Non-Abelian evolution systems with conservation laws
aa r X i v : . [ n li n . S I] O c t Non-Abelian evolution systems with conservation laws
V.E. Adler ∗ , V.V. Sokolov ∗ †
20 August 2020
Abstract
We find noncommutative analogs for well-known polynomial evolution systems withhigher conservation laws and symmetries. The integrability of obtained non-Abeliansystems is justified by explicit zero curvature representations with spectral parameter.Keywords: non-Abelian system, conservation law, symmetry, zero curvature represen-tation
The right-hand side of most popular nonlinear evolution equations integrable by the inversescattering method [1] is a homogeneous differential polynomial. We say that the differentialequation u t = F ( u, u x , u xx , . . . , u n ) , u i = ∂ i u∂x i , (1)is homogeneous with weights µ and ν if it admits the one-parameter scaling group( x, t, u ) −→ ( τ − x, τ − µ t, τ ν u ) . For an N -component system with unknowns u , . . . , u N , the scaling group is of the form( x, t, u , . . . , u N ) −→ ( τ − x, τ − µ t, τ ν u , . . . , τ ν N u N ) . (2)For instance, the celebrated Korteweg–de Vries equation u t = u xxx + 6 u u x (3)is homogeneous with the weights µ = 3 and ν = 2, the modified KdV equation u t = u xxx − u u x (4)is homogeneous with µ = 3 and ν = 1, and for the nonlinear Schr¨odinger equation writtenas the coupled system u t = − u xx + 2 u v, v t = v xx − v u, (5)the weights can be chosen as µ = 2, ν = ε and ν = 2 − ε , where ε is an arbitrary parameter.It turns out that if the weight ν of an integrable homogeneous polynomial equation (1) ispositive then it can be equal only to the values ν = 2, ν = 1 or ν = [2]. There is no suchdescription for systems of several equations. ∗ L.D. Landau Institute for Theoretical Physics, Chernogolovka, Russian Federation.E-mail: [email protected] † Federal University of ABC, Santo Andr´e, Sao Paulo, Brazil. .2 Symmetries and conservation laws Both equations (3) and (5) possess higher infinitesimal symmetries (or higher flows) [3] andhigher conservation laws. Recall that a higher symmetry is an evolution equation u τ = G ( u, u x , u xx , . . . , u m ) , m > , (6)which is consistent with (1). The simplest higher symmetry for the KdV equation is of theform u τ = u xxxxx + 10 uu xxx + 20 u x u xx + 30 u u x . (7)A local conservation law for equation (1) is a relation of the form (cid:0) ρ ( u, u x , u xx , . . . ) (cid:1) t = (cid:0) σ ( u, u x , u xx , . . . ) (cid:1) x , (8)where the t -derivative on the left-hand side is calculated in virtue of the evolution equation(1). For the polynomial homogeneous equations, the functions ρ and σ are homogeneousdifferential polynomials. The function ρ is called the density of conservation law. Fewsimplest densities for the KdV equation are ρ = u, ρ = u , ρ = − u x + 2 u . It is well-known that (8) implies the relation δδu ( ρ t ) = 0 , (9)where δδu = X k ( − k D k ◦ ∂∂u k is the Euler operator or the variational derivative . Here and below, D denotes the total x -derivative D = ∞ X i =0 u i +1 ∂∂u i . (10)The relation (8) can be written as ρ t ∈ Im D . A density ρ of a conservation law is definedup to addition of total x -derivatives. In other words, ρ is an element of the quotient spaceof the algebra of differential polynomials over the subspace Im D .The existence of higher symmetries was adopted as the basis of the symmetry approachto the classification of integrable evolution equations and coupled systems [4, 5]. Another,related and more stringent criterion for integrability is associated with the existence of higherconservation laws for (1). It is well-known [6, 7, 8, 9, 10] that many integrable equations and their symmetries admitmatrix generalizations. For instance, the matrix KdV equation u t = u xxx + 3 uu x + 3 u x u , (11)where u ( x, t ) is an unknown m × m matrix, has infinitely many matrix symmetries for any m . The simplest one is of the form u τ = u xxxxx + 5 ( uu xxx + u xxx u ) + 10 ( u x u xx + u xx u x ) + 10 ( u u x + uu x u + u x u ) . (12)2f m = 1 then equation (11) coincides with (3) and (12) coincides with (7). The mKdVequation (4) admits two different matrix generalizations (see e.g. [8, sect. 3.9]) u t = u xxx − u u x − u x u (13)and u t = u xxx + 3 uu xx − u xx u − uu x u . (14)A matrix generalization of the NLS system (5) is of the form u t = u xx − uvu , v t = − v xx + 2 vuv . (15)In calculations related to matrix equations, we always work not with matrix elements,but with noncommutative associative polynomials. Therefore, it is convenient to adopt aformalized algebraic point of view on matrix evolution equations like (15), which treatsthe variables u , v , u x , v x , . . . as generators of a free associative algebra A over C [11, 12].In this language, an evolution non-Abelian equation is a derivation D t of the algebra A commuting with the derivation D , its symmetry is defined as a derivation D τ such that[ D, D τ ] = [ D t , D τ ] = 0, and a (non-Abelian) density of a conservation law is an element ρ ofthe linear quotient space A / T , where T = [ A , A ] + Im D, such that D t ( ρ ) ∈ T . Often, we denote by ρ ∈ A some representative of the correspondingequivalence class, in hope that this will not lead to a misunderstanding.Less formally, the definition of conserved density in the matrix case means that the valueof the functional I = Z + ∞−∞ trace ρ ( u , v , u x , v x , . . . ) dx (16)does not depend on t for solutions of the system which rapidly decrease for x → ±∞ . It isclear that the value of this functional does not change when the total x -derivative of a matrixpolynomial or the commutator of matrix polynomials is added to ρ . In papers [13, 14], a classification of coupled