is attacked by the naı̈ve approach of taking a general element V in g and subjecting it to various adjoint transformation so as to ”simplify” it as much as possible. Thus we will deal with the construction of the optimal system of subalgebras of g. 6 To compute the adjoint representation, we use the Lie series Ad(exp(εvi )vj ) = vj − ε[vi , vj ] + ε2 [vi , [vi , vj ]] − · · · , 2 (19) where [vi , vj ] is the commutator for the Lie algebra, ε is a parameter, and i, j = 1, 2, 3, 4. Then we have the Table 2. Table 2: Adjoint representation table of the infinitesimal generators vi Ad v1 v2 v3 v4 v1 v1 v1 v1 v1 − ε(v1 + 2v3 ) v2 v2 v2 v2 v2 + ε(3v2 + 2av3 ) v3 v3 v3 v3 v3 + εv3 v4 v4 + ε(v1 + 2v3 ) v4 − ε(3v2 + 2av3 ) v4 − εv3 v4 Theorem 4.3 An optimal system of one-dimensional Lie algebras of the H-R equation is provided by 1) v4 , 2) αv1 + βv2 + v3 , 3) αv1 + v2 , 4) v1 Proof: Consider the symmetry algebra g of the equation (1) whose adjoint representation was determined in table 2 and let Fis : g → g defined by v 7→ Ad(exp(εvi )v) is a linear map, for i = 1, · · · , 4. The matrices Miε of Fiε , i = 1, · · · , 4, with respect to basis {v1 , · · · , v4 } are       1 0  0 1 M1ε =  0 0 0 0 0 0 1 0 −ε 1 0  0  , M2ε =  −2ε  0 1 0 0 1 0 0 0 0 1 0 0 1 3ε  0  , M3ε =  2aε  0 0 1 eε 0  M4ε =  ε e − e−ε 0  0 1 0 0 0 −3 ε e
ae−ε e−2 ε − 1 0 0 0 1 0 0 0  , ε  1 0 0 0 0  0  1 e−ε 0  P Let V = 4i=1 ai vi is a nonzero vector field in g. We will simplify as many of the coefficients ai as possible by acting these matrices on a vector field V alternatively. Suppose first that a4 6= 0, scaling V if necessary we can assume that a4 = 1, then we can make the coefficients of v1 , v2 and v3 vanish by M1ε and M2ε . And V reduced to case 1. If a4 = 0 and a3 6= 0, then we can not make vanish the coefficients of v1 and v2 by acting any matrices Miε . Scaling V if necessary, we can assume that a3 = 1 and V reduced to case 2. 7 If a4 = a3 = 0 and a2 6= 0, then we can not make vanish the coefficient of v1 . Scaling V if necessary, we can assume that a2 = 1 and V reduced to case 3. The remaining one-dimensional subalgebras are spanned by vectors of the above form with a4 = a3 = a2 = 0. If a1 6= 0, we scale to make a1 = 1, and V reduced to case 3. ✷ 5 Symmetry reduction of the H-R equation Lie-group method is applicable to both linear and non-linear partial differential equations, which leads to similarity variables that may be used to reduce the number of independent variables in partial differential equations. By determining the transformation group under which a given partial differential equation is invariant, we can obtain information about the invariants and symmetries of that equation. Symmetry group method will be applied to the (1) to be connected directly to some order differential equations. To do this, a particular linear combinations of infinitesimals are considered and their corresponding invariants are determined. The equation (1) is expressed in the coordinates (x, t, u), so to reduce this equation is to search for its form in specific coordinates. Those coordinates will be constructed by searching for independent invariants (y, v) corresponding to the infinitesimal generator. So using the chain rule, the expression of the equation in the new coordinate allows us to the reduced equation. Here we will obtain some invariant solutions with respect to symmetries. First we obtain the similarity variables for each term of the Lie algebra g, then we use this method to reduced the PDE and find the invariant solutions. 