On the construction of conservation laws: a mixed approach
aa r X i v : . [ m a t h - ph ] D ec On the construction of conservation laws:a mixed approach
M. Ruggieri , M. P. Speciale Faculty of Engineering and Architecture, Kore University of EnnaVia delle Olimpiadi, Cittadella Universitaria, I–94100, Enna, ItalyEmail: [email protected] Department of Mathematical and Computer Sciences,Physical Sciences and Earth Sciences, University of MessinaViale F. Stagno d’Alcontres 31, I–98166 Messina, ItalyEmail: [email protected]
Abstract
A new approach, combining the Ibragimov method and the one byAnco and Bluman, with the aim of algorithmically computing localconservation laws of partial differential equations, is discussed. Someexamples of the application of the procedure are given. The method,of course, is able to recover all the conservation laws found by using thedirect method; at the same time we can characterize which symmetry,if any, is responsible for the existence of a given conservation law.Some new local conservation laws for the Short Pulse equation andfor the Fornberg–Whitham equation are also determined.
Keywords : Conservation laws, Lie point symmetries, a systematicprocedure.
In dealing with differential equations, conservation laws have a deep rele-vance, since they often express the conservation of physical quantities. They1re also important due to their use in investigating integrability, existence,uniqueness and stability of solutions, or in implementing efficient numericalmethods of integration. In 1918, Emmy Noether [1] presented her celebratedprocedure (Noether’s theorem) to find local conservation laws for systemsof differential equations arising from a variational principle. Noether provedthat a point symmetry of the action functional (action integral) provides alocal conservation law through an explicit formula that involves the infinites-imals of the point symmetry and the Lagrangian of the action functional.The most important limitation of Noether’s theorem for the determinationof local conservation laws relies on the fact that it applies to variationalsystems.In recent years, systematic procedures to find conservation laws also fornon variational problems have been introduced. The first one, the “directmethod”, proposed by Anco and Bluman in 1996 [2] (see also [3]), gives thepossibility to generate local as well nonlocal conservation laws. A second ap-proach was introduced by Ibragimov in 2007 [4], and improved in recent years[5]-[7] with the formulation of some theorems allowing for the construction ofconservation laws starting from the symmetries of the differential equationsand the use of a formal
Lagrangian. Recently many authors have been em-ploying these concepts in order to establish conservation laws for equationsand systems which arises in many fields of the applied sciences [8]-[12].In Noether’s theorem, one starts from the Euler-Lagrange equations δ L δu α = 0 , α = 1 , . . . , m, (1)where L ( x , u , u (1) , . . . , u ( k ) ) is a Lagrangian involving the independent vari-ables x = ( x , . . . , x n ), the dependent variables u = ( u , . . . , u m ), and thepartial derivatives up to a fixed order k ; moreover, δδu α = ∂∂u α + ∞ X j =1 ( − j D i D i . . . D i j ∂∂u αi i ...i j are the Euler-Lagrange operators; here and the following we use the Einsteinconvention of sum over repeated indices. Noether’s theorem states that ifthe variational integral with the Lagrangian L is invariant under a group G generated by X = ξ i ∂∂x i + η α ∂∂u α , (2)2hen the vector field C = ( C , . . . , C n ) defined by C i = ξ i L + ( η α − ξ i u αi ) ∂ L ∂u αi (3)provides a conservation law for the Euler-Lagrange equations (1), i.e. , it isdiv C ≡ D i ( C i ) = 0 for all solutions of (1), D i ( C i ) (cid:12)(cid:12) (1) = 0 , (4)where D i is the operator of total differentiation: D i = ∂∂x i + u αi ∂∂u αi + u αik ∂∂u αik + . . . . (5)Any vector field C satisfying (4) is called a conserved vector.The plan of the paper is the following. In Section 2, we sketch brieflyeither the Anco and Bluman direct approach or the Ibragimov method, anddiscuss their main features; it is worth of being remarked that the Ancoand Bluman method (ABM) often finds more conservation laws that thoseprovided by the Ibragimov method (IM), but the latter approach is able toestablish a direct link between the conservation laws and the symmetriesadmitted by the differential equations at hand. In Section 3, we introduceour approach that in some sense merges these two methods. Our method,in fact, is able to recover, at least in the examples we considered, all theconservation laws found by using ABM, and at the same time to show whichsymmetry, if any, is related to a conservation law. Finally, in Section 4, weprovide explicitly some applications of the method. The ABM [2, 3],[13]-[15] does not lead directly to the construction of localconservation laws of a given system of differential equations even if the au-thors suggest several ways of finding the fluxes of local conservation laws.Given a system of m k -th order differential equations F α ( x , u , u (1) , u (2) , . . . , u ( k ) ) = 0 , α = 1 , . . . , m, (6)with m dependent variables u = ( u , . . . , u m ), we have the following funda-mental result. 3 heorem 1 A set of non-singular local multipliers v α = v α ( x , u , u (1) , . . . , u ( ℓ ) ) ℓ ≤ k, α = 1 , . . . , m, yields a divergence expression for the system (6) if and only if the set ofequations δδu α (cid:0) v α F α (cid:1) ≡ , α = 1 , . . . , m, (7) hold for arbitrary function u ( x ) . First one looks for sets of multipliers of the form v α = v α ( x , u , u (1) , . . . , u ( ℓ ) ) α = 1 , . . . , m, ℓ ≤ k, where the dependence of multipliers on their arguments is chosen so thatsingular multipliers do not arise. Then the set of determining equations (7)for arbitrary u ( x ) are solved to find all such sets of multipliers satisfying theidentity v α F α ≡ D i φ i ( x , u , u (1) , . . . , u ( ℓ ) ) . (8) φ i ( i = 1 , . . . , n ) being the fluxes of the conservation law.Once a set of multipliers is known, it is necessary to compute the fluxes ofthe local conservation law (8). One can achieve this task by solving the set ofdetermining equations for the fluxes φ i , or, in the case of complicated formsof multipliers and/or differential equations, through an integral (homotopy)formula.The second approach [4] consists in reformulating the Noether’s theoremfor the determination of conservation laws both in the presence of equationswith a variational structure and in the case of differential equations nothaving variational structure. Theorem 2
If the operator X = ξ i ∂∂x i + η α ∂∂u α (9) is admitted by the Euler-Lagrange equations δ L δu α = 0 , α = 1 , . . . , m, (10) and satisfies X ( L ) + ( D k ξ k ) L = 0 , (11)4 hen the components C i = ξ i L + W α " ∂ L ∂u αi − D j (cid:18) ∂ L ∂u αij (cid:19) + D j D k ∂ L ∂u αijk ! − . . . + D j ( W α ) " ∂ L ∂u αij − D k ∂ L ∂u αijk ! + . . . + D j D k ( W α ) " ∂ L ∂u αijk − . . . , i = 1 , . . . , n, (12) with W α = η α − ξ j u αj , α = 1 , . . . , m, are the fluxes of the conservation law D i ( C i ) (cid:12)(cid:12) ( ) = 0 . The proof of the theorem is based on the operator identity X ( L ) + L D i ( ξ i ) = W α δ L δu α + D i ( N i L ) (13)where N i = ξ i + W α δδu αi + ∞ X s =1 D i . . . D i s ( W α ) δδu αi ...i s , i = 1 , . . . , n ;taking into account (10), it follows C i = N i ( L ) , i = 1 , . . . , n. Notice that the formal Lagrangian L needs to be written in symmetric formwith respect to all mixed derivatives.In the case of a first order Lagrangian, (12) coincides with (3). Moreover,one can omit the term ξ i L when it is convenient [7]. This term provides atrivial conserved vector because L vanishes on the solutions of (6).Ibragimov obtains the explicit formula for constructing the conservationlaws associated with the symmetries of any nonlinearly self-adjoint systemof equations [5]-[7]. 