systems of the form u t = u xx + F ( u, v, u x , v x ) , v t = − v xx + G ( u, v, u x , v x ) , (17)admitting higher conservation laws was obtained. The found integrable systems can besubdivided into those that have a symmetry of order 3 and those that have no third ordersymmetry, but have a symmetry of order 4. We call the first group NLS type systems, afterthe system (5). The conventional name for the second group is the Boussinesq type systems.In the obtained list, many systems have a polynomial right-hand side. In particular, thefollowing statement holds. Theorem 1 ([14, 15]) . i) If a system of the form ( u t = u xx + A ( u, v ) u x + A ( u, v ) v x + A ( u, v ) ,v t = − v xx + B ( u, v ) v x + B ( u, v ) u x + B ( u, v ) (18) admits an infinite sequence of conservation laws then it is polynomial. A homogeneous polynomial non-triangular system (18) admits higher conservation lawsif and only if it belongs to one of the following lists, up to the scaling of the variables x, t, u, v and the interchange ( u, v ) ( v, u ) :1. NLS type systems ( u t = u xx + 2 uvu,v t = − v xx − vuv, S ( u t = u xx + 2 uu x + 2 vu x + 2 uv x ,v t = − v xx + 2 vv x + 2 vu x + 2 uv x , S ( u t = u xx + 2 u x v + 2 uv x ,v t = − v xx + 2 vv x + 2 u x , S ( u t = u xx + 2( u + v ) u x ,v t = − v xx + 2( u + v ) v x , S ( u t = u xx + 2 αu v x + 2 βuvu x + α ( β − α ) u v ,v t = − v xx + 2 αv u x + 2 βuvv x − α ( β − α ) u v ; S
2. Boussinesq type systems ( u t = u xx + ( u + v ) ,v t = − v xx − ( u + v ) , B ( u t = u xx + 2 vv x ,v t = − v xx + u x , B ( u t = u xx + 6( u + v ) v x − u + v ) ,v t = − v xx + 6( u + v ) u x + 6( u + v ) , B ( u t = u xx + 2 vv x ,v t = − v xx + 2 uu x . B Remark . Some of the above systems can be generalized by adding of lower weight termspreserving the integrability. A classification of integrable inhomogeneous polynomial systemscan be found, e.g. in [15].The goal of the present paper is to find all noncommutative generalizations with conser-vation laws for the systems from the above lists. Our approach is similar to the constructionmethod of integrable non-Abelian ODE used in [16]. We postulate that: a noncommutative generalization is polynomial, homogeneous and admits the scalinggroup ( x, t, u , v ) −→ ( τ − x, τ − µ t, τ ν u , τ ν v )with µ , ν and ν which are the same as for the original system. If the weights of thescalar system contain an arbitrary parameter, like in the case of (5), then we assumethat this parameter is preserved also for the noncommutative generalization; the generalization turns into the original system under substitution of commuting vari-ables instead of noncommuting ones (or, less formally, for the case of 1 × . for any homogeneous conserved density of the original system, there exists a homoge-neous conserved density of its non-Abelian analog, which turns into it under substitu-tion of commuting variables instead of noncommuting ones.Of course, in practice we make use only of a finite subset of simplest conservation lawsof the scalar system. Their number depends on the number of indeterminate parametersin the noncommutative generalization which we are looking for. We stop comparing densi-ties as soon as all parameters are fixed; after that we turn to the search of zero curvaturerepresentations for the obtained non-Abelian analogs in order to prove their integrability.Noncommutative analogs exist for almost all systems from the list, with the exceptionsof the system B (see Section 4.8) and the family S with generic values of parameters.Moreover, many systems admit more than one generalization. Remark . The family S ( α, β ) is curious enough and deserves comments. The parametersin this family can be scaled, so that only the ratio τ = α : β is important. It turns out thatnoncommutative generalizations exist only for τ = 1 / τ = 0 and τ = ∞ . In the scalar case,the value τ is changed under a linear transform of the variables p, q related with u, v by thedifferential substitution p x = uv, e q = uv . (19)It is easy to prove that in these variables the system takes the form p t = p x q x + ( α + β ) p x , q t = p xxx p x − p xx p x + q x − α − β ) p x q x − α (2 α − β ) p x and that it is invariant under the change [14, example 8.1]˜ p = p, ˜ q = q − kp, ˜ α = α + k, ˜ β = β + k, where k ∈ C . Applying the transformation inverse to (19), we arrive at the system S (˜ α, ˜ β ) for the variables˜ u, ˜ v , with a new value of τ . The value τ = 1 is invariant and corresponds to a linearizablesystem (cf. Section 5.1). The symmetries and conservation laws corresponding to differentvalues τ = 1 are also related by the above nonlocal transformation. Apparently the values τ = 1 /
2, 0, ∞ may be distinguished from the point of view of some additional structuresbesides the symmetry approach. It should be noted that, historically, exactly these valuesappeared as separate scalar systems DNLS-I, DNLS-II and DNLS-III. For consistency, wekeep the notation S ( α, β ) for their non-Abelian analogs, despite the fact that there are nocommon family in this case.A trivial source of multiple non-Abelian generalizations is related with a discrete symme-try group which leaves the scalar system invariant, but changes its noncommutative analog.A most important example is the C -linear involution ⋆ on A defined as follows: u ⋆ = u , v ⋆ = v , ( a b ) ⋆ = b ⋆ a ⋆ , a, b ∈ A . (20)It is clear that this involution applied to any non-Abelian generalization gives some another,possibly different, generalization of the same scalar system. Another possible transformationmay be related with a scaling of unknown variables. We consider non-Abelian analogs relatedby the involution (20) or by a scaling as equivalent. Theorem 2.