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 v2 + v3 = ∂t + a1 ∂u characteristic equation is dt a du dx = = . 0 1 1 (20) The corresponding invariants are y = x, v = u − at therefore, a solution of our equation in this case is u = v(y) + at . The derivatives of u are given in terms of y and v as ux = vy , ux2 = vyy , ux2 t = 0, ut = 1 . a (21) Substituting (21) into the Eq. (1), we obtain the ordinary differential equation −y +c. Consequently, (a−1)vy +1/a = 0, the solution of this equation is v = a(a−1) we obtain that x t u= + + c. (22) a(1 − a) a All results are coming in the tables 3 and 4. 8 Table 3: Reduction of Eq. (1) operator v1 v2 v4 v1 + av3 v2 + v3 v1 + v2 y t x xt1/3 t x x−t v u u (u − 2x/a)t−1/3 + t2/3 u−x u − t/a u u v(y) v(y) v(y)t1/3 + 2x/a − t v(y) + x v(y) + t/a v(y)/a Table 4: Reduced equations corresponding to infinitesimal symmetries operator v1 v2 v4 v1 + av3 v2 + v3 v1 + v2 6 similarity reduced equations vy = 0 avy = 0 t−2/3 v + (xt−1/3 − 3)vy + xt−2/3 vyy + (at2/3 vy + 2)(6 − t−2/3 v − t−1/3 vy ) = 3 (1 − a)vy + a = 0 (a − 1)vy + 1/a = 0 −vy + vyyy + avy (1 + vy ) = 0 Characterization of differential invariants Differential invariants help us to find general systems of differential equations which admit a prescribed symmetry group. One say, if G is a symmetry group for a system of PDEs with functionally differential invariants, then, the system can be rewritten in terms of differential invariants. For finding the differential invariants of the Eq. (1) up to order 2, we should solve the following systems of PDEs: ∂I 1 ∂I ∂I ∂I 2x ∂I ∂I , , , 3t −x + (2t + u − ) , (23) ∂x ∂t a ∂u ∂t ∂x a ∂u where I is a smooth function of (x, t, u), ∂I1 , ∂x ∂I1 , ∂t 1 ∂I1 , a ∂u 3t ∂I1 ∂I1 ∂I1 , −x + · · · + (2 − 2ut ) ∂t ∂x ∂ut (24) where I1 is a smooth function of (x, t, u, ux , ut ), ∂I2 , ∂x ∂I2 , ∂t 1 ∂I2 , a ∂u 3t ∂I2 ∂I2 ∂I2 ∂I2 − 5utt , −x + · · · − uxt ∂t ∂x ∂uxt ∂utt (25) where I2 is a smooth function of (x, t, u, ux , ut , uxx, uxt , utt ). The solutions of PDEs systems (23),(24) and (25) coming in table 5, where * and ** are refer to ordinary invariants and first order differential invariants respectively. 9 Table 5: differential invariants vector field v1 v2 v3 v4 7 ordinary invariant t, u x, u x, t tx3 , ( −x − t + u)x a 1st order diff. invariant ∗, ux , ut ∗, ux , ut ∗, ux , ut ∗, x2 ux − a1 , utx−1 2 2nd order diff. invariant ∗, ∗∗, uxx , uxt , utt ∗, ∗∗, uxx , uxt , utt ∗, ∗∗, uxx , uxt , utt ∗, ∗∗, x3 uxx , uxxt , uxtt 5 Nonclassical symmetries of the H-R equation In this section we would like to apply the nonclassical method to the H-R equation. The graph of a solution uα = f α (x1 , · · · , xn ), α = 1, · · · , q (26) to the system (2) defines an p-dimensional submanifold Γf ⊂ Rp × Rq of the space of independent and dependent variables. The solution will be invariant under the one-parameter