Strict self-adjointness and shallow water models
aa r X i v : . [ m a t h - ph ] M a r Strict self-adjointness and certain shallowwater models
Priscila Leal da Silva and Igor Leite Freire
Centro de Matem´atica, Computa¸c˜ao e Cogni¸c˜aoUniversidade Federal do ABC - UFABCRua Santa Ad´elia, , Bairro Bangu, . − Santo Andr´e, SP - Brasil
E-mail: [email protected]/[email protected]: [email protected]/[email protected]
Abstract
We consider a class of third order equations from the point of view of strict self-adjointness.Necessary and sufficient conditions for the investigated class to be strictly self-adjoint are obtained.Then, from a strictly self-adjoint subclass we consider those who admit a suitable scaling trans-formation. Consequently, a family of equations including the Benjamin-Bona-Mahony, Camassa-Holm and Novikov equations is deduced. By a suitable choice of the parameters, we deduce anone-parameter family of equations unifying the last two mentioned equations. Then, using somerecent techniques for constructing conserved vectors, we show that from the scale invariance it isobtained, as a conserved density, the same quantity employed to construct one of the well knownHamiltonians for the cited integrable equations.
Introduction
Since the celebrated Korteweg and de Vries paper [34], in which a third order evolution equationwas derived and named after them, a huge number of papers in the literature has been done formodeling, or related with, shallow water equations. During the last century, a sequence of papers,starting with [36], showed and enlightened many properties of such equation. Additionally, the KdVequation u t = u xxx + uu x (1)proved to be a prototype equation for many phenomena, see, for instance, [1].Although its good and versatile properties, the equation was not above criticisms. In the seminalpaper [3], the authors derived a new equation for moderately long wave equations of small amplitudeswhose formal justification is as that for the KdV and from that paper arose the well know Benjamin-Bona-Mahoney (BBM) equation u t = u txx + uu x . (2)However, the differences between both equations are greater than the fact that (1) is an evolutionequation whereas (2) is not. In [3] the authors found three conserved quantities on the solutions of(2). Later in [40], those obtained conservation laws were proved to be the only three admitted by(2). This fact shows a dramatic difference between (2) and (1) since the first one admits an infinitenumber of conserved quantities [37].More recently, Camassa and Holm [8] using Hamiltonian methods derived the famous Camassa-Holm (CH) equation u t − u txx + 3 uu x = 2 u x u xx + uu xxx . (3)The last equation possesses remarkable properties such as solutions with peaks in which the first orderderivatives are discontinuous, called peakon solutions, and it has a bi-hamiltoninan structure, see [8],which implies in the existence of an infinite number of conserved quantities, like the KdV equation[19, 37, 35].Since then, a considerable number of papers have been dedicated to derive third order non-evolutionary dispersive equations having similar properties as those known to KdV and CH equation.To cite a few number of examples, it was derived in [10] an integrable equation having peakon solutionswith first order nonlinearities, while in [11] another integrable equation, combining linear dispersionsuch as the KdV equation and a nonlinear dispersion like the CH equation, was discovered. Morerecently, Novikov [38] has discovered the equation u t − u txx + 4 u u x = 3 uu x u xx + u u xxx , (4)which not only admits peakon solutions and cubic nonlinearities, but it is also integrable [22].In [37] it was shown that the KdV equation possesses infinitely many conservation laws. This wasthe start point of a considerable number of papers dealing with the properties of a certain equation andthe existence of an infinite number of conserved quantities. Then we arrive at the point of integrabilityand the existence of infinitely many conservation laws of an equation.Noether theorem showed a deeper and closer relation between symmetries and conservation lawsfor the Euler-Lagrange equations. From Noether theorem, for each Noether symmetry of the Euler-Lagrange equation, one can establish a conservation law. Although the KdV equation