In this note we discuss some formal properties of universal linearization
operator, relate this to brackets of non-linear differential operators and
discuss application to the calculus of auxiliary integrals, used in
compatibility reductions of PDEs

Kruglikov, Boris

English

arXiv

In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs

NORA - Norwegian Open Research Archives

Anomaly of linearization and auxiliary integrals

Munin - Open Research Archive

arXiv:0712.3497v1  [math.DG]  20 Dec 2007ANOMALY OF LINEARIZATIONAND AUXILIARY INTEGRALS.BORIS KRUGLIKOVAbstract. In this note we discuss some formal properties of uni-versal linearization operator, relate this to brackets of non-lineardifferential operators and discuss application to the calculus of aux-iliary integrals, used in compatibility reductions of PDEs.IntroductionCommutator [∆,∇] of linear differential operators ∆,∇ ∈ Diff(π, π)in the context of non-linear operators F, G ∈ diff(π, π) is up-graded tothe higher Jacobi bracket {F, G}, which plays the same role in com-patibility investigations and symmetry calculus.1The linearization operator relates non-linear operators on a bundleπ with linear operators on the same bundle, whose coefficients shouldbe however smooth functions on the space of infinite jets. The latterspace is the algebra of C -differential operators and we get the mapℓ : diff(π, π) → C Diff(π, π) = C∞(J∞π) ⊗C∞(M) Diff(π, π),defined by the formula [KLV]ℓF (s)h =ddtF (s + th)|t=0, F ∈ diff(π, π), s, h ∈ C∞(π).However it does not respect the commutator:[ℓF , ℓG] 6= ℓ{F,G}.Example: Consider the scalar differential operators on R, so thatπ = 1 and J∞(π) = R∞(x, u, p = p1, p2, . . . ). ChooseF = p2, G = p + c · x; {F, G} = 2c p =⇒ ℓ{F,G} = 2cDx.If we commute ℓF = 2pDx and ℓG = Dx, we get: [ℓF , ℓG] = −2p2 Dx,so that we observe an anomaly.There are two reasons for this. The first is that the operator oflinearization disregards non-homogeneous linear terms, which are im-portant for the Jacobi bracket. The second is the non-linearity itself.1MSC numbers: 35A27, 58A20; 58J70, 35A30.Keywords: Linearization, evolutionary differentiation, compatibility, differentialconstraint, symmetry, reduction, Jacobi bracket, multi-bracket.1The goal of this note is to discuss reasons and consequences of thisanomaly (this also plays a significant role in investigation of coveringsand non-local calculus [KKV]).Acknowledgement. The results were obtained and systematizedduring the research stay in Max Planck Institute for Mathematics inthe Sciences, Leipzig, in April-May 2007.1. Anomaly via HessianThe Jacobi bracket of non-linear operators F, G ∈ diff(π, π) is ex-pressed via linearization as follows:{F, G} = ℓF G − ℓGF.We also consider the evolutionary operators defined by duality:ЗF G = ℓGF.Since ℓG is a derivation in G, ЗF is a derivation (satisfies the Leibnizrule) and their union can be treated as the module of vector fields.These operators have no anomaly, i.e. the map З : C∞(J∞π) →Vect(J∞π) is an anti-homomorphism:[ЗF ,ЗG] = −З{F,G}.This instantly implies Jacobi identity for the bracket {F, G}, so that(diff(π, π), {, })is a Lie algebra [KLV].The operators of universal linearization and evolutionary differenti-ation do not commute and this leads to the followingDefinition. The Hessian operator diff(π, π)×diff(π, π) → C Diff(π, π)is defined by the formulaHessF G = [ЗG, ℓF ].We will also write HessF (G, H) = HessF G(H) for F, G, H ∈ diff(π, π)and note that HessF ≡ 0 for linear operators F , because in this caseℓF = F , which reduces the claim to the commutation of left and rightmultiplications.Next we note that the Hessian HessF is symmetric:Lemma 1. HessF (G, H) = HessF (H, G).Indeed:HessF (G, H) = ЗGℓF