5 efinition 1 A system of m k -th order differential equations F α ( x , u , u (1) , u (2) , . . . , u ( k ) ) = 0 , α = 1 , . . . , m, (14) with m dependent variables u = ( u , . . . , u m ) , is said to be nonlinearly self-adjoint if the adjoint equations F ∗ α ( x , u , u (1) , v (1) , u (2) , v (2) , . . . u ( k ) , v ( k ) ) = δ L δu α = 0 , α = 1 , . . . m, (15) ( L = v β F β , with β = 1 , . . . , m , is the so called formal Lagrangian), aresatisfied for all solutions u ( x ) of the original system (14) upon a substitution v α = ψ α ( x , u ) , α = 1 , . . . , m (16) such that ψ ( x , u ) = 0 . In other words, the following equations hold true F ⋆α ( x , u , u (1) , ψ (1) , u (2) , ψ (2) , . . . u ( k ) , ψ ( k ) )= λ βα F β ( x , u , u (1) , u (2) , . . . , u ( k ) ) , α = 1 , . . . , m, (17)or, equivalently, by (15) and (17), the equations δ ( v β F β ) δu α = λ βα F β ( x , u , u (1) , u (2) , . . . , u ( k ) ) , α = 1 , . . . , m, (18)where λ βα are undetermined coefficients, and ψ ( σ ) = { D i . . . D i σ ( ψ α ( x , u )) } , σ = 1 , . . . , k, α = 1 , . . . , m. Here, v = ( v , . . . , v m ) and ψ = ( ψ , . . . , ψ m ) are m -dimensional vectors,and ψ (x , u) = 0 means that not all components ψ α ( x , u ) ( α = 1 , . . . , m )vanish simultaneously.Ibragimov also considers the case in which the point-wise substitution(16) is replaced by v α = ψ α ( x , u , u (1) , u (2) , . . . , u ( r ) ) , α = 1 , . . . , m, r ≤ k. Then (17) will be written, e.g., in the case r = 1, as follows F ∗ α ( x , u , u (1) , ψ (1) , u (2) , ψ (2) , . . . , u ( k ) , ψ ( k ) ) = λ βα F β + λ jβα D j ( F β ) , (19)6here λ jβα are undetermined coefficients.Using the definition of nonlinear self-adjointness and the theorem 1 onconservation laws proved in [4], the explicit formula for constructing conser-vation laws associated with symmetries of any nonlinearly self-adjoint systemof equations is obtained. Theorem 3
Let a system of m k -th order differential equations F α ( x , u , u (1) , u (2) , . . . , u ( k ) ) = 0 , α = 1 . . . m, (20) with m dependent variables u = ( u , . . . , u m ) , be non linearly self-adjoint.Specifically, let the adjoint system (17) or (18) be satisfied for all solutionsof (20) upon the substitution (16). Then, any Lie point, contact or Lie-B¨acklund symmetry (2), as well as any nonlocal symmetry of (20), leads toa conservation law constructed by the formula (12). Remark 1
The Ibragimov method, that allows to write immediately the fluxes C i ( i = . . . , n ), often obtains a limited set of conservation laws. The con-servation laws obtained are the ones linked to symmetries that are usuallypoint symmetries, since Lie-B¨acklund, generalized and nonlocal symmetriesare not easy to determine. Here we report some examples of applications of the two methods for com-puting local conservation laws.
Example 1 (KdV equation)
Let us consider the Korteweg-deVries equa-tion ∆ ≡ u t − u xxx − uu x = 0 . (21) In [6, 7], it has been proved that KdV equation is nonlinear self-adjointfor ψ = A + A u + A ( x + tu ) , where A , A , A are arbitrary constants; hence, the following three linearlyindependent solutions ψ = 1 , ψ = u, ψ = ( x + tu ) are recovered. sing the symmetries admitted by KdV equation, Ξ = t ∂∂x − ∂∂u , Ξ = 3 t ∂∂t + x ∂∂x − u ∂∂u , Ξ = ∂∂t , Ξ = ∂∂x , (22) some conservation laws are determined. In particular, using the scaling groupand ψ or ψ , D t ( u ) − D x (cid:18) u x + u xx (cid:19) = 0 ,D t (cid:0) u (cid:1) − D x (cid:18) u x − u u xx − u (cid:19) = 0 , (23) whereas it is claimed (see [16], Sect.22.5]) that the Galilei group leads to theconservation law D t (cid:18) tu + xu (cid:19) − D x (cid:18) xu + tuu xx − tu x − xu xx + u x (cid:19) = 0 . (24) The ABM method [13]-[15], assuming ψ ( t, x, u ) , provides the three mul-tipliers ψ = 1 , ψ = u, ψ = ( x + tu ) , whereupon, by the relation (8), the conservation laws (23) and (24) are found[17]. Example 2
