---

INVARIANT INTEGRABILITY CRITERION FOR

arXiv:solv-int/9407003v1 18 Jul 1993INVARIANT INTEGRABILITY CRITERION FORTHE EQUATIONS OF HYDRODYNAMICAL TYPE.Pavlov M.V., Sharipov R.A., Svinolupov S.I.Abstract. Invariant integrability criterion for the equations of hydrodynamicaltype is found.This criterion is written in the form of vanishing for some tensorwhich is derived from the velocities matrix of hydrodynamical equations.1.

Introduction.Systems of quasilinear partial differential equations of the first order arise indifferent models describing the motion of continuous media. Special subclass of suchsystems is known as a systems of equations of hydrodynamical type.

In spatiallyone-dimensional case they are written as follows(1.1)uit =nXj=1Aij(u)ujx, where i = 1, . .

. , nAmong the systems (1.1) man can consider special subclass of systems possessingthe Riemann invariants.

These are the systems which can be transformed to thediagonal form(1.2)uit = λi(u)uix, where i = 1, . .

. , nby means of so called point transformations(1.3)˜ui = ˜ui(u1, .

. .

, un), where i = 1, . .

. , nSystem of equations (1.1) is called the hydrodynamically integrable system if it hasthe continuous set of hydrodynamical symmetries (or hydrodynamical conservationlaws) parameterized by n arbitrary functions of one variable.

For diagonal systems(1.2) with mutually distinct characteristic velocities (λi ̸= λj) one has the well-developed theory of integration (see reviews [1] and [2]). As it was shown in [2] thediagonal system (1.2) is integrable if and only if the following condition is satisfied(1.4)∂i ∂jλkλj −λk= ∂j ∂iλkλi −λk, where i ̸= k ̸= jThis research was supported by Grant #RK4000 from International Science Foundation andby Grant #93-011-16088 from Russian Fund for Fundamental Researches.Typeset by AMS-TEX1

2PAVLOV M.V., SHARIPOV R.A., SVINOLUPOV S.I.where ∂i = ∂/∂ui and ∂j = ∂/∂uj.Such systems are called semi-Hamiltoniansystems, for the property (1.4) itself in Russian papers the term semihamiltonity isused. When the diagonal system (1.2) possess this property it can be integrated bymeans of ”generalized hodograph method” (details see in [2])In section 2 of this paper we consider the problem of hydrodynamical integra-bility for the systems of equations (1.1) with the velocities matrix Aij(u) of generalposition i.e.

eigenvalues of which are mutually distinct. There we managed to provethe following fact: each hydrodynamically integrable system (1.1) with the matrixof general position is necessarily diagonalizable1.

Summing up this fact with theresults of [2] we may state the following theoremTheorem 1. System of equations (1.1) with the matrix of general position is hy-drodynamically integrable if and only if it is diagonalizable and semi-Hamiltonian.Theorem 1 shows that the study of diagonal equations (1.2) is very important.But it doesn’t exclude the necessity of study of general equations (1.1).

Indeedthe integrability test for the equations (1.1) of general position according to thetheorem 1 should include 3 steps(1) test of diagonalizability,(2) diagonalization by means of the point transformation (1.3),(3) test of semihamiltonity (1.4).First of these steps is implemented by means of invariant geometrical criterionfrom [3]. This criterion consists in vanishing of Haantjes’s tensor derived from thevelocities matrix of the equations in question.

This test was first applied to theequations (1.1) in [4].Next step is an actual diagonalization. To pass this step one should calculatethe eigenvalues of the matrix Aij(u), find its eigenvectors, properly normalize themand then one should solve some system of ordinary differential equations definingthe transformation (1.2).

Because of this step the total integrability test may beabsolutely inefficient since the case when the system of differential equations isexplicitly solvable is very rare event. However if we find such solution the thirdstep may have only the difficulties in calculations.The presence of nonefficient step in the above integrability test of the equations(1.1) is due to the absence of the of invariant criterion for testing the semihamil-tonity for these equations.

The main goal of this paper is to eliminate this essentialfault of the theory of such equations. As it was noted in [1]: It was Riemann whofirst recognized that the theory of the equations (1.1) is the theory of tensors sincethe components of matrix Aij(u) are transformed as the components of tensor underthe point transformations (1.3).