is not anEuler-Lagrange equation, it can be transformed in one.2herefore a question naturally arise: would it be possible to derive infinitely many conservedquantities of the KdV equation from its symmetries? This question does not make sense if it isrestricted to Lie point symmetries since the KdV equation (or its “variational form”) admits a finitedimensional symmetry Lie algebra and, therefore, one would expect no more than a finite number ofconservation laws coming from Noether’s theorem. The infinite number of conservation laws of theKdV equation could be interpreted as the existence of higher order symmetries , see [39].And, up to our knownlegement, the first paper relating symmetries (not necessarily Lie pointsymmetries) of the KdV equation and an infinite number of local conservation laws for it was [26], inwhich Ibragimov showed how to construct local conservation laws using symmetries other than theLie point symmetries.In order to construct the conserved vectors, Ibragimov first established a non-local conservedvector. Then he showed that the KdV equation is strictly self-adjoint [25, 26, 32] and, consequently,the non-local conserved quantities can be transformed in locals one. These concepts will be betterdiscussed in section 3.A considerable number of integrable equations has this common property: strict self-adjointness.In fact, Ibragimov [26] showed that KdV is strictly self-adjoint. In [31] it was shown that the CHequation has also the same property, as well as in [6] it was proved that the Novikov equation is strictlyself-adjoint. In particular, with respect to (3) and (4), the obtained results in [9], [31] and [6] showssome common facts:1. both equations are strictly self-adjoint;2. both equations admit the scaling symmetry ( x, t, u ) ( x, λ − b t, λu ), for a certain value of b ,whose corresponding generator is X b = u ∂∂u − bt ∂∂t ; (5)3. from the Lie point symmetry generator (5) and the results proposed in [26], it was obtained theconserved density u + εu x for both equations, see [31, 6]. In fact, let m = u − εu xx and assumethat u ( x, t ) → x → ±∞ . Taking the expression ( uu x ) x = u x + uu xx into account, it iseasily concluded that H = Z + ∞−∞ mudx = Z + ∞−∞ ( u + εu x ) dx. Such conserved quantity provides a Hamiltonian to CH [8], Dullin-Gotwald-Holm (DGH) [11]and Novikov [22] equations.Since Ibragimov’s concepts on self-adjointness [26, 27, 30, 32] have been introduced, a considerablenumber of papers have been dealing with the problem of finding classes of differential equations withsome self-adjoint property, see, for instance, [13, 14, 15, 16, 17, 21, 48].Therefore, motivated by those recent results and provoked by the classification carried out in [38], inwhich certain generalizations of the CH equation possessing infinite hierarchies of higher symmetrieswere considered, in this paper we determine which conditions are necessary and sufficient for theequation u t + εu txx + f ( u ) u x + g ( u ) u x u xx + h ( u ) u xxx = 0 (6)to be strictly self-adjoint. 3nce having carried out the strict self-adjointness classification of (6), we restrict ourselves to findthe subfamily admitting the scale invariance ( x, t, u ) ( x, λ − b t, λu ). It is then obtained the followingfourth-parameter family of strictly self-adjoint equations u t + εu txx + γu b u x = ( b + 1) βu b − u x u xx + βu b u xxx , (7)which includes equations (2), (3) and (4). Moreover, taking b = 1 , ε = − β = α and γ = 3, we arrive,up to a translation u u + u /α , at the DGH equation u t − α u txx + 3 uu x = α ( uu xxx + 2 u x u xx ) + u u xxx , (8)which is also integrable, see [11]. The term u corresponds to the coefficient of the linear dispersionof the equation and when u → α = 1, such equation turns back to the CH equation. However,if u = 0, (8) does not admit the generator (5). We observe that at the limit of the dispersionless u , α →
0, equation (8) is reduced to the Riemann equation u t + 3 uu x = 0. More generally, when thedispersion effects are neglected in (7), that is, ε, β →