H − ℓFЗGH = ЗGЗHF − ℓF ℓHG,so thatHessF (G, H) − HessF (H, G) = [ЗG,ЗH ]F − ℓF{H, G}= −З{G,H}F − ℓF{H, G} = 0.2Now we can express the anomaly of linearization via the Hessian:Proposition 2. [ℓF , ℓG] − ℓ{F,G} = HessG F − HessF G.Indeed we have:[ℓF , ℓG]H = ℓFЗHG − ℓGЗHF= ЗH(ℓFG − ℓGF ) − HessF (H, G) + HessG(H, F )= ЗH{F, G} − HessF (G, H) + HessG(F, H)= ℓ{F,G}H + (HessG F − HessF G)H.Finally let us express the Leibniz identity for non-linear operatorsand the Jacobi bracket. For linear operators it is well-known, but fornon-linear ones there’s an anomaly:Proposition 3. {F, ℓGH} = ℓ{F,G}H + ℓG{F, H} − HessF (G, H).This is obtained as follows:{F, ℓGH} = ℓF ℓGH − ЗF ℓGH= [ℓF , ℓG]H + ℓG(ℓF − ЗF )H − HessG(F, H)= ℓ{F,G}H + ℓG{F, H} − HessF (G, H).2. Coordinate expressionsA local coordinate system (xi, uj) on π induces the canonical coordi-nates (xi, pjσ) on the space J∞π, where σ = (i1, . . . , in) is a multi-indexof length |σ| = i1 + · · ·+ in. The operator of total derivative of multi-order σ (and order |σ|) is Dσ = Di11 · · ·Dinn , where Di = ∂xi+∑pjτ+1i∂pjτ .The linearization of F = (F1, . . . , Fr) is ℓF = (ℓ(F1), . . . , ℓ(Fr)) withℓ(Fi) =∑(∂pjσFi) · D[j]σ ,where D[j]σ denotes the operator Dσ applied to the j-th component ofthe section from C∞(π).The i-th component of the evolutionary differentiation ЗG corre-sponding to G = (G1, . . . , Gn) equalsЗiG =∑(DσGj) · ∂pjσ[i],where ∂pjσ[i] denotes the operator ∂pjσapplied to the i-th component ofthe section from C∞(π).3Then i-th components of the Jacobi bracket is given by{F, G}i =∑(Dσ(Gj) · ∂pjσFi −Dσ(Fj) · ∂pjσGi).These formulas are known [KLV]. It is instructive to demonstrate theJacobi identity in coordinates. For this we need the following assertion.Lemma 4. In canonical coordinates on J∞π:∂piσDτ =∑Dτ−κ∂piσ−κ(the difference of multi-indices σ − κ is defined whenever κ ⊂ σ),the summation is by κ counted with multiplicity. More generally forvector differential operators if D[j]σ is the operator Dσ acting on the j-thcomponent, then the above formula holds true for such specification.This follows from iteration of the formula [∂pjσ,Di] = ∂pjσ−1i. Thus{F, {G, H}} =∑FpσDσ−κ(Gpτ )Dτ+κ(H)−FpσDσ−κ(Hpτ )Dτ+κ(G)− GpσpτDτ (H)Dσ(F ) + HpσpτDτ (G)Dσ(F )− (GpσDσ−κ(Hpτ−κ ) − HpσDσ−κ(Gpτ−κ ))Dτ (F ),which yields∑cyclic{F, {G, H}} = 0.Now we write the Hessian:HessF (G, H) =∑FpσpτDσG · DτH,and its symmetry in G, H and vanishing for linear F is obvious.The compensated Leibniz formula can be written as follows:{F, ℓGH} − ℓ{F,G}H − ℓG{F, H} =∑FpσDσ−κ(Gpτ )Dτ+κ(H)−(GpσpτDτ (H)Dσ(F )+Gpτ ∂pσDτ (H))Dσ(F )−(FpσpτDσ(G)+Fpσ∂pτDσ(G))Dτ(H)+(GpσpτDσ(F )+Gpσ∂pτDσ(F ))Dτ (H)−Gpσ (Dσ−κ(Fpτ )Dτ+κ(H)−Dσ−κ(Hpτ )Dτ+κ(F )) = −HessF (G, H)and the anomaly in commuting linearizations is:[ℓF , ℓG] − ℓ{F,G} =∑FpσDσ−κ(Gpτ )Dτ+κ(H) − GpσDσ−κ(Fpτ )Dτ+κ(H)−(FpσpτDσ(G)+Fpσ∂pτDσ(G))Dτ(H)+(GpσpτDσ(F )+Gpσ∂pτDσ(F ))Dτ (H)= HessG(F, H) − HessF (G, H).This gives an alternative proof of Propositions 3 and 2.43. Auxiliary integralsDefinition. An operator G ∈ diff(π, π) is called an auxiliary integralfor F ∈ diff(π, π) if{F, G} = ℓλF + ℓµGfor some operators λ ∈ diff(π, π) and µ 6∈ C Diff(π, π) · F \ {0}. Theset of such G is denoted by Aux(F ).It is better to denote Auxµ(F ) the space of G satisfying the aboveformula with some fixed µ ∈ diff(π, π), because it is a vector space.Then Aux(F ) = ∪µ Auxµ(F ). We can assume ord(µ) < ord(F ) forscalar operators, i.e. rankπ = 1.With certain non-degeneracy condition for the symbols of F, G thefollowing statement holds:Theorem 5. A non-linear differential operator G is an auxiliary inte-gral for another operator F iff the system F = 0, G = 0 is compatible(formally integrable).The generic position condition for the