Bäcklund transformations: a tool to study Abelian and non-Abelian nonlinear evolution equations
BB¨acklund transformations: a tool to study Abelianand non-Abelian nonlinear evolution equations
Sandra Carillo
Dipartimento Scienze di Base e Applicate per l’IngegneriaSapienza Universit`a di Roma, 16, Via A. Scarpa, 00161 Rome, Italy and
Gr. Roma1, IV - Mathematical Methods in NonLinear PhysicsNational Institute for Nuclear Physics (I.N.F.N.), Rome, Italy
Cornelia Schiebold
Department of Natural Sciences, Engineering, and Mathematics,Mid Sweden University, Sundsvall, Sweden.
Abstract
The KdV eigenfunction equation is considered: some explicit solutions are con-structed. These, to the best of the authors’ knowledge, new solutions represent anexample of the powerfulness of the method devised. Specifically, B¨acklund trans-formation are applied to reveal algebraic properties enjoyed by nonlinear evolutionequations they connect. Indeed, B¨acklund transformations, well known to repre-sent a key tool in the study of nonlinear evolution equations, are shown to allow theconstruction of a net of nonlinear links, termed
B¨acklund chart , connecting Abelianas well as non Abelian equations. The present study concerns third order nonlinearevolution equations which are all connected to the KdV equation. In particular,the Abelian wide B¨acklund chart connecting these nonlinear evolution equationsis recalled. Then, the links, originally established in the case of Abelian equations,are shown to conserve their validity when non Abelian counterparts are consid-ered. In addition, the non-commutative case reveals a richer structure related tothe multiplicity of non-Abelian equations which correspond to the same Abelianone. Reduction from the nc to the commutative case allow to show the connectionof the KdV equation with KdV eigenfunction equation, in the scalar case. Finally,recently obtained matrix solutions of the mKdV equations are recalled.
Keywords:
Nonlinear Evolution Equations; B¨acklund Transformations; RecursionOperators; Korteweg deVries-type equations; Invariances; Cole-Hopf Transformations.
AMS Classification : 58G37; 35Q53; 58F07
The crucial role played by B¨acklund transformation in investigating soliton equations is well known as testified by [1, 7, 56, 53, 57, 32] referring only to some well knownbooks on the subject. Third order nonlinear evolution equations, termed KdV-type,are studied. Indeed, both in the Abelian case [10, 9] as well as in the non-Abelian one[11, 14, 16, 17], a wide net of nonlinear evolution equations turns out to be connected to a r X i v : . [ n li n . S I] J a n he Korteweg-de Vries (KdV) equation or, respectively, to its nc counterpart. The net oflinks, termed B¨acklund chart, allows to reveal many interesting properties enjoyed by thenonlinear evolution equations it connects. Specifically, all algebraic properties which arepreserved under B¨acklund transformations can be transferred from one equation to all theothers within the B¨acklund chart. The nonlinear evolution equations under investigation,further to the KdV equation are the potential Korteweg-de Vries (pKdV), the modifiedKorteweg-de Vries (mKdV), and the KdV eigenfunction (KdV eig.). In addition, in thecommutative case [10, 9] also the Dym equation is included in the B¨acklund chart. Onthe other hand, in the non commutative one, as pointed out in [17], there are two differentmodified Korteweg-de Vries equations which are connected to each other via two furthernonlinear evolution equations: both of them reduce to the KdV eigenfunction when thenon commutativity condition is removed.Section 2 is devoted to briefly recall the definition of B¨acklund transformation to-gether with some remarkable properties. In the subsequent Section 3 the B¨acklund chartsconnecting Abelian KdV-type equations is provided. In particular, the B¨acklund chartin [10, 9] is recalled. The subsequent Section 4 the non trivial