Let us consider the polytropic gas dynamics equations in n spacedimensions ( n ≤ ): ∂ρ∂t + ∇ · ( ρ u ) = 0 ,ρ (cid:18) ∂ u ∂t + ( u · ∇ ) u (cid:19) + ∇ p = 0 ,∂p∂t + u · ∇ p + γp ∇ · u = 0 , (25) where ρ is the mass density, u = ( u , . . . , u n ) the velocity, p the pressure, γ the adiabatic index and x = ( x , . . . , x n ) the rectangular space coordinates. he Lie algebra of point symmetries is spanned by Ξ = ∂∂t , Ξ i = ∂∂x i i = 1 , . . . , n, Ξ = t ∂∂t + x k ∂∂x k , Ξ = t ∂∂t − u k ∂∂u k − p ∂∂p , Ξ = ρ ∂∂ρ + p ∂∂p , Ξ i = t ∂∂x i + ∂∂u i , i = 1 , . . . , n, Ξ i = x k ∂∂x j − x j ∂∂x k + u k ∂∂u j − u j ∂∂u k , j, k = 1 , . . . , n, j < k ;(26) the operators Ξ and Ξ i ( i = 1 , . . . , n ) characterize time and space transla-tions, Ξ , Ξ and Ξ scaling transformations, Ξ i ( i = 1 , . . . , n ) Galileantransformations, and the remaining operators characterize spatial rotations.In the one-dimensional case, Ibragimov proves that the system is nonlin-ear self-adjoint with ψ = ρu, ψ = u , ψ = 1 γ − the same proof applies also in the case of more than one space dimension.In a recent paper [18], the classification of all well known local and non-local classical conservation laws, written in the integral form, is listed: ddt Z Ω( t ) ρdω = 0 , Conservation of mass ,ddt Z Ω( t ) ρ u dω = − ddt Z S ( t ) ρνdS, Momentum ,ddt Z Ω( t ) ρ ( t u − x ) dω = − Z S ( t ) tρ ν dS, Center of mass ,ddt Z Ω( t ) ( 12 ρ | u | + pγ − dω = − Z S ( t ) p u · ν dS, Energy ,ddt Z Ω( t ) ρ ( x × u ) dω = − Z S ( t ) p ( x × ν ) dS, Angular momentum , (27)9 here Ω( t ) is an arbitrary n -dimensional volume, moving with fluid, S ( t ) isthe boundary of the volume Ω( t ) and ν is the unit (outer) normal vector tothe surface S ( t ) .If we write the above conservation laws in the general form ddt Z Ω( t ) T dω = Z S ( t ) ( χ · ν ) dS, then the differential form of these conservation laws follows, say D t ( T ) + ∇ · ( χ + T u ) = 0 . Moreover, if γ = n +2 n , it is proved that the system admits the followingtwo additional conservation laws: ddt Z Ω( t ) [ t ( ρ | u | + np ) − ρ x · u ] dω = − Z S ( t ) p (2 t u − x ) · ν dSddt Z Ω( t ) [ t ( ρ | u | + np ) − ρ x · (2 t u − x )] dω = − Z S ( t ) tp ( t u − x ) · ν dS. (28) We notice that if γ = n +2 n , Euler equations admit an additional Lie pointsymmetry [19]-[21] spanned by Ξ = t ∂∂t + tx i ∂∂x i − ntρ ∂∂ρ + ( x i − tu i ) ∂∂u i − ( n + 2) tp ∂∂p . (29) It was shown in [22] that these two additional conservation laws together withthe previous classical conservation laws provide all pointwise conservationlaws of the system, i.e., the conservation laws whose components depend onthe variables t , x , ρ , u , p , and do not involve derivatives of ρ , u and p . The new approach we propose starts from Ibragimov method even if it over-comes the concept of nonlinearly self-adjointness.Considering the Euler-Lagrange equations δ L δu α = 0 , α = 1 , . . . , m, (30)10e work directly with the condition D i ( C i + H i ) (cid:12)(cid:12) ( ) = 0 , (31)where H i are the components of an additional arbitrary vector field H withzero divergence, and C i are the components of a conserved vector C definedin (12).In fact, if we add an arbitrary vector field whose divergence is zero in(11), we have X ( L ) + D k ξ k L = − D i H i = 0;because of the relation (13) it is W α δ L δu α + D i ( N i L ) = − D i H i , whereupon D i ( C i + H i ) = − W α δ L δu α ; (32)as a consequence, we get (31), where the components C