If a non-Abelian analog of one of the scalar systems S – S , B – B satisfiesthe conditions – then it is equivalent to one of the systems listed in the second column ofthe Table 1, with explicit equations for these systems given in Section 4. weights of u, vS S ′ ν, − ν ) S S ′ , S ′′ , S S ′ , S S ′ , S ′′ , S ( α, β ) 0 ( ν, − ν ) S (1 , S ′ (1 , , S ′′ (1 ,
2) 3 S (0 , S ′ (0 , , S ′′ (0 , , S ′′′ (0 ,
1) 6 S (1 , S ′ (1 , , S ′′ (1 ,
0) 3 B , B B ′ , B B ′ , B ′′ , B B ′ , B ′′ , Table 1.
The notations of noncommutative analogs. The last two columnscontain the total number N of analogs (that is, without taking the involution andthe scaling into account) and the weights of homogeneity. Generally speaking, the existence of several higher conservation laws does not guaranteeintegrability in any sense. In order to make sure that the found noncommutative systems areintegrable indeed, we find for each system a zero curvature representation U t = V x + [ V, U ] , (21)where U and V are homogeneous polynomial matrices depending on the spectral parameter λ . These representations are also given in Section 4.Note that in the scalar case, the zero curvature representations admit transformations ofthe form U U + ρ I, V V + σ I, (22)where ρ t = σ x is any conservation law of the system. However, in the noncommutativesituation this transformation is not generally allowed. In this case the elements on thediagonal turn out to be more rigidly fixed and there is only a finite number of choices whichbring to different noncommutative analogs.In Section 5, we give noncommutative generalizations of linearizable systems of the form(17) and differential substitutions relating them with linear systems. This section containsalso examples of master-symmetries and B¨acklund auto-transformations for some of non-Abelian systems presented in Section 4. In this section, we outline the proof of the Theorem 2 by example of the system B . Thesimplest conserved densities for it are of the form ρ = v, ρ = u, ρ = uv, ρ = 32 u + v − uv x ,ρ = uu xx + u v + 13 v − uvv x + v v xx − uv xxx . Here the subscripts denote the weights of the densities (there are no densities of weights of theform 3 n + 1 for this system). The most general nocommutative system which is homogeneous6ith respect to the same weights as for B and which turns into B in the commutative caseis of the form ( u t = u xx + α vv x + (2 − α ) v x v + β ( uv − vu ) , v t = − v xx + u x . (23)The parameters α and β must be determined by condition that (23) admits non-Abelianconserved densities which turn into the above ρ i in the scalar case.The most general non-Abelian density corresponding to ρ is of the form ¯ ρ = v . It isclear that D t (¯ ρ ) = − v xx + u x ∈ Im D ⊂ T , that is, this gives no information about the parameters α , β . The conditions that ¯ ρ = u and ¯ ρ = u v are non-Abelian densities are satisfied identically as well. For instance, for ¯ ρ we have D t (¯ ρ ) = D ( u x + v ) + ( α − v , v x ] + β [ u , v ] ∈ T . Let us take the next density. One can easily verify that any noncommutative homogeneouspolynomial of weight 6 which coincide with ρ in the scalar case is of the form32 u + k ( u x v − vu x ) + k ( uv x − v x u ) − v x u + k ( vv xx − v xx v ) + v , k i ∈ C . Since non-Abelian densities are defined modulo T , we may set k = k = k = 0 and assumewithout loss of generality that ¯ ρ = 32 u + v − v x u . It is not difficult to prove that D t (¯ ρ ) = D ( K ) + αK + βK + K + 3( α − β − v x uv − v x vu ) , where K = − D ( v ) + 3 v xx u − v x u x + uv + vuv + v u ,K = 3[ v x , v x v ] −
32 [ v x , uv ] + 32 [ v , v x u ] −
32 [ u , v x v ] , K = 32 [ u , v ] ,K = 32 [ u , u xx ] + 2[ v , v x ] − v x , v x v ] + [ u , v x v ] + [ v x , uv ] − [ v , uv x ] − [ v , v x u ] . Let us prove that the polynomial P = v x uv − v x vu does not belong to T . Since all termsin D t (¯ ρ ) except the last one belong to T , this will imply that α = β + 1 . In order to prove this we use the following property of the non-Abelian variational derivatives: δδ u ( a ) = δδ v ( a ) = 0 , ∀ a ∈ T . The variational principle for functionals of the form (16) implies the following definition of thevariational derivatives. One has to perform the substitution u i u i + ε ∆ ui , v i v i + ε ∆ vi into the arguments of the polynomial a and to compute the coefficient at ε (that is, thedifferential ∆ a ). For the polynomial P , we obtain∆ P = ∆ vx uv + v x ∆ u v + v x u ∆ v − ∆ vx vu − v x ∆ v u − v x v ∆ u . a to the form ∆ u R + ∆ v S by subtracting commutators and total x -derivatives. The polynomials R and S ∈ A are uniquely defined and are denoted δaδ u and δaδ v , respectively. In our