invariance exhibited bythe KdV eigenfunction equation is used to construct explicit solution it admits. Thesesolutions are independent on time, but show an explicit dependence on the space variable x . The extension of the B¨acklund chart, induced by the M¨obius invariance enjoyed bythe KdV singularity equation, is given in Section 5. The Section 6 reconsiders the non-Abelian B¨acklund chart, constructed in [11, 14, 16]. The last Section briefly discussesfurther remarkable results B¨acklund transformations allow to achieve. Matrix solutions,generalisation of results to hierarchies of nonlinear evolution equations are mentioned. This Section is devoted to a brief review on B¨acklund transformations aiming to providethe definitions needed in the following restriction the attention on the definitions usedin the following The general definition of B¨acklund transformation in implicit form,according to [53] and references therein, given two non linear evolution equations of thetype u t = K ( u ) , v t = G ( v ) (1)when β ∈ R \ { } denotes the B¨acklund parameter, then the B¨acklund Relations read ∂v∂x (cid:48) =: B (cid:48) ( u, u x ; v ; β ) , , x (cid:48) = x (cid:48) ( x, t ) ∂v∂t (cid:48) =: B (cid:48) ( u, u t ; v ; β ) , t (cid:48) = t (cid:48) ( x, t )Then, compatibility conditions ∂ B (cid:48) ∂t (cid:48) = ∂ B (cid:48) ∂x (cid:48) followed by elimination of x (cid:48) , t (cid:48) , v from thelatter give u t = K ( u ). On the other hand, when we write the B¨acklund Relations in termsf ( x, t ), compatibility condition combined with elimination of x, t, v produce v t = G ( v ).Throughout, the following definition, see [24], is adopted. Definition
Given two evolution equations, u t = K ( u ) , K : M → T M , u : ( x, t ) ∈ R n × R → u ( x, t ) ∈ R m ⊂ M v t = G ( v ) , G : M → T M , v : ( x, t ) ∈ R n × R → v ( x, t ) ∈ R m ⊂ M then B (u , v) = 0 represents a B¨acklund transformation between them whenever giventwo solutions of such equations, say, respectively, u ( x, t ) and v ( x, t ) such that B ( u ( x, t ) , v ( x, t )) | t =0 = 0 (2) it follows that, B ( u ( x, t ) , v ( x, t )) | t =¯ t = 0 , ∀ ¯ t > , ∀ x ∈ R . (3)As usual choice when soliton solutions are considered, it is assumed M := M ≡ M and, in addition, the generic fiber T u M , at u ∈ M , is identified with M itself .As a consequence, solutions of such two equations are linked via the B¨acklund transfor-mation which establishes a correspondence between them: it can graphically representedas can be depicted by the following fugure u t = K ( u ) B –– v t = G ( v ) (4)which shows that the B¨acklund transformation relates the two nonlinear evolution equa-tions. The net of links connecting the many different nonlinear evolution equations can besummarised in a B¨acklund chart; here the latest [17] is reported. Indeed, the constructionis directly related to results in[28, 9] further developments, in the case of nc nonlinearevolution equations are comprised [11, 12, 14, 16] while a comparison between the twodifferent cases is studied in [17]. The links among the various KdV-type equations aresummarised in the following B¨acklund chart: It is generally assumed that M is the space of functions u ( x, t ) which, for each fixed t , belong tothe Schwartz space S of rapidly decreasing functions on R n , i.e. S ( R n ) := { f ∈ C ∞ ( R n ) : || f || α,β < ∞ , ∀ α, β ∈ N n } , where || f || α,β := sup x ∈ R n (cid:12)(cid:12) x α D β f ( x ) (cid:12)(cid:12) , and D β := ∂ β /∂x β ; throughout this article n = 1. dV( u ) ( a ) –– mKdV( v ) ( b ) –– KdV eig.( w ) ( c ) –– KdV sing.