Therefore it is natural to expect that the semi-hamiltonity relationships (1.4) can be rewritten in an invariant tensorial form. Inthe section 4 of this paper we construct the tensor, vanishing of which is equivalentto (1.4).

So invariant integrability criterion for the equations of hydrodynamicaltype is obtained. New integrability test now is absolutely efficient.

It includes twosteps(1) test of vanishing the tensor of Haantjes,(2) test of vanishing the semihamiltonity tensor.In order to construct the semihamiltonity tensor we use the theory developedby Froelicher and Nijenhuis in [5] and [6]. This theory in brief is given in section1May be this is not new fact but we couldn’t find it anywhere.

INVARIANT INTEGRABILITY CRITERION . .

.33. According to the theory of Froelicher and Nijenhuis each smooth manifold isequipped with some Lie superalgebra of tensor fields of type (1, p).Note thatthe paper [6] was written in 1956 but unfortunately it wasn’t known to specialistssince for example in [7] the paper of Martin of 1959 is quoted as the first paper insupermathematics.Authors are grateful to V.E.Adler and I.Yu.Cherdantzev for the fruitful discus-sions.They are also grateful to B.I.Suleymanov for supporting interest to thispaper.2.

Hydrodynamical integrability.Let’s take the system of equations of hydrodynamical type (1.1) and let’s add toit another such system with the dynamics by the variable τ(2.1)uiτ =nXj=1Bij(u)ujx, where i = 1, . .

. , nDefinition 1.

System of equations (2.1) is called the hydrodynamical symmetryfor the equations (1.1) if the equations (1.1) and (2.1) are compatible.Now we shall study the question about the existence and the number of hydro-dynamical symmetries for the system of equations (1.1). Let the equations (1.1)and (2.1) be compatible.

Their compatibility conditions are written in form of therelationshipsnXs=1AisBsj =nXs=1BisAsj(2.2)nXs=1∂sAki Bsj + ∂sAkj Bsi + ∂iBsj Aks + ∂jBsi Aks==nXs=1∂sBki Asj + ∂sBkj Asi + ∂iAsjBks + ∂jAsiBks(2.3)The relationship (2.2) means that the matrices the systems (1.1) and (2.1) arecommuting(2.4)AB = BASecond relationship (2.3) also can be written in an invariant form. In order to doit let’s contract (2.3) with XiXj, where X1, .

. .

, Xn are the components of somearbitrary vector field X. (2.5)[AX, BX] −A[X, BX] −B[AX, X] = 0This result can be stated as a theorem.

4PAVLOV M.V., SHARIPOV R.A., SVINOLUPOV S.I.Theorem 2. System of the equations (2.1) is a hydrodynamical symmetry for thesystem (1.1) if and only if for any vector field X the relationships (2.4) and (2.5)hold.Let the operator field A = A(u) from (1.1) have n mutually distinct eigenvaluesλi = λi(u).

Through X1, . .

. , Xn we denote the frame formed by eigenvectors ofoperator A.

The choice of eigenvectors is not unique, there is the gauge arbitrarinessin n scalar factors(2.6)Xi(u) −→fi(u)Xi(u), where fi ̸= 0For the sake of brevity we introduce the following notations for the Lie derivativesalong the vector fields X1, . .

. , Xn(2.7)Li = LXiMutual commutators of the vector fields X1, .

. .

, Xn are convenient to be expandedin the frame formed by these fields(2.8)LiXj = [Xi, Xj] =nXk=1ckijXkParameters ckij = ckij(u) in (2.8) are to be called the structural scalars of the frameX1, . .

. , Xn.

The term structural constants doesn’t suit since ckij depend on thepoint u.From the algebra we know that the matrix B is commuting with the matrix Ahaving mutually distinct eigenvalues then these two matrices are simultaneouslydiagonalized in the frame X1, . .

. , Xn.

Therefore any operator B satisfying (2.4) iscompletely defined by its eigenvalues µi = µi(u). Note that from (2.5) we have therelationship[AX, BY] + [AY, BX] −A[X, BY]−−A[Y, BX] −B[AX, Y] −B[AY, X] = 0which holds for two arbitrary vector fields X and Y. Let’s substitute X = Xi andY = Xj into the above relationship.