subgroup generated by vector (4) if and only if Γf is an invariant submanifold of this group. By applying the well known criterion of invariance of a submanifold under a vector field we get that (26) is invariant under vector (4) if and only if f satisfies the first order system EQ of partial differential equations Qα (x, u, u(1) ) = ϕ(α) (x, u) − p X ξ i (x, u)uα i = 0, α = 1, · · · , q (27) i=1 known as the invariant surface conditions. The q-tuple Q = (Q1 , · · · , Qq ) is known as the characteristic of the vector field (4). In what follows, the n-th (n) prolongation of the invariant surface conditions (27) will be denoted by EQ , which is a n-th order system of partial differential equations obtained by appending to (27) its partial derivatives with respect to the independent variables of orders j ≤ n − 1. For the system (2), (27) to be compatible, the n-th prolongation Pr(n) v of (n) the vector field v must be tangent to the intersection E ∩ EQ Pr(n) v(∆ν )|E∩E (n) = 0, ν = 1, · · · , l. (28) Q If the equations (28) are satisfied, then the vector field (28) is called a nonclassical infinitesimal symmetry of the system (2). The relations (28) are generalizations of the relations (5) for the vector fields of the infinitesimal classical symmetries. A similar procedure is applicable to the case of the nonclassical infinitesimal symmetries with an evident difference that in general one has fewer 10 determining equations than in the classical case. Therefore, we expect that nonclassical symmetries are much more numerous than classical ones, since any classical symmetry is clearly a nonclassical one. The important feature of determining equations for nonclassical symmetries is that they are nonlinear, this implies that the space of nonclassical symmetries does not, in general, form a vector space. For more theoretical background see [18, 1]. Consider the system E of second order equations ut − vt + aux (1 − ut ) = 0, uxx − v = 0 (29) obtained from the H-R equation. If we assume that the coefficient of ∂t of the vector field (4) does not identically equal zero, then for the vector field v = ξ(x, t, u, v)∂x + ∂t + ϕ(x, t, u, v)∂u + ψ(x, t, u, v)∂v (30) the invariant surface conditions are ut + ξux = ϕ, vt + ξvx = ψ (31) The equations (28) take the forms (2 (ut − 1/2)u2xa − ut (ux − vx ))ξu − ut ψu − ψt +((1 − 2 ut )aux + ut )ϕu + (ux (1 − ut )a − ut )ψv +(−u3x (ut − 1)a2 + 2 (ut − 1/2)u2xa − ut (ux − vx ))ξv +(u2x (ut − 1)a2 + ((1 − 2 ut )ux − vx (ut − 1))a + ut )ϕv +(1 − aux )ϕt − a(ut − 1)ϕx + ux (ut − 1)aξx + (u2x a − ux + vx )ξt = 0, (32) and −ψ − ux ξxx − 2 u2x ξxu + u2x ϕuu − u3x ξuu + uxx ϕu − 2 uxxξx + vx2 ϕvv +vxx ϕv + 2 uxϕxu + 2 vx ϕxv − 2 ux vx ξxv + 2 ux vx ϕuv − 2 u2x vx ξuv −3 uxxux ξu − 2 uxxvx ξv − ux vx2 ξvv − vxx ux ξv + ϕxx = 0. (33) After inserting ψ and its derivatives, as determined by the equation (33), in to (32) and substituting vx = uxxx, vxx = uxxxx, and equating the coefficients of the various monomials in partial derivatives with respect to x and various power of u, we can find the determining equations. Solving this equations, we get four Algebraic equations equal to zero. This means that no supplementary symmetries, of non-classical