0, one obtains a family of Riemann equationsgiven by u t + γu b u x = 0.Finally, if we choose ε = − γ = β ( b + 2), equation (7) can be rewritten as u t − u txx + β ( b + 2) u b u x = ( b + 1) βu b − u x u xx + βu b u xxx . (9)Therefore, defining m = u − u xx , (9) is equivalent to m t − βu b m x − β ( b + 1) u b − u x m = 0 . (10)Equation (9), or its equivalent form (10), still contains the CH (3) and Novikov (4) equations, but notBBM (2).Local conserved currents for the family of strictly self-adjoint scale-invariant equations found arepresented in section 5. Finally, in section 6 we discuss the obtained results. In section 3 we presentsome basic facts about Lie symmetries, conservation laws and strict self-adjointness. Then, in section4 it is obtained the family (7) from (6). Here we present some basic facts regarding Lie symmetries of differential equations. In whatfollows, we shall only consider single partial differential equations, however, the discussed theory canbe used for ordinary differential equations and also for systems of differential equations. Further andbetter discussions on this subject can be found in [4, 5, 23, 24, 43].Since in our main problem we shall consider time dependent equations, in what follows such variablewill be represented with either symbols x or t , depending on the situation. Let x = ( x , · · · , x n ) ∈ X ⊆ R n , u = u ( t, x ) ∈ U ⊆ R and u ( j ) be, respectively, n independent and a dependent variable, whilethe set of all jth derivatives of u . Hereafter, the summation over repeated indices is presupposed. Allfunctions here are assumed to be smooth. In particular, u i ··· i j = D i · · · D i j ( u ) , where D i = ∂∂x i + u i ∂∂u + u ij ∂∂u j + · · · , i = 0 , · · · , n are the total derivative operators. 4et A be the set of all locally analytic functions of a finite number of the variables x, u and u ( j ) .Let F ∈ A and consider an equation F ( x, u, u (1) , · · · , u ( k ) ) = 0 . (11)Let M = X × U ≈ R n +1 be the space of the independent and dependent variables and considera local group of transformations G acting on an open subset of M . Then the action on such spaceinduces a local action on the jet space ( x, u, u (1) , u (2) , · · · , u ( k ) ), called k − jet space. For further details,see [43], Chapter 2.Let X = ξ i ( x, u ) ∂∂x i + η ( x, u ) ∂∂u (12)be a generator of the local one-parameter group ε exp εX ( x, u ) acting on M . One says that the X is a Lie point symmetry generator of the equation (11) if X ( k ) F = λF, (13)for a certain function λ depending on x, u, u (1) , · · · . Equation (13) is called invariance condition and X ( k ) = ξ i ( x, u ) ∂∂x i + η ( x, u ) ∂∂u + ζ i ∂∂u i + ζ ij ∂∂u ij + · · · + ζ i ··· i k ∂∂u i ··· i k , (14)where ζ i = D i ( η ) − D i ξ j u j , · · · , ζ i ··· i k = D i · · · D i ( η ) − D i k ξ j u i ··· i k − j , is the k − th prolongation of the vector field X . In this case, we say that( x, u ) exp εX ( x, u ) := (cid:18) x + εX ( x ) + ε X ( X ( x )) + · · · , u + εX ( u ) + ε X ( X ( u )) + · · · (cid:19) is a Lie point symmetry of (11).Consider equation (6) and the transformation( x, t, u ) ( x, λ − b t, λu ) , λ > . (15)Let us determine for which functions f = f ( u ) , g = g ( u ) and h = h ( u ) such transformation is a Liepoint symmetry of (6).Firstly we observe that (5) is the generator of the transformation (15). Then, its third extensionis X (3) = u ∂∂u − bt ∂∂t + ( b + 1) u t ∂∂u t + u x ∂∂u x + u xx ∂∂u xx + u xxx ∂∂u xxx + ( b + 1) u txx ∂∂u txx . A simple calculation shows that X (3) ( u t + εu txx + f ( u ) u x + g ( u ) u x u xx + h ( u ) u xxx ) =( b + 1)( u t + εu txx ) + ( uf ) ′ u x + [( ug ) ′ + g ] u x u xx + ( uh ) ′ u xxx , where the prime ′ means derivative with respect to u . Therefore, condition (13) gives( b + 1)( u t + εu txx ) + ( uf ) ′ u x + [( ug ) ′ + g ] u x u xx + ( uh ) ′ u xxx = λ ( u t + εu txx + f ( u ) u x + g ( u ) u x u xx + h ( u ) u xxx ) . u t , u x , u x u xx and u xxx we respectively obtain: λ = b + 1 , ( uf ) ′ = λf, ( ug ) ′ + g = λg, ( uh ) ′ = λh, which reads f ( u ) = γu b , g ( u ) = σu b − and h ( u ) = δu b , where γ, σ and δ are arbitrary constants.Then we conclude that the transformation (15) is a Lie point symmetry of the equation (6) if andonly if the equation takes the form u t + εu txx + γu b u x + σu b − u x u xx + δu b u xxx = 0 . (16) Here we present some elements regarding conservation laws. However, the interested reader isrefereed to [2, 26, 30, 32, 40, 41, 42, 49] for further details. We also guide the curious reader to[24, 33, 43, 44, 45, 46] for additional readings.