As a result we obtain that it is equivalent tothe pair of sets of relationships. First set is algebraic with respect to the eigenvaluesµi of the matrix B(2.9)cijk(λj −λk)µi + cijk(λk −λi)µj++ cijk(λi −λj)µk = 0, for i ̸= j, j ̸= k, k ̸= iSecond set contains the partial differential equations with respect to µi(2.10)Liµj = λijµi −µjλi −λj, for i ̸= jHere and everywhere below we use the notations λij = Liλj in terms of Lie deriva-tives (2.7).

INVARIANT INTEGRABILITY CRITERION . .

.5System of differential equations (2.10) is overdetermined. When it is compatiblethe maximal degree of arbitrariness for its solutions is n functions of one variables.Let this degree of arbitrariness be actually realized.

Then(2.11)µi = µi(f1, . .

. , fn, u)Let’s substitute (2.11) into (2.9).

This leads to the functional dependence for theparameters f1(z1), . .

. , fn(zn) from (2.11), which contradicts their arbitrariness.Therefore the relationships (2.9) should be trivial.

For the structural scalars (2.8)this gives(2.12)ckij = 0, for i ̸= j, j ̸= k, k ̸= iThe relationships (2.12) have the important consequences which are due to thefollowing lemma.Lemma 1. The linear operator of the general position A = A(u) is diagonalizableby means of the transformation (1.3) if and only if the relationships (2.12) hold forthe frame of its eigenvectors.We give the sketch of proof of this lemma.

Operator A(u) is diagonal in theframe of its eigenvectors. For A(u) to be diagonalizable by the transformation(1.3) this frame should be the coordinate frame i.e.

structural scalars of this frameshould be identically zero ckij = 0. The relationship (2.12) provides vanishing mostof these scalars.

One can reach vanishing the rest of these scalars by use of thegauge arbitrariness (2.6).Because of lemma 1 the further analysis of the compatibility conditions for theequations (2.10) becomes unnecessary. For the diagonal systems (1.2) such analysiswas done by S.P.Tsarev in [2].

Note only that as result of such analysis we add to(2.12) the following relationships(2.13)Liλjkλj −λk−Ljλikλi −λk++cjjiλjkλj −λk−ciijλikλi −λk= 0which hold for i ̸= j, j ̸= k, k ̸= i.The relationships (2.13) are the same asthe semihamiltonity relationships (1.4) but written in the frame of eigenvectors ofdiagonalizable operator A(u). The above considerations prove the theorem 1 inthe following form.Theorem 3.

System of the equations (1.1) with the matrix of general positionpossess the continuous set of hydrodynamical symmetries with functional arbitrari-ness given by n functions of one variable if and only if it is diagonalizable andsemi-Hamiltonian.Now let’s study the similar question about the conservation laws for (1.1). Onthe set of their solutions we define the integral functionals of the following form(2.14)F =Zf(u)dx

6PAVLOV M.V., SHARIPOV R.A., SVINOLUPOV S.I.Functional (2.14) is called the hydrodynamical conservation law or the first integralfor the equations (1.1) if ˙F = 0 when time derivative ˙F is calculated according tothe dynamics given by (1.1). This derivative is the following integral functional(2.15)G = ˙F =Z nXi=1nXj=1∂ifAijujxdxFrom vanishing the functional (2.15) we get the vanishing of its variational deriva-tivesδGδui =nXj=1nXk=1∂i(∂kfAkj ) −∂j(∂kAki )ujx = 0This leads to the following relationship for the density f(u) of the primary func-tional (2.14)(2.16)nXk=1∂i(∂kfAkj ) −∂j(∂kAki )= 0It is equivalent to the existence of the function T (u) such that(2.17)G =ZnXi=1nXj=1∂ifAijujxdx =Z ∂T∂x dxBecause of (2.17) the condition (2.16) is exactly the condition of vanishing thefunctional G = ˙F = 0.

In order to write the relationships (2.16) in an invariantform let’s choose two arbitrary vector fields X and Y. After contracting (2.16) withXi and Y j we can write the result of such contraction through the Lie derivatives(2.18)LXLAY −LYLAX −LA[X,Y]f = 0This result can be stated in form of the following theorem.Theorem 4.

Integral functional (2.14) is hydrodynamical conservation law for thesystem of equations (1.1) if and only if for any choice of vector fields X and Y theequations (2.18) hold.The relationship (2.18) is the system of differential equations with respect tothe unknown function f(u). One should investigate it for the compatibility.

Let’sdenote Lif = ϕi. Substituting frame vectors X1, .

. .