type, are specific for our model. Now assume that the coefficient of ∂t in (30) equals zero and try to find the infinitesimal nonclassical symmetries of the form v = ∂x + ϕ(x, t, u, v)∂u + ψ(x, t, u, v)∂v (34) for which the invariant surface conditions are the following ones ux = ϕ, vx = ψ 11 (35) Relations (28) lead to the system of equations for the functions ϕ and ψ (u2x (ut − 1)a2 + ((−vx − 2 ux)ut + vx + ux )a + ut )ϕv +((ux − 2 ux ut )a + ut )ϕu + (ux (ut − 1)a − ut )ψv +(1 − aux )ϕt − a(ut − 1)ϕx − ψt − ut ψu = 0, (36) −ψ + ϕxx + 2 uxϕxu + 2 ϕxv vx + u2x ϕuu + 2 ux vx ϕuv +uxx ϕu + vx2 ϕvv + vxx ϕv = 0. (37) and Similar the previous case, we can find determining equations. Solving this equations, we get the following form of the coefficient functions ϕ = c1 u − c1 x + t(1 − 2ut )c1 + c2 a (38) where c1 and c2 are arbitrary constant. So the system (29) admits the classical symmetry v1 and nonclassical symmetry v2 : v1 = ∂x + ∂u , 8 v2 = ∂x − ( x − u − t + 2 tut )∂u a (39) Conservation laws of the H-R equation Many methods for dealing with the conservation laws are derived, such as the method based on the Noether’s theorem, the multiplier method, by the relationship between the conserved vector of a PDE and the Lie-Bcklund symmetry generators of the PDE, the direct method, etc.[17, 2, 3, 19]. Now, we derive the conservation laws from the multiplier method. Definition 8.1 A local conservation law of the PDE system (2) is a divergence expression Di Φi [u] = D1 Φ1 [u] + · · · + Dn Φn [u] = 0 (40) holding for all solutions of the system (2). In (40), Φi [u] = Φi (x, u, ∂u , · · · , ∂ur ), i = 1, · · · , n, are called fluxes of the conservation law, and the highest-order derivative (r) present in the fluxes Φi [u] is called the order of a conservation law. [3] In particular, a set of multipliers {Λν [U ]}lν=1 = {Λν (x, U, ∂U , · · · , ∂Ur )}lν=1 yields a divergence expression for the system ∆ν (x, u(n) ) such that if the identity Λν [U ]∆ν [U ] ≡ Di Φi [U ] (41) holds identically for arbitrary functions U (x). Then on the solutions U (x) = u(x) of the system (2), if Λν [U ] is non-singular, one has local conservation law Λν [u]∆ν [u] = Di Φi [u] = 0. 12 Definition 8.2 The Euler operator with respect to U j is the operator defined by EU j = ∂ ∂ ∂ + ··· − Di + · · · + (−1)s Di1 · · · Dis ∂U j ∂U j ∂Uij ···i 1 (42) s for j = 1, · · · , q. [3] Theorem 8.3 The equations EU j F (x, U, ∂U , · · · , ∂Us ) ≡ 0, j = 1, · · · , q hold for arbitrary U (x) if and only if F (x, U, ∂U , · · · , ∂Us ) ≡ Di Ψi (x, U, ∂U , · · · , ∂Us−1 ) holds for some functions Ψi (x, U, ∂U , · · · , ∂Us−1 ), i = 1, · · · q. [3] ✷ Theorem 8.4 A set of non-singular local multipliers {Λν (x, U, ∂U , · · · , ∂Ur )}lν=1 yields a local conservation law for the system ∆ν (x, u(n) ) if and only if the set of identities EU j (Λν (x, U, ∂U , · · · , ∂Ur )∆ν (x, u(n) )) ≡ 0, j = 1, · · · q, holds for arbitrary functions U (x). [3] (43) ✷ The set of equations (43) yields the set of linear determining equations to find all sets of local conservation law multipliers of the system (2). Now, we seek all local conservation law multipliers of the form Λ = ξ(x, t, u, ux , ut , uxx , uxt , utt ) of the