( ϕ ) ( d ) –– int. sol KdV( s ) ( e ) –– Dym( ρ ) Figure 1: KdV-type equtions B¨acklund chartwhere, in turn, the linked nonlinear evolution equations are: u t = u xxx + 6 uu x (KdV) ,v t = v xxx − v v x (mKdV) ,w t = w xxx − w x w xx w (KdV eig.) ,ϕ t = ϕ x { ϕ ; x } , where { ϕ ; x } := (cid:18) ϕ xx ϕ x (cid:19) x − (cid:18) ϕ xx ϕ x (cid:19) (KdV sing.) ,s s t = s s xxx − ss x s xx + 32 s x (int. sol KdV) ,ρ t = ρ ρ ξξξ (Dym) . that is, in order, the Korteweg-de Vries (KdV), the modified Korteweg-de Vries (mKdV),and the KdV eigenfunction (KdV eig.), the Korteweg deVries interacting soliton (int.sol.KdV),the Korteweg deVries singuarity manifold (KdV sing.) and the Dym equations. Respec-tively, in the B¨acklund chart, ( a ) , ( b ) , ( c ) , ( d ) , ( e ) denote the following B¨acklund trans-formations ( a ) u + v x + v = 0 , ( b ) v − w x w = 0 , (5)( c ) w − ϕ x = 0 , ( d ) s − ϕ x = 0 , (6)and ( e ) ¯ x := D − s ( x ) , ρ (¯ x ) := s ( x ) , where D − := (cid:90) x −∞ dξ, (7)so that ¯ x = ¯ x ( s, x ) and, hence, ρ (¯ x ) := ρ (¯ x ( s, x )). The transformation (e) is termed reciprocal transformation since it interchanges the role of the dependent and independentvariables . The
KdV eigenfunction equation, for sake of brevity denoted as KdV eig., is included ina wide study by Konopelchenko in [38] where, among many other ones, it is proved to see, for instance, [56] where reciprocal transformations are defined and applications are provided.The transformation ( e ) is analysed in [10, 28] where it is shown to represent a B¨acklund transformationbetween the extended manifold consisting of the both the dependent and the independent variables. e integrable via the inverse spectral transform (IST) method. Indeed, this equation wasfirstly derived in a founding article of the IST method [48] and also [62], later furtherinvestigated in [38, 44] wherein a wide variety of nonlinear evolution equations is studied.Nevertheless, the KdV eigenfunction equation does not appear in subsequent classifica-tion studies of integrable nonlinear evolution equations, such as [5, 51, 63, 45, 46] untilvery recently when, in [4], linearizable nonlinear evolution equations are classified. TheKdV eigenfunction equation is a third order nonlinear equation of KdV-type since it isconnected via B¨acklund transformations with the Korteweg deVries (KdV), the modifiedKorteweg deVries (mKdV), the Korteweg deVries interacting soliton (int.sol.KdV) [27]and the
Korteweg deVries singuarity manifold (KdV sing.), introduced by Weiss in [64]via the
Painlev`e test of integrability.
This section is concerned only about the an invariance property enjoyed by the KdVeigenfunction equation. It can be trivially checked to be scaling invariant since onsubstitution of αw, ∀ α ∈ C , to w it remains unchanged. In addition, according to [9],see prop. 4 therein, the following proposition shows further nontrivial invariances. Proposition 4.1
The KdV eigenfunction equation w t = w xxx − w x w xx w is invariant under the transfor-mation I : ˆ w = ad − bc ( cD − ( w ) + d ) w , a, b, c, d ∈ C s.t. ad − bc (cid:54) = 0 , (8)where D − := (cid:90) x −∞ dξ is well defined since so called soliton solutions are looked for in the Schwartz space S ( R n ) . The proof, according to [9], is based on the invariance under the M¨obius group oftransformationsM : ˆ ϕ = aϕ + bcϕ + d , a, b, c, d ∈ C such that ad − bc (cid:54) = 0 . (9)of the KdV singularity manifold equation ϕ t = ϕ x { ϕ ; x } , where { ϕ ; x } := (cid:18) ϕ xx ϕ x (cid:19) x − (cid:18) ϕ xx ϕ x (cid:19) . (10)Combination of such an invariance with the connection between the KdV eigenfunctionand the KdV singularity manifold equation allows to prove the proposition. Indeed, letM : ˆ ϕ = aϕ + bcϕ + d , ∀ a, b, c, d ∈ C | ad − bc (cid:54) = 0 (11) t = w xxx − w x w xx w B –––––– ϕ t = ϕ x { ϕ ; x }(cid:108) I (cid:108) Mˆ w t = ˆ w xxx − w x ˆ w xx ˆ w (cid:98) B –––––– ˆ ϕ t = ˆ ϕ x { ˆ ϕ ; x } Figure 2: Induced invariance B¨acklund chart.the following B¨acklund chart where the B¨acklund transformations B and (cid:98)
B are, respec-tively: B : w − ϕ x = 0 and (cid:98) B : ˆ w − ˆ ϕ x = 0 . The invariance I follows via combination of the M¨obius transformation M with the twoB¨acklund transformations B and (cid:98)