example, we have∆ P − D (∆ v uv ) − [ v x , ∆ u v ] − [ v x u , ∆ v ] + D (∆ v vu ) + [ v x , ∆ v u ] + [ v x v , ∆ u ] == ∆ u [ v , v x ] + ∆ v (cid:0) [ v , u x ] + 2[ v x , u ] (cid:1) . Therefore, δδ u ( P ) = [ v , v x ] , δδ v ( P ) = [ v , u x ] + 2[ v x , u ] . Since the variational derivatives do not vanish, P does not belong to T . Remark . The result of the above computation procedure for the variational derivativescan be given by an explicit expression. Let P be an arbitrary polynomial of noncommutingvariables u i , . . . , u Ni (where subscripts denote the order of derivatives with respect to x ). Letus denote, for any monomial p = u s i . . . u s m i m , L k ( p ) = u s i . . . u s k − i k − , C k ( p ) = u s k i k , R k ( p ) = u s k +1 i k +1 . . . u s m i m , so that always p = L k ( p ) C k ( p ) R k ( p ). Then, given a polynomial P = P j α j p j , we have δPδ u s = X j α j X k,i : C k ( p j )= u si ( − D ) i ( R k ( p j ) L k ( p j )) , s = 1 , . . . , N. The parameter β is determined at the next step. It is not difficult to prove that the mostgeneral ansatz for the density ¯ ρ can be taken in the form¯ ρ = u xx u − v xxx u + vu + ζ v x uv − (2 + ζ ) v x vu + v xx v + 13 v . Notice that a free parameter ζ appeared in this density for the first time. The presence of suchparameters leads to nonlinear algebraic relations for the coefficients of the noncommutativegeneralization that we are looking for. A direct computation shows that vanishing of thevariational derivatives of the expression D t (¯ ρ ) is equivalent to the system of equations ζ + β + 1 = 0 , β (1 + ζ ) = 1 . From here we obtain β = ± i √ /
3. Two non-Abelian systems corresponding to different signsof β are related by involution (20).As the weights of u and v decrease, the number of monomials participating in the right-hand sides of the noncommutative system and in expressions for their densities increasesand the calculations become more cumbersome. However, they remain fairly straightforwardand, what is important, a large part of the relations for the unknown coefficients are linearequations. As a rule, quadratic relations arise only at the last step and are easily analyzed. In this section we present all non-Abelian analogs admitting conserved densities for systemsfrom the lists S n , B n . In order to justify their integrability we provide the zero curvaturerepresentations (21). The search of the matrices U and V was performed under assumption8hat their entries are homogeneous polynomials from A . Another assumption was that, as inthe scalar case, the size of the matrices is 2 × × U and V . It turns out thatthis system is solvable for all systems under considerations, although bringing the answer toan ‘elegant’ form may require some efforts.In what follows, I denotes the unity of the algebra A and I denotes the 2 × × I on its diagonal. S The system S admits one noncommutative analog: ( u t = u xx + 2 uvu , v t = − v xx − vuv . S ′ It is clear that this system is invariant with respect to the involution (20). The matrices ofthe zero curvature representation are of the form (see e.g. [18]) U = (cid:18) λ I − vu − λ I (cid:19) , V = − λU + (cid:18) − vu v x u x uv (cid:19) . S The system S [19] admits two very similar, yet non-equivalent noncommutative generaliza-tions (and two more are obtained from them by involution): ( u t = u xx + 2 u x u + 2( vu ) x + 2[ vu , u ] , v t = − v xx + 2 v x v + 2( uv ) x + 2[ v , uv ] , S ′ ( u t = u xx + 2 uu x + 2( vu ) x + 2[ u , vu ] , v t = − v xx + 2 v x v + 2( vu ) x + 2[ v , vu ] . S ′′ The difference between these two systems can be observed under the reduction v = − u in their third order symetries, which leads to two different generalizations of the mKdVequation: the symmetry of S ′ turns into (14) and the symmetry of S ′′ turns into (13).The matrices ot the zero curvature representation for S ′ are U = (cid:18) I λ v λ I + u − v (cid:19) , V = (2 u I − U ) U + (cid:18) − λ v x u x + v x (cid:19) , the matrices for the systems S ′′ are U = (cid:18) − u I λ v λ I − v (cid:19) , V = − U − (cid:18) u x + 2 vu λ v x − v x + 2 vu (cid:19) . S The system S has, up to the involution, one noncommutative analog (see e.g. [8]): ( u t = u xx + 2( vu ) x , v t = − v xx + 2 v x v + 2 u x + 2[ v , u ] . S ′ ψ xx + ( v − λ ) ψ x + u ψ = 0 , ψ t = ψ xx + 2 v ψ x . The consistency condition for these equations is equivalent to S ′ . This can be cast into theform of representation (21) with the matrices U = (cid:18) I − u λ I − v (cid:19) , V = (cid:18) − u λ I + v − λ u − u x − vu λ I + v x − u − v (cid:19) . S Up to the involution, the system S admits two noncommutative analogs: ( u t = u xx + 2 u x ( u + v ) , v t = − v xx + 2( u + v ) v x , S ′ ( u t = u xx + 2( u + v ) u x + 2[ v x , u ] − u v + 4 uvu − vu , v t = − v xx + 2 v x ( u + v ) + 2[ v , u x ] + 2 uv − vuv + 2 v u . S ′′ The zero curvature representation matrices for S ′ are of the form U = (cid:18) u ( λ I − u )( λ I − v ) I v (cid:19) ,V = U + 2 λU − λ I + (cid:18) u x λ ( v x − u x ) + u x v − uv x − v x (cid:19) ;the matrices for S ′′ read U = (cid:18) − v ( λ I − u )( λ I − v ) I − u (cid:19) ,V = − U + 2 λU + ( λ I + 2[ u , v ]) I + (cid:18) v x λ ( v − u ) x + u x v − uv x − u x (cid:19) . S (1 , The system S (1 ,