, Xn for X and Y into therelationships (2.18) we get the following equations(2.19)Liϕj =nXk=1Bkijϕk, for i ̸= jwhere the functions Bkij are defined by the formulae(2.20)Bkij = ckijλk −λiλj −λi+ λjiδkiλj −λi−λijδkjλj −λi

INVARIANT INTEGRABILITY CRITERION . .

.7The equations (2.19) are analogous to the equations (2.10). When they are com-patible their solutions have the arbitrariness in n functions of one variable.

Letsuch arbitrariness be actually realized. We look for the differential consequences ofthe equations (2.19).

Among them we find the following relationships(2.21)BijkLiϕi −BjikLjϕj −ckijLkϕk = −nXq=1Rqkijϕqwhich hold for i ̸= j, j ̸= k, k ̸= i. The values of Rqkij for i ̸= j, j ̸= k, k ̸= i arecalculated according to the formula(2.22)Rqkij = LiBqjk +nXs̸=iBqisBsjk−−LjBqik −nXs̸=jBqjsBsik −nXs̸=kcsijBqskThe derivatives Liϕi, Ljϕj and Lkϕk aren’t defined by the equations (2.19).

Thisgives the arbitrariness in n functions for the solutions of (2.19). When they arenontrivial the relationships (2.21) define the functional dependence between thesederivatives.

Therefore they diminish the degree of arbitrariness. In case of maximalarbitrariness the relationships (2.12) should be trivial(2.23)Bijk = Bjik = ckij = 0, for i ̸= j, j ̸= k, k ̸= iFrom (2.23) due to the lemma 1 we get the diagonalizability of the operator A(u)by means of point transformations from (1.3).

Due to (2.20) the equations (2.19)are rewritten as follows(2.24)Liϕj = −λjiϕi −λijϕjλi −λj+ cjijϕj, for i ̸= jThe compatibility conditions for (2.24) are defined by the quantities from (2.22)as Rqkij = 0.On taking into account (2.23) these compatibility conditions areexactly coincide with (2.13).In spite of the fact that the equations (2.10) and(2.24) are different their compatibility conditions are the same and have the formof semihamiltonity condition written in the frame of eigenvectors of the operatorA(u). The above considerations prove the following version of the theorem 1.Theorem 5.

System of the equations (1.1) with the matrix of general positionpossess the continuous set of conservation laws parameterized by n functions of onevariable if and only if it is diagonalizable and semi-Hamiltonian.3. The Froelicher-Nijenhuis bracket and the Liesuperalgebra of vector-valued differential forms.Let A be the tensor field of the type (1, p) and let it be skew symmetric in co-variant components.

Then A defines the vector-valued p-form A = A(X1, . .

. , Xp).Here X1, .

. .

, Xp are some arbitrary vector fields. Let B be the second tensor field

8PAVLOV M.V., SHARIPOV R.A., SVINOLUPOV S.I.of the type (1, q) which define vector valued q-form B(X1, . .

. , Xq).

We shall callA and B the vector fields of rank 1 if they are of the following form(3.1)A = a ⊗αB = b ⊗βwhere a and b are vector fields while α and β are the differential forms.Forthe tensor fields of the form (3.1) we define the pairing {A, B} (it is known asFroelicher-Nijenhuis bracket)(3.2){A, B} = [a, b] ⊗α ∧β −a ⊗Lbα ∧β + b ⊗α ∧Laβ++ (−1)pa ⊗ιbα ∧dβ + (−1)pb ⊗dα ∧ιaβVia ιa and ιb in formula (3.2) we denote the differentiations of substitution. Forthe r-form ω and for the vector field c the expression ιcω is a r −1-formιcω(X1, .

. .

, Xr−1) = rω(c, X1, . .

. , Xr−1)The operation ιc is also known as inner product with respect to the vector field c(see [8]).Theorem 6.

The bracket {A, B} defined for the tensor fields A and B of rank 1by the formula (3.2) is uniquely continued for the arbitrary tensor fields of the types(1, p) and (1, q) skew symmetric in their covariant components.PROOF. Each tensor field of the type (1, p) can be written as a sum of tensorfields of rank one as follows(3.3)A =XiAi =Xiai ⊗αiThe analogous formula can be written for the field B.