B. An application of the invariance I is indicates howto construct solutions of the KdV eigenfunction equation.
In this subsection an example of solutions admitted be the KdV eigenfunction equationis constructed on the basis of the invariance in the previous subsection. Indeed, it iseasily checked that w ( x, t ) = k , ∀ k ∈ R represents a solution of the KdV eigenfunctionequation w t = w xxx − w x w xx w . (12)When, in the M¨obius group the parameters are set to be a = d = 0 , b = 1 , c = − I indicates that alsoˆ w ( x, t ) = ( k x + k ) − , ∀ k ∈ R (14)represent solutions of the KdV eigenfunction equation.Further solutions can be obtained in the same way. Remarkably, also in the nc casesolutions can be constructed. Note that the whole B¨acklund chart in Fig. 1 can be extended, as indicated in thefollowing figure. see footnote on page 2. dV( u ) ( a ) –– mKdV( v ) ( b ) –– KdV eig.( w ) ( c ) –– KdV sing.( ϕ ) ( d ) –– int. sol KdV( s ) ( e ) –– Dym( ρ ) AB (cid:108) AB (cid:108) AB (cid:108) M (cid:108) AB (cid:108) AB (cid:108) KdV(˜ u ) ( a ) –– mKdV(˜ v ) ( b ) –– KdV eig.( ˜ w ) ( c ) –– KdV sing.( ˜ ϕ ) ( d ) –– int. sol KdV(˜ s ) ( e ) –– Dym(˜ ρ ) Figure 3: Abelian KdV-type hierarchies B¨acklund chart: induced invariances.that is, since the KdV singularity manifold equation (KdV sing.) is invariant un-der the M¨obius group of transformations, all the the auto-B¨acklund transformations AB k , k = 1 . . . and AB are, respectively, the well known KdVand the mKdV auto-B¨acklund transformations [47, 7, 28]. The invariance of the KdVeigenfunction equation is I ≡ AB [9] and, according to [28], auto-B¨acklund transforma-tions of the the int. sol. KdV and Dym equations are also obtained, denoted as AB ,and AB . In this section the attention is focussed on non-Abelian equations. Specifically, accordingto [43, 2, 21] nonlinear evolution equations in which the unknown is an operator ona Banach space are studied. These, for short, are termed operator equations . Crucialto the present study, both in the Abelian as well as in the non-Abelian setting, is thatthe algebraic properties of interest nonlinear evolution equations enjoy are preservedunder B¨acklund transformations. To stress the distinction between scalar and operatorunknown functions, they are, respectively, denoted via lower and upper-case letters.Taking into account the results in [3, 11, 14, 16], a B¨acklund chart which connectsoperator KdV-type equations, can be constructed: it is depicted in the following Fig. 4.
KdV( U ) mKdV( V ) meta-mKdV( Q ) mirrormeta-mKdV( (cid:101) Q ) alternativemKdV( (cid:101) V ) Int So KdV ( S ) KdV Sing ( φ ) (cid:27) ( a ) (cid:0)(cid:0)(cid:0)(cid:18) V = Q x Q − (cid:64)(cid:64)(cid:64)(cid:82) (cid:101) V = − (cid:101) Q x (cid:101) Q − (cid:64)(cid:64)(cid:64)(cid:73) V = − (cid:101) Q − (cid:101) Q x (cid:0)(cid:0)(cid:0)(cid:9) (cid:101) V = − (cid:101) Q x (cid:101) Q − (cid:27) ( b ) (cid:27) ( d ) Figure 4: KdV-type equations B¨acklund chart: the non-Abelian case.In Fig. 4, the third order nonlinear operator evolution equations where, as pointed out,all unknowns are denoted via capital case letters with the only exception of the KdVing. equation. The KdV-type operator equations are, in turn, U t = U xxx + 3 { U, U x } (KdV) ,V t = V xxx − { V , V x } (mKdV) ,Q t = Q xxx − Q xx Q − Q x (meta-mKdV) , (cid:101) Q t = (cid:101) Q xxx − (cid:101) Q x (cid:101) Q − (cid:101) Q xx (mirror meta-mKdV) , (cid:101) V t = (cid:101) V xxx + 3[ (cid:101) V , (cid:101) V xx ] − (cid:101) V (cid:101) V x (cid:101) V (amKdV) ,φ t = φ x { φ ; x } , where { φ ; x } = (cid:0) φ − x φ xx (cid:1) x − (cid:0) φ − x φ xx (cid:1) (KdV sing.) ,S t = S xxx − (cid:0) S x S − S x (cid:1) x (int. sol KdV) . where the B¨acklund transformations ( a ) , ( b ) , ( c ) linking the KdV with the mKdV, theamKdV with the int.sol KdV and the latter with the KdV sing. are the following, ones,the non-commutative counterparts of those in the commutative B¨acklund chart, see Fig.1 and Fig. 3. U = − ( V + V x ) ( a ) (cid:101) V = 12 S − S x ( b ) S = φ x . ( c ) . Notably, both the two equation named mirror meta-mKdV and meta-mKdV [16] coincidewith the KdV eigenfunction equation when the nc condition is removed. Similarly, alsothe mKdV and amKdV equations when commutativity is assumed reduce to the usualmKdV equation and, therefore the box in Fig. 4 collapses to a line and, therefore theB¨acklund chart in Fig. 1 can be recovered. The following Fig. 6 is esplicative.where note that, in the commutative case, the composition of the transformations V = Q x Q − and (cid:101) V = − Q − Q x reduce to V → − V , trivial sign invariance admitted by themKdV equation; correspondingly, the two forms of modified KdV equations (amKdVand mKdV) coincide with the (Abelian) mKdV equation. This closing section aims to give a brief overview on two different lines of results con-cerning further developments that is, on one side the determination of explicit solutionsnd on te other one, the extension of the B¨acklund chart from the considered nonlin-ear evolution equations to corresponding hierarchies. Further perspectives are finallymentioned.