2) (DNLS-I or the Kaup–Newell system [20]) admits two noncommutativegeneralizations (the first one is symmetric with respect to the involution and the second oneis not): ( u t = u xx + 2( uvu ) x , v t = − v xx + 2( vuv ) x , S ′ (1 , ( u t = u xx + 2 u v x + 2 u x uv + 2 u x vu + 2[ u v , u ] , v t = − v xx + 2 u x v + 2 uvv x + 2 vuv x − v , u v ] . S ′′ (1 , S ′ (1 , U = 2 λ (cid:18) λ I u − v − λ I (cid:19) , V = 4 λ U + 2 λ (cid:18) λ uv u x + 2 uvuv x − vuv − λ vu (cid:19) ;10or S ′′ (1 , U = (cid:18) λ I + uv λ u − λ v − λ I + uv (cid:19) ,V = U + (4 λ uv + u x v − uv x + 2 u v ) I + 2 λ (cid:18) λ I λ u + u x − λ v + v x − λ I (cid:19) . S (0 , The system S (0 ,
1) is known as DNLS-II or the Chen–Lee–Liu system [21]. It has therenoncommutative analogs (and three more are obtained by involution): ( u t = u xx + 2 uvu x , v t = − v xx + 2 v x uv , S ′ (0 , ( u t = u xx + 2 vuu x + 2[ v x u − v [ v , u ] u , u ] , v t = − v xx + 2 v x vu + 2[ v , vu x + v [ v , u ] u ] , S ′′ (0 , ( u t = u xx + 2 u x vu + 2[ uv , u x ] + 2[ uv x , u ] − u ( u v − uvu + vu ) v , v t = − v xx + 2 vuv x + 2[ v x , uv ] + 2[ v , u x v ] + 2 u ( uv − vuv + v u ) v . S ′′′ (0 , S ′ (0 , U = (cid:18) − λ I λ u λ v − vu (cid:19) , V = − U − λ U + (cid:18) λ u x − λ v x v x u − vu x (cid:19) ;for S ′′ (0 , U = (cid:18) − λ I + [ u , v ] λ u λ v − vu (cid:19) ,V = − U − λ U + 2 v [ u , v ] u I + (cid:18) [ u x , v ] − [ u , v x ] λ u x − λ v x v x u − vu x (cid:19) ;and for S ′′′ (0 , U = (cid:18) − λ I λ u λ v − uv (cid:19) ,V = − U − λ U + (cid:18) λ u x + 2 λ [ uv , u ] − λ v x − λ [ uv , v ] − λ [ u , v ] + uv x − u x v + 2 u [ u , v ] v (cid:19) . It is easy to see that in the scalar case all three representations are equivalent up to thediagonal shift (22) corresponding to the conservation law with the density uv . S (1 , The system S (1 ,
0) is known as DNLS-III or the Gerdjikov–Ivanov system [22]. It has twononcommutative generalizations: ( u t = u xx + 2 uv x u − uvuvu , v t = − v xx + 2 vu x v + 2 vuvuv , S ′ (1 , ( u t = u xx + 2 u v x − u v + 2[ u x + u v , uv ] , v t = − v xx + 2 u x v + 2 u v − v x − uv , uv ] . S ′′ (1 , S ′ (1 ,
0) is defined by the matrices U = (cid:18) − uv λ u − λ v λ I + vu (cid:19) ,V = − λ U + (cid:18) uv x − u x v − ( uv ) λ u x λ v x vu x − v x u + ( vu ) (cid:19) ;similarly, for S ′′ (1 ,
0) we have U = (cid:18) λ u − λ v λ I + uv + vu (cid:19) ,V = − λ U + (cid:18) λ uv λ u x + λ [ u , uv ] λ v x + λ [ v , uv ] λ uv + u x v − uv x + vu x − v x u + 2 u v − [ u , v ] (cid:19) . B The Boussinesq type system B has one noncommutative analog, up to the involution (adetailed analysis of this example is in Section 3): u t = u xx + v x v + vv x + i √
33 [ u − v x , v ] , v t = − v xx + u x . B ′ Remark . Although there are no local non-Abelian analogs for B , this system admits thenonlocal generalization ( u t = u xx + ( u + v ) + w , v t = − v xx − ( u + v ) − w , w x = i √
33 [ u x − v x , u + v ] , (24)which is related with B ′ by non-invertible differential substitution u = 4˜ u x , v = 2˜ u + 2˜ v (variables with tilde correspond to (24)). A systematic description of differential substitutionsconnecting the non-Abelian systems presented in the paper is a separate interesting problem.The zero curvature representations for all Boussinesq type systems are more convenientlywritten in some another variables related with u and v by some invertible linear changes.Such changes lead out of the class of systems (17), because they do not preserve the separant of the system (the matrix at the second order derivatives on the right-hand side). For thesystem B ′ , the transformation ∂ t = − i √ ∂ T , p = i √ u + 3 − i √ v x , q = 16 v (25)brings to the system p T = p xx − q xxx − qq x + 2[ q , p ] , q T = − q xx + 23 p x , (26)which serves as the compatibility condition for the third order spectral problem [23] ψ xxx + 3 q ψ x + p ψ = λψ, ψ T = ψ xx + 2 q ψ. The representation (21) for (26) is given by the matrices U = I
00 0 I λ I − p − q , V = q I λ I + 2 q x − p − q q xx − p x λ I + q x − p − q .