equation (1). The determining equations (43) become EU [ξ(x, t, U, Ux , Ut , Uxx , Uxt , Utt )(Ut − Ux2 t + aUx (1 − Ut ))] ≡ 0, (44) where U (x, t) are arbitrary function. Equation (44) split with respect to third order derivatives of U to yield the determining PDE system whose solutions are the sets of local multipliers of all nontrivial local conservation laws of second order of H-R equation. The solution of the determining system (44) given by 1 c2 (2 tUtt + Ut − 1) + c3 utt , (45) 2 where c1 , c2 and c3 are arbitrary constant. So local multipliers given by 1 1 1) ξ = Uxx , 2) ξ = Utt , 3) ξ = tUtt + Ut − , (46) 2 2 Each of the local multipliers ξ determines a nontrivial two-order local conservation law Dt Ψ + Dx Φ = 0 with the characteristic form c1 Uxx + Dt Ψ + Dx Φ ≡ ξ(Ut − Ux2 t + aUx (1 − Ut )), (47) To calculate the conserved quantities Ψ and Φ, we need to invert the total divergence operator. This requires the integration (by parts) of an expression in multi-dimensions involving arbitrary functions and its derivatives, which is a difficult and cumbersome task. The homotopy operator [20] is a powerful algorithmic tool (explicit formula) that originates from homological algebra and variational bi-complexes. 13 Definition 8.5 The 2-dimensional homotopy operator is a vector operator  (t) (x) with two components, Hu(x,t) f, Hu(x,t) f , where (x) Hu(x,t) f = Z 1 0 q X  (x) Iuj f j=1 (x) The x-integrand, Iuj dλ [λu] λ and (t) Hu(x,t) f Z = 0 1 q X  dλ (t) Iuj f [λu] . (48) λ j=1 f, is given by (x,t) j j M1 M2  k1 −1 k2 X X X X (x) Iuj f = k1 =1 k2 =0 B (x) ujxi1 ti2 (−Dx )k1 −i1 −1 (−Dt )k2 −i2 i1 =0 i2 =0  ∂f ∂ujxk1 tk2 , (49) where M1j , M2j are the order of f in u to x and t respectively and combinatorial coefficient    B (x) = B(i1 , i2 , k1 , k2 ) = (t) Similarly, the t-integrand, Iuj i1 + i2 i1 k 1 + k 2 − i1 − i2 − 1 k 1 − i1 − 1   k1 + k2 k1 . (50) f, is defined as (x,t) j (t) Iuj f = j M1 M2  k1 k2 −1 X X X X k1 =0 k2 =1 B (t) ujxi1 ti2 (−Dx )k1 −i1 (−Dt )k2 −i2 −1 i1 =0 i2 =0  ∂f ∂ujxk1 tk2 , (51) where B (t) (i2 , i1 , k2 , k1 ). For instance we apply homotopy operator to find conserved quantities Ψ and Φ which yield of multiplier ξ = utt . We have f = utt (ut − ux2 t + aux (1 − ut )), (52) the integrands (49) and (51) are (x) Iuj f = auut2 − auut ut2 − 23 uuxt3 + 13 ut uxt2 + 31 ux ut3 − 32 uxt ut2 , (t) Iuj f = u2t − ut ux2 t + aux ut − aux u2t + 23 uux2 t2 − auuxt + auuxt ut + 31 ux uxt2 − 31 ux2 ut2 , (53) apply (48) to the integrands (53), therefore (x) Hu(x,t) f = 12 auut2 − 31 auut ut2 − 31 uuxt3 + 16 ut uxt2 + 61 ux ut3 − 31 uxt ut2 , (t) Hu(x,t) f = 12 u2t − 21 ut ux2 t + 21 aux ut − 31 aux u2t + 13 uux2 t2 − 12 auuxt + 31 auuxt ut + 16 ux uxt2 − 61 ux2 ut2 , 14 (54) so, we have the first conservation low of the H-R equation respect to multiplier ξ = utt
Dx 12 auut2 − 31 auut ut2 − 31 uuxt3 + 16 ut uxt2 + 61 ux ut3 − 13 uxt ut2 +Dt 1 2 1 1 1 2 u2t + 13 uux2 t2 2 ut − 2 ut ux t + 2 aux ut − 3 aux
1 1 1 + 3 auuxt ut + 6 ux uxt2 − 6 ux2 ut2 = 0. − 21 auuxt (55) Similarly, conservation law respect to multiplier ξ = uxx is Dx 21 ux ut + 21 au2x − 31 au2x ut − 21