Mathematically speaking, one can define a conservation law for (11) starting from the expression
Div ( C ) := D t C + D i C i = λF, (17)for a certain vector field C := ( C , C x ), where C x := ( C , · · · , C n ), and functions λ = λ ( t, x, u, · · · ).Equation (17) is called characteristic form of the conservation law D t C + D i C i = 0, while λ is thecharacteristic of it.A vector field C = ( C , C x ) provides a trivial conservation laws if Div ( C ) ≡
0. Such a vector C is, therefore, called trivial conserved vector . Otherwise C is called nontrivial conserved vector .If C is a trivial conserved vector, for any differential equation (11), equation (17) holds with thecharacteristic λ = 0. Two conserved vectors are said to be equivalent if they differ by a trivial conservedvector. Clearly two equivalent conserved vectors possess the same characteristic λ . A conservationlaw of (17) can now be rigorously defined as follows.By conservation law of (11) we mean the equivalence class of conserved vectors of (11). Then, theset of all conservation laws is a vector space whose the identity is the equivalence class of the trivialconserved quantities.On (11), the relation (17) becomes D t C + D i C i ≡
0. From the physical point of view, the vectorfield C = ( C , C , · · · , C n ) is usually a density and it is called conserved vector or conserved current of the modeled phenomena by (11). The component C is the conserved density while the remainingcomponents are the conserved flux. Being a density, restricting x to a fixed domain Ω ⊆ R n , with asmooth, constant, boundary ∂ Ω, and defining Q Ω = Z Ω C dx, application of the divergence theorem gives dQ Ω dt = Z Ω D t C dx = − Z Ω D i C i dx = − Z ∂ Ω C x · dS. A more general discussion can be found in [33]. Here we employ such definition of trivial conserved vector becauseit is enough for our purposes, mainly because we will use it in Section 5. Q Ω depends only on the behavior of the solutions on theboundary ∂ Ω and it is equal to the total flux over it. For non-dissipative physical model, this factprovides the general form of a conservation law.
Although its importance, the construction of conserved quantities is a sensitive point. There area lot of techniques for dealing with this matter [2, 24, 26, 43, 49]. In this paper we will use the ideasintroduced by Ibragimov in [26] for constructing local conserved currents for equations of the type (6).Firstly, let (11) be a given differential equation, L := vF be another differential function, called formal Lagrangian , and δδw = ∂∂w + ∞ X j =0 ( − j D i · · · D i j ∂∂w i ··· i j be the Euler-Lagrange operator. Taking the formal Lagrangian into account, from the Euler-Lagrangeequations δ L /δu = 0 , δ L /δv = 0, it is obtained the system F = 0 and F ∗ = 0, where F ∗ := δ L δu = 0 (18)is the adjoint equation to (11).In [26] Ibragimov showed that if X = ξ i ( x, u ) ∂∂x i + η α ( x, u ) ∂∂u (19)is a Lie point symmetry generator of (11), then C i = ξ i L + W (cid:20) ∂ L ∂u i − D j (cid:18) ∂ L ∂u ij (cid:19) − D j D k ∂ L ∂u ijk − · · · (cid:21) + D j ( W ) (cid:20) ∂ L ∂u ij − D k (cid:18) ∂ L ∂u ijk (cid:19) + · · · (cid:21) + D j D k ( W ) (cid:20) ∂ L ∂u ijk − · · · (cid:21) + · · · , (20)where W = η − ξ j u j , provides a conserved vector for the system formed by (11) and (18). We guidethe interested reader to [32] for a better and deeper discussion about this subject.In particular, for equations of the type (6) admitting a Lie point symmetry generator X = τ ∂∂t + ξ ∂∂x + η ∂∂u , C = τ L + W (cid:20) ∂ L ∂u t + D x (cid:18) ∂ L ∂u txx (cid:19)(cid:21) − D x ( W ) D x (cid:18) ∂ L ∂u txx (cid:19) + D x ( W ) ∂ L ∂u txx ,C = ξ L + W (cid:20) ∂ L ∂u x − D x (cid:18) ∂ L ∂u xx (cid:19) + D x (cid:18) ∂ L ∂u xxx (cid:19) + D x D t (cid:18) ∂ L ∂u xxt (cid:19) + D t D x (cid:18) ∂ L ∂u xtx (cid:19)(cid:21) − D x ( W ) D x (cid:20) ∂ L ∂u xx − D x (cid:18) ∂ L ∂u xxx (cid:19) − D t (cid:18) ∂ L ∂u xxt (cid:19)(cid:21) − D t ( W ) D x (cid:18) ∂ L ∂u xtx (cid:19) + D x ( W ) ∂ L ∂u xxx + D t D x ( W ) ∂ L ∂u xtx + D x D t ( W ) ∂ L ∂u xxt , (21)where the formal Lagrangian is given by L = v (cid:20) u t + ε u txx + u xtx + u xxt f ( u ) u x + g ( u ) u x u xx + h ( u ) u xxx (cid:21) . (22) Considering (21) and (22) one can easily conclude that the quantity (21) is not a local conservedvector for (6) since the components (21) depend on the nonlocal variable v . In [26], see [30, 32] fora deeper discussion, Ibragimov introduced the concept of strictly self-adjoint differential equations.Actually, an equation (11) is said to be strictly self-adjoint if and only if its adjoint equation satisfiesthe relation F ∗ | v = u = λF, (23)for a certain function λ ∈ A . If equation (11) is strictly self-adjoint, then one can eliminate thenon-physical variable v from (20) obtaining, therefore, a conserved quantity depending only on t, x, u and derivatives of u . It means that for strictly self-adjoint differential equations, the adjoint equationis equivalent to the original one, hence the new conserved quantity is a local conserved current forthe original equation and, therefore, it provides a local conservation law for it. Further details andapplications of this techniques can be found in [13, 14, 15, 16, 17, 18, 21, 27, 28, 29, 48].Let F := u t + εu txx + f ( u ) u x + g ( u ) u x u xx + h ( u ) u xxx . Then F ∗ = v ( f ′ ( u ) u x + g ′ ( u ) u x u xx + h ′ ( u ) u xxx ) − D t ( v ) − D x [ v ( f ( u ) + g ( u ) u xx )]+ D x ( vg ( u ) u x ) − D x D t ( εv ) − D x ( vh ( u )) . (24)After a calculation of the terms involving the total derivatives in (24), an explicit expression forthe adjoint equation is given by F ∗ = − v t − εv txx − h ( u ) v xxx + v ( g ′′ ( u ) u x + 3 g ′ ( u ) u x u xx − h ′′′ ( u ) u x − h ′′ ( u ) u x u xx )+ v x ( − f ( u ) − g ( u ) u xx + 2 g ′ ( u ) u x + 2 g ( u ) u x x − h ′′ ( u ) u x − h ′ ( u ) u xx )+ v xx ( g ( u ) u x − h ′ ( u ) u x ) . (25)8etting v = u in (25) and then equaling it to λF to use the definition of strict self-adjointness (23),we obtain F ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v = u = − u t − εu txx − h ( u ) u xxx − f ( u ) u x + u x ( ug ′′ ( u ) − uh ′′′ ( u ) + 2 g ′ ( u ) − h ′′ ( u ))+ u x u xx (3 ug ′ ( u ) − uh ′′ ( u ) − h ′ ( u ) + 2 g ( u ))= λu t + λεu txx + λf ( u ) u x + λg ( u ) u x u xx + λh ( u ) u xxx . (26)From (26) it is concluded, from the coefficient of u t , that λ = −
1. From the coefficients of u txx , u x , u xxx , u x , u x u xx , respectively, one obtains − ε = λε, − f ( u ) = λf ( u ) , − h ( u ) = λh ( u ) , ( ug ) ′′ − ( uh ) ′′′ = 0 , ( ug ) ′ − ( uh ) ′′ = 0 . (27)Integrating the coefficient of u x u xx once, the condition g ( u ) = ( uh ) ′ u + cu , (28)where c is an arbitrary constant, is obtained. It is important to observe that the coefficients of theother derivatives in (27) do not provide any new information about the functions f ( u ) , g ( u ) and h ( u ).This implies that (6) is strictly self-adjoint if and only if g and h are related by (28).In section 2 we showed that equation (6) forms a family of scale-invariant equations if and onlyif it takes the form of (16). Let us now determine under what condition the scale-invariant family ofequations (16) is strictly self-adjoint. Substituting g ( u ) = σu b − and h ( u ) = δu b into (28), one arrivesat σu b − = δ ( b + 1) u b − + cu , whose solutions are σ = δ + c , whenever b = 0, and σ = δ ( b + 1) and c = 0, for b = 0. Defining δ = − β ,we obtain the following family of strictly self-adjoint scale-invariant equations • for b = 0, we have (7) and, • for b = 0, we have u t + εu txx + γu x = β u x u xx u + ( β − c ) u xxx . (29)Considering physical applications, we can assume that the constants β and γ are positive.As we have already pointed out, if we choose ε = − γ = β ( b + 2), equation (7) becomes u t − u txx + β ( b + 2) u b u x = (1 + b ) βu b − u x u xx + βu b u xxx . (30)Note that under the change t β − t , equation (30) is then reduced to u t − u txx + ( b + 2) u b u x = ( b + 1) u b − u x u xx + u b u xxx . (31)For b = 1, equation (31) is the well known Camassa-Holm [8] equation, while for b = 2 it is theNovikov equation [38]. Up to our knowledgement, it is the first time that an one-parameter family ofequations connecting both Camassa-Holm and Novikov equations is reported.9 Local conserved currents