12n order to obtain a representation for B ′ , one has just to replace p and q according to (25).One can see that the appearance of √ B ′ is not related with the passage to thenoncommuting variables, rather this is a price we pay for bringing the system (26) to thecanonical form (the third derivative is eliminated and the matrix at second derivatives ismade diagonal). In fact, the radicals appear already in the scalar case B , hidden in thezero curvature representation, but they cancel in the system itself. In the noncommutativeversion no cancellation occurs and the radicals become visible. B For B , there are two noncommutative analogs, up to the involution (which amounts to thecomplex conjugation, like in the previous example): ( u t = u xx + 3( u + v ) v x + 3 v x ( u + v ) − u + v ) + i √ u + v , v x ] , v t = − v xx + 3( u + v ) u x + 3 u x ( u + v ) + 6( u + v ) − i √ u + v , u x ] , B ′ u t = u xx + 3( u + v ) v x + 3 v x ( u + v ) − u + 3 uvu + uv + vuv + v u + v )+ i √ (cid:0) [ v x , u − v ] + 2[ u + v , u x ] + 6[ u , v ] + 6[ u , v ] (cid:1) , v t = − v xx + 3( u + v ) u x + 3 u x ( u + v )+ 6( u + u v + uvu + vu + 3 vuv + v )+ i √ (cid:0) [ u x , u − v ] − u + v , v x ] − u , v ] − u , v ] (cid:1) . B ′′ The system B ′ admits the representation (21) with U = i √ u + v ) I i √ ε u + ε v ) I λ I i √ ε vu − ε uv ) i √ ε u + ε v ) ,V = − i √ U + i √ u x − v x ε u x − ε v x a ε u x − ε v x , where we denote ε = − / i √ / a = 3 ε ( uv x − u x v ) + 3 ε ( vu x − v x u ) − i √ u + v ) . The matrices for B ′′ are of the form e U = U − i √ u + v ) I, e V = V − i √ u x − v x + 3( u + v ) ) I − u , v ] 00 b [ u , v ] , where b = uvu + ε u v + ε vu + vuv + ε uv + ε v u . .10 System B The system B admits two non-equivalent generalizations: u t = u xx + vv x + v x v + i √
33 [ v , u − v x ] −
13 ( u v − uvu + vu ) , v t = − v xx + uu x + u x u + i √
33 [ u , v + u x ] + 13 ( uv − vuv + v u ) , B ′ u t = u xx + vv x + v x v + u v − uvu + vu − i √ (cid:0) u + v , u x ] + [ v − u , v x ] + [ u , v ] + 2[ u , v ] (cid:1) , v t = − v xx + uu x + u x u − uv + 2 vuv − v u + i √ (cid:0) [ u − v , u x ] + 2[ u + v , v x ] + 2[ u , v ] + [ u , v ] (cid:1) . B ′′ The scalar system is invariant with respect to the scaling with the cubic root of 1: u → ε u , v → ε v , ε = − / i √ / . The system B ′ also does not change under this transformation, while B ′′ turns into twoanother systems with different coefficients. Applying the involution (20) we obtain in total8 noncommutative analogs for B .The linear change ∂ T = i √ ∂ t , p = 12 ( u − v ) , q = i √ u + v )brings to radical-free systems with a non-standard separant: B ′ and B ′′ take, respectively,the form ( p t = 3( q x − pq − qp ) x + [ p , p x − q ] − q , q x ] − p q − pqp + qp ) , q t = ( − p x − p + 3 q ) x + [ p x + 2 p , q ] − [ p , q x ] + 2( pq − qpq + q p ) , (27) ( p t = 3( q x − pq − qp ) x + 3[ p , p x − q ] − q , q x ] + 6( p q − pqp + qp ) , q t = ( − p x − p + 3 q ) x − [5 p x + 2 p , q ] − p , q x ] − pq − qpq + q p ) . (28)A scalar version of these systems was obtained in [24] from a spectral problem with a thirdorder differential operator in factorized form. This spectral problem can be generalized fornoncommuting variables. As the result, we arrive at the zero curvature representation (21)for the system (27), with matrices U = q + p I q − p I λ I − q ,V = − U + q x − p x − q x − p x
00 0 2 p x + 2( p + [ p , q ] + 3 q ) I. The matrices for (28) differ from them in diagonal elements: e U = U + 2 q I, e V = V + diag(0 , , p , q ]) − p x + p + 4[ p , q ] − q ) I. Symmetries of non-Abelian systems
It is well-known that higher symmetries exist not only for equations which are integrable bythe inverse scattering method, but also for the Burgers type equations which admit lineariza-tion by differential substitutions (see e.g. [4]). If we relax our requirements and replace theexistence of conservation laws with the existence of symmetries then the list from Section 2is extended by linearizable systems.
Theorem 3 ([15]) . i) Any system of the form (18) which admits a symmetry of the form ( u τ = u xxxx + f ( u, v, u x , v x , u xx , v xx , u xxx , v xxx ) ,v τ = − v xxxx + g ( u, v, u x , v x , u xx , v xx , u xxx , v xxx ) (29) is polynomial. ii) A homogeneous polynomial non-triangular system (18) admits a symmetry (29) if andonly if it belongs to the lists from Theorem 1 or coincides with one of the systems listed below,up to a scaling of the variables x, t, u, v and the interchange ( u, v ) ( v, u ) : ( u t = u xx + 2 uu x + 2 vu x + 2 uv x + 2 u v + 2 uv ,v t = − v xx − vv x − vu x − uv x − u v − uv , L ( u t = u xx + 2 u x v + 2 uv x + 2 uv + u ,v t = − v xx − vv x − u x , L ( u t = u xx + 2 αu v x + 2 αuvu x − αβu v ,v t = − v xx + 2 βv u x + 2 βuvv x + αβu v , αβ = 0 , L ( u t = u xx + 4 uvu x + 4 u v x + 3 vv x + 2 u v + uv ,v t = − v xx − v u x − uvv x − u v − v , L ( u t = u xx + 4 uu x + 2 vv x ,v t = − v xx − vu x − uv x − u v − v . L In order to obtain an alternative verification of the results presented in Theorem 2 and tofind noncommutative analogs for the systems L – L we use a criterion based on the existenceof symmetries. Theorem 4.