Therefore the bracket (3.2)for the arbitrary A and B can be redefined as(3.4){A, B} =XiXj{Ai, Bj}However the expansion (3.3) is not unique.Therefore the definition {A, B} bymeans of (3.4) should be tested for the correctness. The arbitrariness in the expan-sion (3.3) for A is defined by the following identities in tensor algebra(a + ˜a) ⊗α = a ⊗α + ˜a ⊗αa ⊗(α + ˜α) = a ⊗α + a ⊗˜α(3.5)(fa) ⊗α = a ⊗(fα)(3.6)where f is an arbitrary scalar field.

The arbitrariness due to (3.5) does not influenceto the value of bracket (3.4) since the relationship (3.2) is additive with respect toa and α. Let’s ensure that the arbitrariness due to (3.6) also doesn’t make theinfluence to the value of {A, B}. In order to do it we calculate this bracket by (3.2)first for A = (fa) ⊗α then for A = a ⊗(fα) and after all we compare the results.All these calculations are based on the following formulae from [8][fa, b] = f[a, b] −aLbfLfaβ = fLaβ + df ∧ιaβLb(fα) = fLbα + Lbfαιfaβ = fιaβSince these calculations are standard we did not write them here.Theorem isproved □Note that for p = q = 0 the bracket (3.2) coincides with the ordinary commutatorof vector fields.

For the arbitrary values of p and q the algebraic properties of thisbracket are given by the following theorem.

INVARIANT INTEGRABILITY CRITERION . .

.9Theorem 7. The bracket {A, B} for the tensor fields of rank 1 defined by (3.2)and then generalized by (3.4) satisfies the relationships{A, B} + (−1)pq{B, A} = 0{{A, B}, C}(−1)rp + {{B, C}, A}(−1)pq + {{C, A}, B}(−1)qr = 0because of which it defines the structure of graded Lie superalgebra in tensor fieldsof type (1, m) skew symmetric in covariant components.Let A and B be tensor fields of the type (1.1) i.e.

operator fields. Tensor fieldS = 2{A, B} is the vector valued 2-form.

Its values may be calculated by thefollowing formula(3.7)S(X, Y) = [AX, BY] + [BX, AY]++ AB[X, Y] + BA[X, Y] −A[X, BY]−−A[BX, Y] −B[X, AY] −B[AX, Y]Tensor S is known as the torsion of Nijenhuis for the operator fields A and B (see[5], [6] and [8]).4. The construction of semihamiltonity tensor.First let’s recall the classical invariant criterion of diagonalizability for the op-erator field A.

We mentioned this criterion in section 1 (see also [3], [4] and [6]).Let’s consider the particular form of tensor (3.7)N = {A, A}It is usually called the tensor of Nijenhuis. From (3.7) we obtain(4.1)N(X, Y) = [AX, AY] + A2[X, Y]−−A[X, AY] −A[AX, Y]Tensor of Haantjes is defined via the tensor of Nijenhuis (4.1) according to theformula(4.2)H(X, Y) = N(AX, AY) + A2N(X, Y)−−AN(X, AY) −AN(AX, Y)It has the same type as the tensor N. It is also the vector-valued skew symmetric2-form.Theorem 8 (criterion of diagonalizability).

The operator A(u) of general positionwith mutually distinct eigenvalues is diagonalizable by means of point transforma-tions (1.3) if and only if its tensor of Haantjes (4.2) is identically zero.The criterion of diagonalizability in form of this theorem was first proved in [3].It was applied to the systems of equations (1.1) in [4].

10PAVLOV M.V., SHARIPOV R.A., SVINOLUPOV S.I.PROOF. Because of skew symmetry of bilinear form (4.2) it’s enough to testvanishing this form only for vector fields X = Xi and Y = Xj from the frame ofeigenvectors of the operator A(u) with i ̸= j.

By direct calculations we obtain(4.3)H(Xi, Xj) =nXk=1(λi −λk)2(λj −λk)2ckijXkBecause of (4.3) the equality H(Xi, Xj) = 0 is equivalent to (2.12). Then we areonly to apply the lemma 1.

Criterion is proved. □Practical use of this criterion for the testing the diagonalizability is based on thefollowing formulae for the components of tensors N and H expressing them throughthe components of matrix A(u)N kij =nXs=1Asi ∂sAkj −Asj∂sAki + Aks∂jAsi −Aks∂iAsj(4.4)Hkij =nXs=1nXr=1AksAsrN rij−−AksN srjAri −AksN sirArj + N ksrAsi Arj(4.5)Let B be the operator field i.e the tensor field of type (1, 1) and let Q be theskew symmetric tensor field of type (1, 2).