Assuming b = − , C = u − εu x + D x (cid:20) bε tu t u x − bb + 2 γtu b +2 + 23 εuu x − bβtu b +1 u xx − bε tuu tx (cid:21) ,C = 22 + b γu b +2 + 2 βu b +1 u xx + 2 εuu tx − D t (cid:20) bε tu t u x − bb + 2 γtu b +2 + 23 εuu x − bβtu b +1 u xx − bε tuu tx (cid:21) We observe that the term D x ( · · · ) in C along with the term − D t ( · · · ) in C are the components ofa trivial conserved vector. Therefore, according to the discussion presented in Section 3, the mentionedterms are not needed for finding a really useful conserved vector and we should simplify the establishedcomponents. For this reason, after transferring the terms D x ( · · · ) from C to C and eliminating thenull divergence, it is obtained the components C = u − εu x , C = 22 + b γu b +2 − βu b +1 u xx + 2 εuu tx . (32)Components (32), for a fixed ε , provide a three-parameter family of components of conservedvectors to the family of equations (7). On Table 5 one finds some shallow water models belonging tothe family (7) and their corresponding conservation laws associated with the scale symmetry ( x, t, u ) ( x, λ b t, λu ). ε b γ β Equation Conserved density Conserved flux − − u + u x u − uu tx − u + u x u − u u xx − uu tx − u + u x u − u u xx − uu tx = − ∀ u b γu b +2 ∀ ∀ ∀ − u − εu x γu − β − c ) uu xx + 2 εuu tx − cu x ∀ − ∀ ∀ − u − εu x − β u xx u + 2 γ ln u + 2 εuu xt Table 1:
In this table it is presented some equations of the type (7) as well as some conserved currents andthe corresponding symmetry from which the conserved vector was obtained. For further details about the lasttwo cases presented in this table, see Remarks 1 and 2, respectively, below. On the penultimate line, c is anarbitrary constant. Using the same approach, some of the listed conservation laws were obtained in the last 5 years.In fact, the conservation laws for the CH equation were obtained in [31]. In [13, 16] conservation lawsfor Riemman equations were established using the same approach and, more recently, the conservationlaw for the Novikov equation was derived in [6].Although the conservation law found for the Benajamin-Bona-Mahony is well known, see [3, 40],up to our knowledge it is the first time that it is obtained via the results introduced in [31].
Remark 1:
Regarding the case b = 0, it is interesting to observe that the scaling transformation( x, t, u ) ( x, t, λu ) provides a nontrivial conservation law for the equation (29).10 emark 2: Similarly as in Remark 1, for b = − x, t, u ) ( x, λ − t, λu )also gives a nontrivial conserved quantity for u t + εu txx + γ u x u + β u x u xx u − β u xxx u = 0 . In this paper we considered the subclasses of equation (6) having two properties: strict self-adjointness and admitting the Lie point symmetry generator (5). As a consequence we obtained thefamily (7) and, after a suitable choice of the arbitrary constant, we arrived at the equivalent equations(9) and (10), which includes the CH, DGH and Novikov equations.Moreover, using some recent techniques [30, 32] due to Ibragimov, we established conservation lawsfor some members of the obtained classes, as it is shown on the Table 1. It is interesting to observethat the obtained conserved quantities are those employed in the literature of completely integrableequations of the type (9) to construct a first Hamiltonian for these equations. Then the results obtainedin this paper suggest a connection between strict self-adjointness