All non-equivalent non-Abelian analogs of the systems S – S , B – B and L – L , which satisfy the following assumptions:— the non-Abelian analog admits a symmetry of the same minimal order as the originalsystem;— the non-Abelian analog and its symmetry are polynomial, homogeneous and admit thescaling group ( x, t, u , v ) −→ ( τ − x, τ − µ t, τ ν u , τ ν v ) with the same µ , ν and ν as for the original system and its symmetry;— the non-Abelian analog and its symmetry turn into the original system and its symmetryunder substitution of commuting variables instead of noncommuting ones ( or, in thematrix language, for the × matrices ) , re exhausted by the systems from Theorem 2 and the systems L ′ , L ′ , L ′ (1 , and L ′′ (1 , given below. The system L has, up to the involution, one noncommutative generalization ( u t = u xx + 2 u x u + 2( uv ) x + 2 uv ( u + v ) , v t = − v xx − v x v − vu ) x − vu ( u + v ) . L ′ It is related with the system ˜ u t = ˜ u xx , ˜ v t = − ˜ v xx by the differential substitution u = ˜ u x (˜ u + ˜ v ) − , v = ˜ v x (˜ u + ˜ v ) − . Here we assume that the algebra A is extended by additional element ( u + v ) − .The system L also has one noncommutative analog, ( u t = u xx + 2( uv ) x + u + 2 uv , v t = − v xx − u x − v x v − [ u , v ] , L ′ which is linearized by substitution u = ˜ u x ˜ u − , v = ˜ v x ˜ u − , where ˜ u t = ˜ u xx , ˜ v t = − ˜ v xx − ˜ u x . For the family L ( α, β ) of the Eckhaus equation type [25, 26] one can assume, taking thescaling into account, that α = 1 and β is a free parameter. It turns out that non-Abeliangeneralizations exist only for β = 0, that is, when the scalar system becomes triangular.Although this degenerate system is not considered in Theorem 3, it still has higher symmetriesand can be treated in the same manner as other examples. For this system there exist twonon-Abelian analogs (up to the involution), moreover, one of them is not triangular: ( u t = u xx + 2 uv x u + 2 u x vu , v t = − v xx , L ′ (1 , ( u t = u xx + 2 u x uv + 2 u v x − u [ u , v ] v , v t = − v xx + 2[ uv , v x − uv ] + 2[ u x v + u v , v ] . L ′′ (1 , L ′ (1 ,
0) admits the following substitution ( u , v ) → ( w , v ): w = u x u − + uv , where w t = w xx + 2 w x w , v t = − v xx , and the Burgers equation for w is linearized by the non-Abelian Cole–Hopf substitution w = ˜ u x ˜ u − , where ˜ u t = ˜ u xx . This define the transform ( u , v ) ↔ (˜ u , v ) which is implicitin both directions (B¨acklund type transformation or correspondence). Similarly, the system L ′′ (1 ,
0) admits the substitution w = u x u − + u vu − , p = uvu − , which brings to the system w t = w xx + 2 w x w , p t = − p xx + 2[ w , p ] x + 2[ w , p ] w , and the latter is linearized by the substitution w = ˜ u x ˜ u − , p = ˜ u ˜ v ˜ u − , where ˜ u t = ˜ u xx , ˜ v t = − ˜ v xx . The systems L (1 , β ) with β = 0, L and L have no noncommutative generalizations ad-mitting the higher symmetries. 16 .2 Examples of master-symmetries and B¨acklund transformations Some of non-Abelian systems presented in the paper admit master-symmetries or B¨acklundtransformations in the form of integrable lattice equations. We have not investigated theseadditional structures in full generality and will only give a few examples that we have beenable to find.For the sake of simplicity we restricted ourselves by consideration of local master-symmetries,although they usually involve nonlocalities even in the scalar case. For instance, a localmaster-symmetry exists for the system S (see e.g. the review article [5] where master-symmetries were given for some of NLS type systems). We have found a generalizationonly for S ′ : ( u τ = x ( u xx + 2 u x u + 2( vu ) x + 2[ vu , u ]) + 2 u x + u + 3 vu , v τ = x ( − v xx + 2 v x v + 2( uv ) x + 2[ v , uv ]) − v x + v + 3 uv . (30)A master-symmetry for the system S ′′ remains unknown (it may exist, but it may require theintroduction of some nonlocality, which disappears in the scalar case). Another well-knownresult is that the system S admits the B¨acklund-Schlesinger transformation in the formof the Volterra lattice [19]. A noncommutative generalization of this fact is known for thesystem S ′′ : one can prove by a direct computation that the non-Abelian Volterra lattice [27] q n,x = q n q n +1 − q n − q n (31)possesses the symmetry q n,t = q n q n +1 q n +2 + q n q n +1 + q n q n +1 − q n − q n − q n − q n − q n − q n − q n (32)and that for any n the variables u = q n +1 , v = q n satisfy the system S ′′ in virtue of thesetwo lattice equations.The system S ′ (1 ,
2) has the local master-symmetry ( u t = x u xx + 2( x uvu ) x , v t = − x v xx + 2( x vuv ) x − v x . (33)The B¨acklund-Schlesinger transformation for the DNLS-I systems is defined by the modifiedVolterra lattice. It is known that it admits two non-Abelian generalizations. We write downboth the lattice and its symmetry of second order. The pair of equations q n,x = q n ( q n +1 − q n − ) q n , q n,t = q n q n +1 ( q n +2 + q n ) q n +1 q n − q n q n − ( q n + q n − ) q n − q n is consistent and the variables u = q n +1 , v = q n satisfy the system S ′ (1 ,
2) [28]. Similarly,for the consistent pair q n,x = q n +1 q n − q n q n − , q n,t = q n +2 q n +1 q n + q n +1 q n q n +1 q n + q n +1 q n q n − q n − q n q n +1 q n q n − − q n q n − q n q n − − q n q n − q n − , the variables u = q n +1 and v = q n satisfy S ′′ (1 ,
2) (notice that the substitution ˜q n = q n +1 q n leads to the Volterra lattice (31), up to the involution).It would be interesting to find lattice representations for other non-Abelian systems.However, such lattice equations can be not polynomial even in the scalar case, and theirgeneralization may require consideration of noncommutative Laurent polynomials.17 Conclusion and perspectives
In this paper, we have found non-Abelian generalizations for some key integrable systemsof NLS and Boussinesq types. Basically, we restricted ourselves to the case of homogeneouspolynomial quasilinear systems of the form (18). For this class of equations we were able togive complete classifications of non-Abelian analogs based both on the existence of higherconservation laws (Theorem 2) and the higher symmetries (Theorem 4). While some of thegeneralizations obtained are well-known, there were also surprisingly many new examples.The integrability of each equation is justified either by explicit zero-curvature representationor by linearizing substitution.Our approach is suitable not only for quasilinear systems. The reason for this restrictionwas only that all such systems are polynomial and their complete list was already obtainedin [15]. It would be quite possible to find noncommutative generalizations for any integrablepolynomial homogeneous systems (17) with positive weights, however one should start thisproject by compiling a complete list of such scalar systems.