i = N i ( L ) ( i =1 , . . . , n ) are given by (12).In our approach, when we consider a system of partial differential equa-tions of order k , F α ( x , u , u (1) , u (2) , . . . , u ( k ) ) = 0 , α = 1 , . . . , m, (33)the vector field ψ involved in the expression of the formal Lagrangian L , L = X ψ α F α , (34)is an unknown arbitrary function of x , u , and possibly also of the partialderivatives of dependent variables up to a finite order.In fact, if we start from (32), and use [7] δ L δu α = λ βα F β + λ jβα D j ( F β ) + . . . = 0 , α = 1 , . . . , m, (35)we obtain a unique condition that overcomes the condition of nonlinear self-adjointness required as a prerequisite to determine the components of theconserved vector in Ibragimov approch: D i ( C i + H i ) = − W α (cid:16) λ βα F β + λ jβα D j ( F β ) + . . . (cid:17) , (36)11hat evaluated on the solutions of the system of partial differential equations(20), gives us: D i ( C i + H i ) (cid:12)(cid:12) F α =0 = 0 . (37)The above relations is a differential system in ψ , H and all their derivativeswith respect to the independent and dependent variables. Forcing to zerothe coefficient of all derivatives u α (1) , u α (2) , . . . , we get a set of differential con-straints for ψ α and H i ( α = 1 , . . . , m , i = 1 , . . . , n ); by solving this set ofdifferential conditions, we may obtain the explicit expression of ψ and H ,and simultaneously get the components of the conserved vector.In this context, the presence of the derivatives of the functions H i H ( σ ) = { D i . . . D i σ ( H j ( x , u )) } , σ = 1 , . . . , k, j = 1 , . . . , n, in the coefficients of u α (1) , u α (2) , . . . in (37) provides the differential constraintslinking ψ α to H i that obviously are not independent.Moreover, the components of the new conserved vector T = C + H contain the components C i ( i = 1 , . . . , n ), linked to the symmetries, and H i that may be independent of the symmetries; in this way, we recover thecomponents of the conserved vector of Ibragimov and an additional termthat, sometimes is independent of the symmetries. In this Section, we present some applications of the method outlined in theprevious section.
Example 3 (KdV equation)
Let us consider the KdV equation (21) andthe formal Lagrangian L = ψ ( u t − u xxx − uu x ) , where ψ = ψ ( t, x, u ) .Let us take a linear combination of the admitted Lie point symmetries(22), say Ξ = a Ξ + a Ξ + a Ξ + a Ξ , , a , a and a being arbitrary constants. Now, let us write T and T byusing the expressions of C and C given by (12) T = C + H = W ∂ L ∂u t + H ,T = C + H == W (cid:20) ∂ L ∂u x − D xx (cid:18) ∂ L ∂u xxx (cid:19)(cid:21) − D x W (cid:20) D x (cid:18) ∂ L ∂u xxx (cid:19)(cid:21) + D xx W (cid:20) ∂ L ∂u xxx (cid:21) + H , (38) where H = H ( t, x, u ) , H = H ( t, x, u ) , and W = − ( a + 2 a u ) − ( a + a t + a x ) u x − ( a + 3 a t ) u t . By requiring [ D t ( T ) + D x ( T )] | ∆=0 = 0 , and forcing to zero the coefficients of the derivatives of u , we get the differ-ential constraints for the functions ψ , H and H . f a + 3 a t = 0 we obtain ψ = c a + c u + c ( tu + x ) + g ( ξ )( a + 3 a t ) / − c (cid:18) a a − a (6 a − t ( a + 2 a u ))6 a (cid:19) − c (cid:18) a a − a (3 a + a u + 3 a ( tu + x ))9 a (cid:19) log | a + 3 a t | ,H = (cid:0) ( c ξ + a g ( ξ ) − g ′′′ ( ξ ))( a + 3 a t ) / − (cid:18) a + a t + a x + a + 3 a t u (cid:19) g ′ ( ξ ) (cid:19) u + a ( a + 3 a t ) / g ( ξ ) + a (cid:18) c a − c x (cid:19) + ∂ x h ( t, x ) − c a a (cid:16) a a + a t ) − a a + (cid:16) a a − a ( a + a x ) (cid:17) log( a + 3 a t ) (cid:17) ,H = (cid:0) ( c ξ + a g ( ξ ) − g ′′′ ( ξ ))( a + 3 a t ) / − (cid:18) a + a t + a x + a + 3 a t u (cid:19) g ′ ( ξ ) (cid:19) × a + a t + a xa + 3 a t u − a ( a + 3 a t ) / g ′′ ( ξ ) + h ( t ) − ∂ t h ( t, x ) , (39) where ξ = − a ( a + a t ) + 2 a ( a + a x )2 a ( a + 3 a t ) / , while g ( ξ ) , h ( t, x ) and h ( t ) are arbitrary functions of their arguments and c i for i = 0 , . . . , , arbitrary constants.We remark that if c = g ( ξ ) = 0 , h ( t, x ) = a (cid:18) c x − c a (cid:19) x + h ( t ) , and h ( t ) = h ′ ( t ) , we obtain H = H = 0 , h ( t ) arbitrary function), and we recover exactly Ibragimov expression of ψ .The density and the flux, without the additional terms that give trivialcontributions to the conservation laws, read T = u (cid:18) − c + c (cid:16) x + t u (cid:17) − c (cid:18) a + 32 a u (cid:19) + c (cid:16) a + a u (cid:17)(cid:19) ,T = c (cid:18) u + u xx (cid:19) + c (cid:18) − (cid:18) x + 13 tu (cid:19) u + (cid:18) u x (cid:19) u x − ( x + tu ) u xx (cid:19) + c (cid:18)(cid:16) a a u (cid:17) u − a u x + ( a + 3 a u ) u xx (cid:19) + c (cid:18) − (cid:18) a + 23 a u (cid:19) u + 12 a u x − ( a + a u ) u xx (cid:19) + h ( t ) . (40) By taking a c = 1 and all the remaining coefficients vanishing (or a c = 1 ,and all the remaining coefficients vanishing), we get T = − u, T = (cid:18) u u xx (cid:19) (41) related to galilean group (space translation, respectively). Moreover, by taking a c = 1 and all the remaining coefficients vanishing (or a c = 1 , and allthe remaining coefficients vanishing), we get T = − u , T = − (cid:18) u x − uu xx − u (cid:19) , (42) related to the scaling group (the time translation, respectively).If only c = 0 , we recover the same conservation law linked to the galileangroup, whereas if only c = 0 we find the conserved vector T = (cid:16) x + t u (cid:17) u, T = (cid:18) u x + t (cid:18) u x − uu xx − u (cid:19) − x (cid:18) u u xx (cid:19)(cid:19) not related to a Lie symmetry but obtained in [3, 15]. Example 4 (The Fornberg–Whitham equation)
Let us consider the Forn-berg –Whitham equation ∆ = u t − u txx − uu xxx − u x u xx + uu x + u x = 0 . (43)15 n a recent paper [23], using the IM, it was proved that (43) is nonlinear self-adjoint with ψ = k ( k constant), even if only trivial conservation laws havebeen obtained [24]. Now we show that the method proposed in the previous sec-tion is able to provide nontrivial conservation laws for the Fornberg-Whithamequation (43).The equation (43) admits the following symmetries: Ξ = ∂∂t , Ξ = ∂∂x , Ξ = t ∂∂x + ∂∂u . Starting from a linear combination of them,
Ξ = a Ξ + a Ξ + a Ξ ( a , a , a arbitrary constants), we have W = a − a u t − ( a + a t ) u x . Assuming ψ , H and H functions of t, x, u , through the condition [ D t ( T ) + D x ( T )] (cid:12)(cid:12) ∆=0 = 0 , where we force to zero the coefficients of all derivatives of u , we obtain ψ = c t + c x + g ( x ) ,H = a ( u ( c + c + g ′ ( x )) + u c + g ′ ( x ) − g ′′′ ( x ))) ,H = 0 , (44) with a = a = 0 and a = 0 (it means that the galilean group and spacetranslation do not induce local conservation laws), whereas g ( x ) is an arbi-trary function of its arguments while c and c are arbitrary constants.Eliminating the terms that lead to trivial contributions in the conservationlaw, we get T = a (cid:18) c u + 53 c u (cid:16) u (cid:17) −
53 ( c t + c x ) u t (cid:19) ,T = − c a (cid:18) u (cid:18) u (cid:19) − u x − u tx (cid:19) + 53 c a ( u t u x + u xx + uu tx ) −
53 ( c t + c x ) a ( u t (1 + u − u xx ) − u x u tx − u ttx − uu txx ) . (45)16 y taking a c = 1 and all the remaining coefficients vanishing, we obtain T = u − tu t ,T = − (cid:16) u (cid:16) u − u xx (cid:17) − u x − u tx (cid:17) − t ( u t (1 + u − u xx ) − u x u tx − uu txx − u ttx ) . (46) While if a c = 1 and all the remaining coefficients vanishing, we get T = 53 ( u (cid:16) u (cid:17) − xu t ) ,T = 53 ( u t u x + u xx + uu tx ) − x ( u t (1 + u − u xx ) − u x u tx − uu txx − u ttx ) . (47) We observe that, when g ( x ) = k = const., c = c = 0 , we obtain H = H = 0 , and the trivial conservation law reported in [24]arises. Example 5 (Short Pulse equation)