and integrable equations. Moreover,previous results [14, 15, 18, 26] had shown that the KdV equation also possesses this same property.Although there are some known examples of integrable equations that are not strictly self-adjoint,such as the Harry–Dym (HD) and Krichever–Novikov (KN) equation, they are nonlinearly self-adjoint [27, 30, 32, 18, 48]. However Ibragimov proved [32] that, up to a multiplier, all nonlinearly self-adjoint equations are strictly self-adjoint. Moreover, it is well known that most of integrable equationsare nonlinearly self-adjoint, see, for instance [15, 17, 18]. This reinforces the suspicion of a closerelationship, not clear yet, between strict self-adjointness and integrability.On the other hand, our results also show that strict self-adjointness does not imply in the integra-bility, as one can easily see from the fact that the BBM equation is strictly self-adjoint, but it is notcompletely integrable [40]. However, noticing that under the change t β − t , equation (10) is thenreduced to m t − u b m x − ( b + 1) u b − u x m = 0 , (33)which gives the CH equation for b = 1 and the Novikov equation for b = 2, but not the BBM equation.If the class (33) admits more completely integrable equations, we do not know, since we are notspecialists in this field. But we suspect that the answer is positive and we wait for a confirmation fromwell versed researchers in integrability. Moreover, we do not know, in the literature, an one-parameterfamily of equations unifying the CH and Novikov equations.Additionaly, it is not clear if, from the family (33), it is possible, up to the mentioned cases, to findequations admitting either soliton or peakon solutions or when the inverse scattering method can beapplied [1]. We hope that some enlightening about this unclear point to us could be soon presented.We would like to do some comparisons between (33) and the b − equation (see [10, 12]) u t − u txx + ( B + 1) uu x = Bu x u xx + uu xxx . (34)Clearly (34) admits the scale invariance ( x, t, y ) ( x, λ − t, λu ), since it can be obtained from thefamily (16) choosing b = 1, γ = B + 1 , σ = − B and δ = 1. However, comparing (34) with (31) or (7),we conclude that b = 1 and B = 2, which means that (34), with these values, is the Camassa-Holmequation. In fact, in the references the equation is denoted by u t − u txx + ( b + 1) uu x = bu x u xx + uu xxx . However, here, inorder to avoid confusion, we use the form (34).
11n [10] it was shown that (34) is integrable if B = 3. However, such equation is not a member of ourfamily (33). In fact, comparing (34) with (6) it is easy to conclude that f ( u ) = ( B + 1) u, g ( u ) = − B and h ( u ) = − u . From the condition (28) we obtain B = 2 + c/u , which implies that B = 2 and c = 0.Then we realise that (34) is strictly self-adjoint if and only if B = 2, which is the CH equation.Although (34) possesses among its members, the CH and the Degasperis-Procesi equation (case B = 3), which are both integrables, it is not strictly self-adjoint, for any B . Moreover, these are theonly integrable equations of the type (34), see [10]. On the other hand, our new equation (33) alsoconnects at least two integrable equations, namely, Camassa-Holm and Novikov. But, differently from(34), every member of (33) is also strictly self-adjoint.Further investigations, incorporating more general classes, will be considered in forthcoming pa-pers. Particularly we would like to consider wider classes of equations, in order to incorporate otherwell known equations, such as the Qiao’s equation [47], which does not fit in the class investigated inthis work. Acknowledgements
The authors would like to thank FAPESP for financial support (grant n. 2011/19089-6 and schol-arship n. 2012/22725-4). We are grateful to Professor Valery Shchesnovich for having guided us tothe book [1]. I. L. Freire is also grateful to CNPq for financial support (grant n 308941/2013-6).