This subsection is devoted to the special case when the operator is finite dimensional sothat it admits a matrix representation. Thus, the aim is to emphasise the importance ofB¨acklund transformations also when solutions admitted by non-Abelian soliton equationsare looked for. Solutions admitted by the matrix equations are a subject of interest inthe literature. The study presented, based on previous results [11, 12] further developedin [18, 19], take into account multisoliton solutions of the matrix KdV equation obtainedby Goncharenko [31], via a generalisation of the Inverse Scattering Method. Accordingto [12], and in particular Theorem 3 therein, generalises Goncharenko’s multisolitonsolutions which follow as special ones. Solutions of matrix mKdV equation are discussedand obtained in [18], where some 2 × × d × d, d ∈ N and further solutions are produced to give an ideaof the results in this direction and currently under investigation [20]. Further matrixsolutions are obtained in [23, 31, 41, 60, 61, 35, 59]. Finally, some further observations deserve attention. First of all, one of the propertiesof B¨acklund transformations crucial in the present research project, which involves notonly the two authors, but also further collaborators, is the notion of recursion operator .Indeed the existence of a hereditary recursion operator admitted by a nonlinear evolutionequation is a remarkable algebraic property [25]. Such a property, on one side, allowsto construct a whole hierarchy of nonlinear evolution equations associated, for instance,to the KdV equation. On another side, the algebraic properties which characterise ahereditary recursion operator are preserved under B¨acklund transformations. Hence,since the KdV equation admits a hereditary recursion operator, the B¨acklund chart inFig.1 indicates the way to construct the recursion operators of all the nonlinear evolutionequations it connects. In addition, such a B¨acklund chart can be naturally extended tothe whole hierarchies of all the nonlinear evolution equations therein [28], in the Abeliancase.The corresponding non-Abelian B¨acklund chart is constructed in [11, 12] and furtherextended in [14, 16]. Also the case of non-Abelian Burgers equation [40, 34, 36, 13, 15]shows a richer structure with respect to the corresponding Abelian one.Furthermore, a 2 + 1- dimensional [49] B¨acklund chart which links the Kadomtsev-Petviashvili (KP), the modified Kadomtsev-Petviashvili (mKP) and other 2 + 1- dimen-sional soliton equations, such as the KP singularity manifold equation and the 2 + 1-dimensional version of the Dym equation. The link, obtained by Rogers [52], betweenthe KP and the 2 + 1-dimensional Dym equation indicates the way to the constructionof solutions of the Dym equation in 2 + 1- dimensions. Notably, as shown in [49], theC in Fig. 3 follows to represent a constrained version of the KP B¨acklund chart [50].In addition, as shown in [28], the Hamiltonian and bi-Hamiltonian structure admittedby the KdV equation, since these properties are preserved under B¨acklund transforma-tions [25], all the nonlinear evolution KdV-type equations in the B¨acklund chart, in Fig.3, are all proved to admit a bi-Hamiltonian structure [42, 26, 29, 30]. As discussedin [8], a B¨acklund chart connects the Caudrey-Dodd-Gibbon-Sawata-Kotera and Kaup-Kupershmidt hierarchies [22, 58, 37]. All the involved equations are 5th order nonlinearevolution equations; notably, the B¨acklund chart linking them all shows an impressiveresemblance to the one connecting KdV-type equations. Again, such B¨acklund chartcan be extended to the corresponding whole hierarchies [54, 10]. The study on 2 + 1-dimensional non-Abelian equations seems of interest.
Acknowledgments
The financial support of G.N.F.M.-I.N.d.A.M., I.N.F.N. and
Sapienza
University ofRome, Italy are gratefully acknowledged. C. Schiebold thanks Dipartimento Scienzedi Base e Applicate per l’Ingegneria and
Sapienza
University of Rome for the kindhospitality.
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