Let us consider Short Pulse equation[25]-[27]: ∆ ≡ u tx − u − u u xx − uu x = 0 , (48) which describes the propagation of linearly polarized ultra-short light pulsesin a one-dimensional medium with assuming that the light propagates in theinfrared range.The Lie point symmetries admitted by (48) are spanned by: Ξ = ∂∂t , Ξ = ∂∂x , Ξ = t ∂∂t − x ∂∂x − u ∂∂u . In this case, we consider only the operator Ξ so that W = − a ( u + tu t − xu x ); we neglect time and space translations (operators Ξ and Ξ ), since we caninsert their contributions at the end of the procedure by replacing a t with a t + a and a x with a x + a . t is well known that the ABM and IM, when ψ depends on the inde-pendent, the dependent variables and the first derivatives, lead to obtainsome nontrivial conservation laws [7]; here, we consider ψ = ψ ( t, x, u, u t , u x ) , H = H ( t, x, u, u t , u x ) and H = H ( t, x, u, u t , u x ) .According to the procedure, we impose the constraint [ D t ( T ) + D x ( T )] (cid:12)(cid:12) ∆=0 = 0 , and solving the differential conditions obtained by requiring that the coeffi-cients of all derivatives of u greater than or equal to are zero, we get ψ = ( c + 3 a f ( t )) ua t + (2 u t − u u x ) f ( t ) + c u x √ ux − uf ′ ( t ) ,H = c p u x + ( c + a (3 f ( t ) − tf ′ ( t ))) (cid:18) u + xu x t − u u x (cid:19) u x − a u (( tu + 2 u x ) f ′ ( t ) − tu x f ′′ ( t )) ,H = − u (cid:18) c p u x + ( c + a (3 f ( t ) − tf ′ ( t ))) u x (cid:18) u + xu x t − u u x (cid:19) − a (cid:18) u ( tu + 2 u x ) f ′ ( t ) − tuu x f ′′ ( t ) + f ′ ( t ) t (2 + t u − f ′′ ( t ) + tf ′′′ ( t ) (cid:19) +2( c + a (3 f ( t ) − tf ′ ( t ))) (cid:18) t + u − xt (cid:19)(cid:19) , (49) with c , c and c arbitrary constants, and f ( t ) arbitrary function of t .When c = c = 0 , f ( t ) = c ( c constant) and c = − a c , we get H = H = 0 , and we recover the expression of ψ for which equation (48) isnonlinear self-adjoint [5]-[7]; this is also the form of the multiplier that leadsto get a set of conservation laws by using ABM.Finally, neglecting the terms leading to trivial conservation laws, we ob- ain T = c p u x − c u + a c ( u + tu t − xu x )(1 + u x ) p u x u xx + 2 a tf ( t ) uu t ,T = − c u p u x + c − a ( f ( t ) + tf ′ ( t ))4 ( u + ( u u x − u t ) ) − a tf ( t ) (cid:18) u u t + ( u u x − u t )( uu x u t − u tt + u u tx )) (cid:19) − a c p u x × (cid:18) ( u + tu t − xu x ) u tx u x − ( u + tuu t )(1 + u x ) + u x (2 u t + tu tt − t u u tx ) (cid:19) ;(50) moreover, we can replace a t → a t + a and a x → a x + a and obtainricher forms.The expressions of density and flux include the known results obtained byapplying IM and ABM that we recover if c = c = 0 , f ( t ) = c ( c con-stant) and c = − a c ; in fact, with these assumptions we get the followingconservation law D t ( u ) + 14 D x ( u + ( u u x − u t ) ) = 0 . Moreover, we get a new conservation law by taking c = 0 and all remain-ing coefficients vanishing, [ D t ( p u x ) − D x ( u p u x )] (cid:12)(cid:12)(cid:12)(cid:12) ∆=0 = 0 . Another conservation law, linked to the scaling group Ξ , is characterized bythe following expressions of density and flux T = c ( u + tu t − xu x )(1 + u x ) p u x u xx + 2 tf ( t ) uu t T = − ( f ( t ) + tf ′ ( t ))4 ( u + ( u u x − u t ) ) − tf ( t ) (cid:18) u u t + ( u u x − u t )( uu x u t − u tt + u u tx )) (cid:19) − c p u x × (cid:18) ( u + tu t − xu x ) u tx u x − ( u + tuu t )(1 + u x ) + u x (2 u t + tu tt − t u u tx ) (cid:19) . (51)19 xample 6 (Euler equations of gas dynamics) Let us again consider thepolytropic gas dynamics equations (25) with γ = nn , whose symmetries arelisted in section 2.1 (formulas (26), (29)). As usually, we consider a linearcombination of all operators admitted by the system, Ξ = i X k =0 a k Ξ k , i = 1 , . . . , n. Following