Through K we denote the Froelicher–Nijenhuis bracket of these two fields K = 3{Q, B}. For the 3-form K we have(4.6)K(X, Y, Z) = B[X, Q(Y, Z)] −[BX, Q(Y, Z)]++ BQ(X, [Y, Z]) + Q(X, B[Y, Z]) −Q(X, [BY, Z])−−Q(X, [Y, BZ]) + .

. .Dots in (4.6) denote 12 summand that can be obtained from explicitly writtensummands in (4.6) by means of the cyclic transposition of X, Y and Z.Starting with deriving the invariant semihamiltonity criterion we should notethat everywhere above (see theorems 1, 3 and 5) the semihamiltonity comes togetherwith diagonalizability.

It doesn’t play the separate role. Therefore we will use thefollowing scheme of action: we will construct the tensor vanishing of which gives theequations (1.4) after bringing this tensor to the coordinates where the matrix A(u)is diagonal.

The equations (1.4) are rational. Let’s rewrite them in polynomialform.

In order to do it we introduce the following quantities(4.7)αkkij = −(λi −λj)(λi −λk)(λj −λk)∂ijλk−−(λi −λj)(λi + λj −2λk)∂iλk∂jλk++ (λi −λk)2∂iλj∂jλk −(λj −λk)2∂jλi∂iλkThe semihamiltonity (1.4) then is written as the condition of vanishing the quan-tities (4.7)(4.8)αkkij = 0, for i ̸= k ̸= j

INVARIANT INTEGRABILITY CRITERION . .

.11Partial derivatives in (4.7) as in (1.4) are calculated with respect to the vari-ables u1, . .

. , un for which the matrix A(u) is diagonal.

The frame of eigenvectorsX1, . .

. , Xn is chosen to be the coordinate frame for such variables.Using the Froelicher–Nijenhuis bracket we construct the tensor K from the ma-trix A as follows(4.9)K = 3{{A, A}, A2} = 3{N, A2}Tensor K defines the vector-valued 3-form the values of which for the frame vectorscan be calculated according to (4.6).

As a result from (4.9) we have(4.10)K(Xk, Xi, Xj) = KkkijXk + KikijXi + KjkijXj(there is summation by i, j, k here). We are interested only in one group of com-ponents of tensor K. Others can be obtained by cyclic transpositions of indicesin Kkkij.

The values of components from this group in (4.10) are defined by thefollowing formula(4.11)Kkkij −αkkij =2(λi −λj)(λi −λk) + (λj −λk)(∂iλk∂jλk −∂iλj∂jλk −∂jλi∂iλk)From (4.11) we see that the difference Kkkij−αkkij contains only the derivatives of thefirst order. Now taking the tensor N we construct another tensor M. Correspondingpolylinear form is the following(4.12)M(X, Y, Z) = N(X, AN(Y, Z))++ N(AX, N(Y, Z)) −N(N(X, Z), AY)++ N(N(X, Y), AZ) −N(X, N(AY, Z))−−N(X, N(Y, AZ))For the tensor M from (4.12) computed in the frame of eigenvectors of the operatorA(u) we have(4.13)M(Xk, Xi, Xj) = M kkijXk + M ikijXi + M jkijXjFor the of components components of M in (4.13) we get(4.14)M kkij = −(λi −λj)(λi −λk)(λj −λk)(∂iλk∂jλk −∂iλj∂jλk −∂jλi∂iλk)The coefficients M ikij and M jkij aren’t of interest for us now.

Comparing (4.11)with (4.14) we define another tensor Q(4.15)Q(X, Y, Z) = K(AX, AY, Z) −K(A2X, Y, Z)−−K(X, AY, AZ) + K(AX, Y, AZ) + 4M(AX, Y, Z)−−2M(X, AY, Z) −2M(X, Y, AZ)

12PAVLOV M.V., SHARIPOV R.A., SVINOLUPOV S.I.For the components of this tensor in the frame of eigenvectors of operator A(u) weget the relationship(4.16)Q(Xk, Xi, Xj) = QkkijXk + QikijXi + QjkijXjwhich is analogous to (4.10) and (4.13). The components in (4.16) which are ofinterest for us are expressed through (4.7).