uuxt + 13 auux uxt ) +Dt ( 21 uuxx − 13 auux ut2 − 21 uxx = 0. and conservation law respect to multiplier ξ = tutt + 21 ut − 1 2 (56) is Dx − 21 au + 12 auut + 12 atuut2 − 31 atuut ut2 − 61 auu2t − 12 uuxt2 1 1 1 1 1 2 2 3 − 13 tuuxt3 + 6 tut uxt − 12 ut uxt + 4 ux ut + 6 tux ut + 3 uxt
− 13 tuxt ut2 = 0. 1 +Dt − 12 u + 16 uux2 t + 31 tuux2 t2 + 16 tux uxt2 + 12 ux uxt + 61 ux2 1 − 12 ux2 ut − 12 atuuxt + 31 atuut uxt
+ 21 tu2t − 12 tut ux2 t − 61 tux2 ut2 + 21 atux ut − 31 atux u2t = 0. (57) References [1] N. Bı̂lǎ, J. Niesen, On a new procedure for finding nonclassical symmetries, Journal of Symbolic Computation, 38 (2004) 1523-1533. [2] G.W. Bluman, New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl. 322 (2006), 233250. [3] G.W. Bluman, A.F Cheviacov, S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, vol. 168 Springer, New York, 2010. [4] G.W. Bluman, J.D. Cole, Similarity Methods for Differential Equations, Applied Mathematical Sciences, No.13, Springer, New York, 1974. [5] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989. [6] G.W. Bluman, JD. Cole, The general similarity solutions of the heat equation, Journal of Mathematics and Mechanics, 18(1969) 1025-1042. [7] PA. Clarkson, Nonclassical symmetry reductions of the Boussinesq equation, Chaos, Solitons, Fractals 5(1995) 2261-2301. 15 [8] PA. Clarkson, EL. Mansfield, Algorithms for the nonclassical method of symmetry reductions, SIAM Journal on Applied Mathematics 55(1994) 1693-1719. [9] Wu Guo-cheng and Xia Tie-cheng, A new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations, Phys.Lett. A 372 (2008) 604-609. [10] H. Hirota and A. Ramani, The Miura transformations of Kaups equation and of Mikhailovs equation, Phys. Lett. A 76 (1980) 95-96. [11] N.H. Ibragimov, (Editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, Symmetries, Exact Solutions and Conservation Laws, CRC Press, Boca Raton, 1994. [12] Jie Ji, A new expansion and new families of exact traveling solutions to Hirota equation, Appl. Math. Comput, 204 (2008) 881-883. [13] D. Levi D, P. Winternitz, Nonclassical symmetry reduction: example of the Boussinesq equation, Journal of Physics A 22(1989) 2915-2924. [14] S. Lie, Theories der Tranformationgruppen, Dritter und Letzter Abschnitt, Teubner, Leipzig, 1893. [15] M. Nadjafikhah, Lie Symmetries of Inviscid Burgers Equation, Advances in Applied Clifford Algebras, 19(1)(2008), 101-112. [16] M. Nadjafikhah, Classification of Similarity Solutions for Inviscid Burgers Equation, Advances in Applied Clifford Algebras, 20(1)(2010), 71-77. [17] P.J. Olver, Applications of Lie Groups to Differential Equations, In: Graduate Texts in Mathematics, vol. 107. Springer, New York, 1993. [18] P.J. Olver, E.M. Vorob’ev, Nonclassical and conditional symmetries, in: CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, N.H. Ibragimov, ed., CRC Press, Boca Raton, Fl., 1996, pp. 291-328. [19] D. Poole, W. Hereman, Symbolic computation of conservation laws of nonlinear partial differential equations using homotopy operators, Ph.D. dissertation, Colorado School of Mines, Golden, Colorado, (2009). [20] D. Poole, W. Hereman, The homotopy operator method for symbolic integration by parts and inversion of divergences with applications, Appl. Anal. 87(2010), 433-455. 16