symbols of F, G is essential. Ifπ = 1 is the trivial one-dimensional bundle, this condition is just thetransversality of the characteristic varieties CharC(F ) and CharC(G)in the bundle PCT ∗M (after pull-back to the joint system F = G = 0in jets); in this form it is a particular form of the statement provedin [KL2]. For rankπ > 1 the condition is more delicate and will bepresented elsewhere.Notice that Aux0(F ) = Sym(F ) is the space of symmetries of F .This is a Lie algebra with respect to the Jacobi bracket. It can berepresented as a union of spacesSymθ(F ) = {H : ℓFH = ℓθ+HF}, θ ∈ diff(π, π),which are modules over Sym0(F ). More generally we have the gradedgroup: Symθ′(F ) + Symθ′′(F ) ⊂ Symθ′+θ′′(F )Let us assume G ∈ Auxµ(F ), H ∈ Symθ(F ), i.e.{F, G} = ℓλF + ℓµG, {F, H} = ℓθF.Then denoting adH = {H, ·} = ℓH − ЗH we get:5adF{G, H} = {adF G, H} + {G, adF H}= −{H, ℓλF + ℓµG} + {G, ℓθF}= ℓ{λ,H}F +ℓλ{F, H}+HessH(λ, F )+ℓ{µ,H}G+ℓµ{G, H}+HessH(µ, G)− ℓ{θ,G}F − ℓθ{F, G} − HessG(θ, F )= (ℓ{λ,H} + [ℓλ, ℓθ] − ℓ{θ,G} + HessH λ − HessG θ)F + ℓµ{G, H}+ (ℓ{µ,H} − ℓθℓµ + HessH µ)G.Thus {G, H} is an auxiliary integral for F if ℓθℓµ = ℓ{µ,H} +HessH µ(the ”iff” condition means the difference annihilates G), which can bewritten asµ ∈ Ker[(ℓθ + ℓadH − HessH) ◦ ℓ ].Such a pair θ ∈ sym∗(F ) = Sym(F )/ Sym0(F ), H ∈ Symθ(F ) deter-mines the action of the second componentadH : Auxµ(F ) → Auxµ(F ).Also sinceℓ{µ,H}G = ЗG{µ, H} = ЗGЗH(µ) − ЗGℓH(µ) = (ЗH − ℓH)ЗG(µ)−З{G,H}µ−HessH(G, µ) = − adH ЗG(µ)−HessH(µ, G)− ℓµ{G, H},we have:ℓµ{G, H} + (ℓ{µ,H} − ℓθℓµ + HessH µ)G = −(adH +ℓθ)ℓµG.Thus if H ∈ Symθ(F ), i.e. (adH +ℓθ)F = 0, and µ ∈ Ker[(adH +ℓθ)◦ℓ ],i.e. (adH +ℓθ)ℓµ = 0, thenadH : Auxµ(F ) → Sym(F ).4. Symmetries and compatibilityIt has been a common belief that if G ∈ Sym(F ), then the systemF = 0, G = 0 is compatible, which forms the base of investigation forauto-model solutions. This is however not always true.Example: Let F, G be two linear diagonal operators with constantcoefficients. Then {F, G} = 0 (in this case the Jacobi bracket is thestandard commutator), so that G is a symmetry of F . However thesystem F = 0, G = 0 is usually incompatible: for generic F, G ofthe considered type the only solution will be the trivial zero vector-function.More complicated non-diagonal operators are possible, but it wouldbe better to consider non-homogeneous linear operators. Then if the6coefficients are constant and generic, the linear matrix part commute,but the system F = 0, G = 0 may have no solutions at all.For instance if we takeF =[(D2x −Dy) 00 (DxDy + 1)]·[uv]−[10],G =[(DxDy − 1) 00 (D2y −Dx)]·[uv]+[01],then {F, G} = 0, so that G ∈ Sym(F ), while the system F = 0, G = 0is not compatible, and moreover its solutions space is empty.Thus the flow ut = G(u) on the equation F = 0 has no fixed points(no auto-model solutions). Here t is an additional variable (x is thebase multi-variable for PDEs F = 0 and G = 0), so that G ∈ Sym(F )can be expressed as compatibility of the systemF (u) = 0, ut = G(u),while symmetric solutions correspond to the stationary case ut = 0, i.e.compatibility of the system F (u) = 0, G(u) = 02.However if the non-degeneracy condition assumed in Theorem 5 issatisfied, then auto-model (or invariant) solutions exist in abundance,namely they have the required functional dimension and rank as Hilbertpolynomial (or Cartan test [C]) predicts, see [KL4].Remark. Symmetric solutions are the stationary points of the evolu-tionary fields and they are similar to the fixed points of smooth vectorfields on Rn, which must exist provided the vector field is Morse atinfinity. The non-degeneracy condition plays a similar role.Many examples of auto-model solutions and their generalizations