They have the form(4.17)Qkkij = −(λi −λk)(λj −λk)αkkijNow on a base of (4.17) we are able to construct the semihamiltonity tensor whichis the main goal of the whole paper. It is defined by the following formula(4.18)P(X, Y, Z) = AQ(X, AY, Z)++ AQ(X, Y, AZ) −A2Q(X, Y, Z) −Q(X, AY, AZ)It is easy to check that for arbitrary three vectors from the frame X1, .

. .

, Xn onehas the relationship(4.19)P(Xk, Xi, Xj) = (λi −λk)2(λj −λk)2αkkijXkAs a result we proved the following theorem (the invariant criterion of semi-hamiltonity).Theorem 9. The diagonalizable operator of general position A with mutually dis-tinct eigenvalues is semi-Hamiltonian if and only if when associated tensor P from(4.18) is identically zero.The proof of this theorem follows directly from (4.8) and (4.9).

It doesn’t requireany comments. Concluding all above considerations we give the formulae whichenable us to calculate the components of the tensor of semihamiltonity P from thematrix A(u)(4.20)P skij =nXp=1nXq=1AspQpkqjAqi + AspQpkiqAqj−−AsqAqpQpkij −QskpqApi AqjThis formula is derived from (4.18).

Components of the tensor Q in the formula(4.20) are calculated on a base of (4.15)Qskij =nXp=1nXq=1ApkKspqjAqi + ApkKspiqAqj −ApqAqkKspij−−KskpqApi Aqj+nXp=14ApkM spij −2M skpjApi −2M skipApjComponents of M in the above formula are found from (4.12)M skij =nXp=1nXq=1N skpApqN qij + N spqApkN qij−−N spqN pikAqj −N spqN pkjAqi −N skpN piqAqj −N skpN pqjAqi

INVARIANT INTEGRABILITY CRITERION . .

.13Tensor K are computed through tensor of Nijenhuis with the use of the bracket ofFroelicher and Nijenhuis on a base of formula (4.9). Let’s take B = A2.

Then forthe components of K we have(4.21)Kskij =nXp=1Bsp∂kN pij −Bpk∂pN sij++ N pij∂pBsk −N skp∂iBpj + N skp∂jBpi+ . .

.By dots in (4.21) we denote 10 summand that can be obtained from 5 explicitsummands by cyclic transposition of indices i, j and k.The above formulae huge enough for direct calculations. However modern com-puter systems for analytical calculations solve this problem for any particular equa-tions of hydrodynamical type in applications.References1.

Dubrovin B.A., Novikov S.P., Hydrodynamics of weakly deformed soliton lattices. Differentialgeometry and Hamilton’s theory., Uspehi Mat.

Nauk 44 (1989), no. 6, 29–98.2.

Tsarev S.P., Geometry of hamiltonian systems of hydrodynamical type. Generalized methodof hodograph., Izvestiya AN SSSR, ser.

Metem. 54 (1990), no.

5, 1048–1068.3. Haantjes A., On Xn−1-forming sets of eigenvectors., Indagationes Mathematicae 17 (1955),no.

2, 158–162.4. Ferapontov E.V., Tsarev S.P., Systems of hydrodynamical type appearing in chromatography.Riemann invariants and exact solutions., Mathematical modeling 3 (1991), no.

2, 82–91.5. Nijenhuis A., Xn−1-forming sets of eigenvectors., Indagationes Mathematicae 13 (1951),no.

2, 200–212.6. Froelicher A., Nijenhuis A., Some new cohomology invariants for complex manifolds., Proc.Koninkl.

nederl. akad.

wetensch. A59 (1956), no.

5, 540–564.7. Berezin F.A., Introduction into the algebra and analysis with anticommutating variables.,Moscow State University publishers, Moscow, 1983.8.

Kobayashi Sh., Nomizu K., Foundations of differential geometry. Vol.

1., Interscience Pub-lishers, New York London, 1963.Landau Institute for Theoretical Physics, Kosigina street 2, 117940 Moscow,Russia; Department of Mathematics, Bashkir State University, Frunze street 32,450074 Ufa, Russia; Institute of Mathematics, Chernishevsky street 112, 450000 Ufa,RussiaE-mail address: pavlov@cpd.landau.free.net; root@bgua.bashkiria.su; sersv@nkc.bashkiria.su

---

출처: arXiv:9407.003 • 원문 보기