the method above proposed, assuming ψ k , k = 1 , . . . , n , and H i , i = 1 , . . . , n , functions of all dependent and independent variables,imposing the condition [ D t ( T ) + D x k ( T k )] (cid:12)(cid:12) ( ) = 0 , k = 1 , . . . , n, and solving the differential constraints, obtained requiring the coefficients ofall derivatives to be zero, we get explicit forms of H i ( i = 1 , . . . , n ), and,consequently, some conservation laws.In the following, we consider the -dimensional case that includes thesubcases - and -dimensional ones if x = u = 0 and all dependent variablesfunctions of t, x , x , or x = x = u = u = 0 with dependent variablesfunction of t, x , respectively; so, we get: H = − a ( ψ p + ψ ρ ) + ((2 k t + k ) t + k )( 2 γ − p + ρ ( u + v + w ))+ ρt ( k u + k v + k w − k ( ux + vy + wz ))+ ρ ( k u + k v + k w − k ( ux + vy + wz ) − k ( wx + uz ) − k ( wy − vz ) − k ( uy − vx )+ k ( x + y + z ) − k x − k y + k z + f ( pρ − γ ) (cid:1) ,H = uH − a ψ p + p ( t ( k − k x + k y ) + k − k x + k z ) ,H = vH − a ψ p + p ( t ( k − k x − k y ) + k − k y + k z ) ,H = wH − a ψ p + p ( t ( k − k z ) + k − k x − k y − k z ) , (52) with a i = 0 for all i = 1 , . . . , and i = 6 ; for the sake of simplicity, werenamed the variables as follows: x = x , x = y , x = z , u = u , u = v and u = w .In previous formulas, all ψ i ( i = 1 , . . . , ) continue to be arbitrary func-tions of their arguments, f ( pρ − γ ) is an arbitrary function of its arguments,and k j for j = 1 , . . . , are arbitrary constants. s a result, we have T = ((2 k t + k ) t + k )( 2 γ − p + ρ ( u + v + w ))+ ρt ( k u + k v + k w − k ( ux + vy + wz ))+ ρ ( k u + k v + k w − k ( ux + vy + wz ) − k ( wx + uz ) − k ( wy − vz ) − k ( uy − vx )+ k ( x + y + z ) − k x − k y + k z + f ( pρ − γ ) (cid:1) ,T = uT + p ( t ( k − k x + k y ) + k − k x + k z ) ,T = vT + p ( t ( k − k x − k y ) + k − k y + k z )) ,T = wT + p ( t ( k − k z ) + k − k x − k y − k z ) , (53) where we neglected the terms giving trivial contributions to T , T , T and T .Splitting the coefficients of the integration constants k , . . . , k , we gettwelve forms of the conserved vectors.Integrating the density and the fluxes of (53) on Ω( t ) , an arbitrary –dimensional volume, moving with the fluid, we obtain all well known classicalconservation laws listed in [18], and reported in section 2.1 (formulas (27)and (28)); when all but one coefficient k i ( i = 1 , . . . , ) and f ( pρ − γ ) vanish,we get the conservation of the components of angular momentum ( k = 0 , k = 0 , or k = 0 ), the conservation of energy ( k = 0 ), the two “additional”laws ( k = 0 , or k = 0 ), the laws of center of mass ( k = 0 , k = 0 , or k = 0 ), the conservation of the components of linear momentum ( k = 0 , k = 0 , or k = 0 ), and we get the conservation of mass (all k i vanishingand f ( pρ − γ ) = const. ). In addition, a new conservation law arises if f ( pρ − γ ) is not constant. In this paper, we introduced a new mixed method for the construction of con-servation laws of differential equations. The technique, in some sense, mergesthe well known Ibragimov method and the one by Anco and Bluman; ourmethod, in fact, is able to recover, at least in the examples we considered, allthe conservation laws found by using ”the direct method”, and at the sametime to show which symmetry, if any, is related to a conservation law. In21articular, in section 4, we have found with our method new explicit con-servation laws for the Short Pulse equation and for the Fornberg–Whithamequation.
Acknowledgments
The authors acknowledge the financial support by G.N.F.M. of I.N.d.A.M.through the project ”Formazione di pattern, insorgenza di fenomeni oscilla-tori e soluzioni localizzate in sistemi reazione-diffusione con diffusione nonlineare”, 2015.
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