References [1] M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering,Cambridge University Press, 1991.[2] S. C. Anco and G. Bluman, Direct constrution method for conservation laws for partial differentialequations part II: general treatment, Euro. J. Appl. Math., vol. 41, 567–585, (2002).[3] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlineardispersive systems, Philos. Trans. Roy. Soc. London, vol. 272, 47–78, (1972).[4] G. W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations,Springer, New York, (2002).[5] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied MathematicalSciences 81, Springer, New York, (1989).[6] Y. Bozhkov, I. L. Freire and N. H. Ibragimov, Group analysis of the Novikov equation, Comp.Appl. Math., (2013), DOI 10.1007/s40314-013-0055-1.[7] R. W. Atherton and G. M. Homsy, On the existence and formulation of variational principles fornonlinear differential equations, Stuides in Applied Mathematics, LIV(1), 31–60, (1975).[8] T. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys.Rev. Lett., vol. 71, 1661–1664, (1993).[9] P. A. Clarkson, E. L. Mansfield and T. J. Priestley, Symmetries of a class of nonlinear third-orderpartial differential equations, Math. Comput. Modelling, vol. 25, 195–212, (1997).1210] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,Theor. Math. Phys., vol. 133, 1463–1474, (2002).[11] R. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linearand nonlinear dispersion, Phys. Rev. Lett., vol. 87, 194501, 4pp., (2001).[12] H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and otherasymptotically equivalent equations for shallow water waves, Fluid Dynamics Research, vol. 333,73–95, (2003).[13] I. L. Freire, Conservation laws for self-adjoint first order evolution equations, J. Nonlin. Math.Phys., vol. 18, 279–290, (2011).[14] I. L. Freire, Self-adjoint sub-classes of third and fourth-order evolution equations, Appl. Math.Comp., vol. 217, 9467–9473, (2011).[15] I. L. Freire and J. C. S. Sampaio, Nonlinear self-adjointness of a generalized fifth-order KdVequation, J. Phys. A: Math. Theor., vol. 45, 032001, (2012).[16] I. L. Freire, New conservation laws for inviscid Burgers equation, Comp. Appl. Math., vol. 31,559–567, (2012).[17] I. L. Freire, New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order,Commun. Nonlinear Sci. Numer. Simul., vol. 18, 493–499, (2013).[18] I. L. Freire and J. C. Santos Sampaio, On the nonlinear self-adjointness and local conservationlaws for a class of evolution equations unifying many models, Commun. Nonlinear Sci. Numer.Simul., vol. 19, 350–360, (2014).[19] C. S. Gardner, Kortewerg-de Vries equation and generalizations IV. The Korteweg-de Vries equa-tion as a Hamiltonian system, J. Math. Phys., vol. 12, 1548–1551, (1971).[20] L. R. Galiakberova and N. H. Ibragimov, Nonlinear self-adjointness of the Krichever–Novikovequation, Commun. Nonlin. Sci. Num. Simul., vol. 19, 361–363, (2014).[21] M. L. Gandarias, Weak self-adjoint differential equations, J. Phys. A, vol. 44, 262001 (6pp),(2011).[22] A. N. W. Hone and J. P, Wang, Integrable peak on equations with cubic nonlinearities, J. Phys.A: Math. Theor., vol. 41, 372002, 10 pp., (2008).[23] N. H. Ibragimov, Transformation groups applied to mathematical physics, Translated from theRussian Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht,(1985).[24] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, JohnWiley and Sons, Chirchester (1999).[25] N. H. Ibragimov, Integrating factors, adjoint equations and Lagrangians, J. Math. Anal. Appl.,vol. 318, 742–757, (2006). 1326] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., vol. 333, 311–328, (2007).[27] N. H. Ibragimov, Quasi-self-adjoint differential equations, Archives of ALGA, vol. 4, 55–60,(2007).[28] N. H. Ibragimov, M. Torrisi and R. Tracin`a, Quasi self-adjoint nonlinear wave equations, J. Phys.A: Math. Theor., vol. 43, 442001-442009, (2010).[29] N. H. Ibragimov, M. Torrisi and R. Tracin`a, Self-adjointness and conservation laws of a generalizedBurgers equation, J. Phys. A: Math. Theor., vol. 44, 145201-145206, (2011).[30] N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor., vol.44, 432002, 8 pp., (2011).[31] N.H. Ibragimov, R.S. Khamitova, A. Valenti, Self-adjointness of a generalized Camassa-Holmequation, Appl. Math. Comp., vol. 218, 2579–2583, (2011).[32] N. H. Ibragimov, Nonlinear self-adjointness in constructing conservation laws, Archives of ALGA,vol. 7/8, 1–90, (2011).[33] N. M. Ivanova and R. O. Popovych, Equivalence of conservation laws and equivalence of potentialsystems, Int. J. Theor. Phys., vol. 46, 2658–2668, (2007).[34] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangularcanal, and on a new type of long stationary waves, Phil. Mag., vol. 39, 422–443, (1895).[35] M. D. Kruskal, R. M. Miura, C. S. Gardner and N. J. Zabusky, Korteweg-de Vries equation andgeneralizations. V. Uniqueness and nonexistence of polynomial conservation laws, J. Math. Phys.,vol. 11, 952–960, (1970).[36] R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlineartransformation, J. Math. Phys., vol. 9, 1202–1204, (1968).[37] R. M. Miura, C. S. Gardner and M. D. Kruskal, Korteweg-de Vries equation and generalizations.II. Existence of conservation laws and constants of motion, J. Math. Phys., vol. 9, 1204–1209,(1968).[38] V. S. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor,42