canbe found in [BK, Ol, Ov], non-local analogs use the same technique andsimilar theory [KLV, KK, KKV].Compatible systems correspond to reductions of PDEs and are some-times called conditional symmetries by analogy with finite-dimensionalintegrable systems on one isoenergetic surface [FZ]. But the rigorousresult must rely on certain general position property for the symbolof differential operators, otherwise it can turn wrong [KL2, KL3]. Themethod based on this approach makes specification of the general ideaof differential constraint and is described in [KL1].2I am grateful to S.Igonin and A.Verbovetsky for an enlightening discussionabout the results of [KL2, KL3] and the symmetry condition.75. ConclusionIn this note we described the higher-jets calculus corresponding tosymmetries and compatible constraints, basing on the Jacobi brackets.Another approach to integrability of vector systems is given by minimaloverdetermination and it uses multi-brackets of differential operators{· · · } : Λm+1 diff(m · 1, 1) → diff(m · 1, 1)introduced in [KL3], which are governed by the non-commutative Plückeridentity.Following this approach a minimal generalization of symmetry forF = (F1, . . . , Fm) ∈ diff(π, π) with π = m ·1 is such G ∈ diff(π, 1) that{F1, . . . , Fm, G} = ℓθ1F1 + · · ·+ ℓθmFm.With certain non-degeneracy assumption [KL3] this implies that theoverdetermined system F = 0, G = 0 is compatible (formally inte-grable).A more advanced algebraic technique would yield another higher-jets calculus producing anomaly that manifests in non-vanishing of theexpression{ℓF1, · · · , ℓFm+1} − ℓ{F1,··· ,Fm+1}.Implications for vector auxiliary integrals and generalized Lagrange-Charpit method follow the same scheme.References[BK] G. W. Bluman, S. Kumei, Symmetries and differential equations , Appl.Math. Sci. 81, Springer, 1989.[C] E. Cartan, Les systèmes différentiels extérieurs et leurs applicationsgéométriques (French), Actualités Sci. Ind. 994, Hermann, Paris (1945).[FZ] W.I. Fushchych, R.Z. Zhdanov, Conditional symmetry and reduction of par-tial differential equations , Ukrain. Math. J. 44 (1992), 970–982.[KKV] P. Kersten, I. S. Krasilschik, A. Verbovetsky, Hamiltonian operators andℓ∗-coverings, J. Geom. and Phys., 50 (2004) 273–302.[KLV] I. S. Krasilschik, V. V. Lychagin, A. M. Vinogradov, Geometry of jet spacesand differential equations , Gordon and Breach (1986).[KL1] B. S. Kruglikov, V. V. Lychagin, A compatibility criterion for systems ofPDEs and generalized Lagrange-Charpit method , A.I.P. Conference Proceed-ings, Global Analysis and Applied Mathematics: International Workshop onGlobal Analysis, 729, no. 1 (2004), 39–53.[KL2] B. S. Kruglikov, V. V. Lychagin, Mayer brackets and solvability of PDEs –II , Trans. Amer. Math. Soc. 358, no.3 (2005), 1077–1103.[KL3] B. S. Kruglikov, V. V. Lychagin, Compatibility, multi-brackets and inte-grability of systems of PDEs, prepr. Univ. Tromsø 2006-49; ArXive:math.DG/0610930.8[KL4] B. S. Kruglikov, V. V. Lychagin, Geometry of Differential equations, in:D. Krupka, D. Saunders, Handbook of Global Analysis (2007); prepr.IHES/M/07/04.[KK] I.S. Krasilshchik, P.H.M. Kersten, Symmetries and recursion operators forclassical and supersymmetric differential equations, Kluwer (2000).[Ol] P. Olver, Applications of Lie groups to differential equations, Graduate Textsin Mathematics, 107, Springer-Verlag, New York (1986).[Ov] L. V. Ovsiannikov, Group analysis of differential equations, Russian: Nauka,Moscow (1978); Engl. transl.: Academic Press, New York (1982).Institute of Mathematics and Statistics, University of Tromsø, Tromsø90-37, Norway.E-mail address : kruglikov@math.uit.no9

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