Example . The well-known scalar system [14] ( u t = u xx + 6 v x + 18 v v x − uv x ,v t = − v xx + u x , (34)which contain a quadratic term in derivatives, has the following non-Abelian analogs: u t = u xx + 6 v x + 6( v ) x − uv x − v x u + i √ (cid:0) v , v xx ] + 3[ v , v x − u ] + [ u , v ] x (cid:1) , v t = − v xx + u x + i √ v , v x − u ] , (35)and u t = u xx + 6 v x + 18 vv x v − uv x − v x u − i √ (cid:0) v , v xx ] + 3[ v , v x − u ] − [ u , v ] x (cid:1) , v t = − v xx + u x + i √ v , v x − u ] . (36)The differential substitution ˜u = 2 u x + 3( u + v ) , ˜v = u + v relates the systems (35) (written for the variables with tilde) with the system B ′ .As already noted, some of the systems from Theorem 1 admit the addition of terms oflower weight preserving the integrability property. Non-commutative generalizations mayexist for such systems as well. Example . It is well-known that the scalar system S (1 ,
2) (DNLS-I) admits the followinggeneralization by adding the linear terms [14, 15]: ( u t = u xx + 2( u v ) x + αv x ,v t = − v xx + 2( uv ) x + βu x (37)(the constants α and β can be scaled either to 1 or 0, without loss of generality, so that,taking the interchange of u and v into account, we have here just two essentially differentversions in addition to the homogeneous case). There exist two noncommutative analogs for18his system which coincide with S ′ (1 ,
2) and S ′′ (1 ,
2) for the zero values of parameters. Thenon-homogeneous extension of the system S ′ (1 ,
2) is of the form u t = u xx + 2( uvu ) x − γ v x − γ [ u , v ] x − δ [ u , u x + 2 uvu ]+ 4 γδ ( u v − uvu + vu ) + 4 γ δ [ u , v ] , v t = − v xx + 2( vuv ) x − δ u x + 2 δ [ u , v ] x − γ [ v , v x − vuv ] − γδ ( uv − vuv + v u ) + 4 γδ [ u , v ] , (38)where α = − γ and β = − δ . The signs of γ and δ can be chosen arbitrarily, resulting infour extensions of the scalar system (as usual, the involution halves this number). For thenon-Abelian system S ′′ (1 , ( u t = u xx + 2 u v x + 2 u x uv + 2 u x vu + 2[ u v , u ] + α ( v x + [ v , u ]) , v t = − v xx + 2 u x v + 2 uvv x + 2 vuv x − v , u v ] + β ( u x + [ u , v ]) . (39)The statement is that the systems (38) and (39) give all non-Abelian generalizations of (37),up to the involution, and that both systems admit higher symmetries and conservation lawsfor any values of parameters.A limitation of our approach is the assumption that the weights of u and v should bepositive. This is always true for quasilinear systems, but there exist also integrable equationsof more general form for which the weights may be zero or even negative. A simple exampleis given by the potential DNLS system ( u t = u xx + 2 u x v x ,v t = − v xx + 2 u x v x with ν + ν = −
1. In such cases, it is necessary to introduce additional selection rules inorder to bound the number of monomials under consideration.The setting of classification problems for the noncommutative systems also requires pon-dering. The main problem is that the set of admissible transformations becomes significantlynarrower compared to the commutative case. For instance, it was proved in [13] that if theseparant of a scalar integrable system depends on u and v then this system can be broughtto the form (17) by some change of variables. The examples (35) and (36) suggest that thismay be not true in for non-Abelian systems. Because of this, the scalar systems related bypoint changes (not saying about more general transformations) with systems from Theorem 1may have noncommutative generalizations which are not equivalent to the systems obtainedin our paper.Although it is clear that the theory of transformations must be changed in the noncommu-tative setting, at the moment it is difficult to say how this affects the choice of the canonicalforms of integrable equations. The examples related with systems of the Boussinesq typesuggest that in some aspects other canonical forms may turn more convenient.The generalization of the method to various classes of systems with rational right-handside also requires additional research. An example of such a system is the non-AbelianHeisenberg equation [9] ( u t = u xx − u x ( u + v ) − u x , v t = − v xx + 2 v x ( u + v ) − v x . Thus, we see that in the theory of noncommutative integrable equations, there are manyunsolved problems, with potentially rich and interesting answers. Their study should be thesubject of further research. 19 cknowledgements
This work was carried out under the State Assignment 0033-2019-0006 (Integrable systemsof mathematical physics) of the Ministry of Science and Higher Education of the RussianFederation.
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