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Proc. NATO ARW ”Applications of analitic and geometric methods to nonlinear

arXiv:hep-th/9303092v1 16 Mar 1993hep-th/9303092Proc. NATO ARW ”Applications of analitic and geometric methods to nonlineardifferential equations, 14-19 July 1992, Exeter, UK)CLASSICAL DIFFERENTIAL GEOMETRY AND INTEGRABILITY OFSYSTEMS OF HYDRODYNAMIC TYPESERGUEI P. TSAREVSteklov Mathematical Institute117966, Vavilova,42,Moscow, GSP-1, USSRe-mail: tsarev@top.mian.suAugust 25, 1992ABSTRACT.

Remarkable parallelism between the theory of integrable systems offirst-order quasilinear PDE and some old results in projective and affine differen-tial geometry of conjugate nets, Laplace equations, their Bianchi-B¨acklund trans-formations is exposed.These results were recently applied by I.M.Krichever andB.A.Dubrovin to prove integrability of some models in topological field theories.Within the geometric framework we derive some new integrable (in a sense to bediscussed) generalizations describing N-wave resonant interactions.Ten years ago [10] a natural hamiltonian formalism was proposed for the class ofhomogeneous systems of PDE(1+1,h)u1t...unt=v11(u)· · ·v1n(u)·· · ··vn1 (u)· · ·vnn(u)u1x...unx,ui = ui(x, t),i = 1, · · · , n(called ”one-dimensional systems of hydrodynamic type”). Later (see [11]) it wasgeneralized for the class of multidimensional1

(N+1,h)u1t...unt=v(1)11(u)· · ·v(1)1n(u)·· · ··v(1)n1(u)· · ·v(1)nn(u)u1x1...unx1+ . .

.. . .

+v(N)11(u)· · ·v(N)1n(u)·· · ··v(N)n1(u)· · ·v(N)nn(u)u1xN...unxN,ui = ui(t, x1, . .

. , xN),i = 1, .

. .

, nand non-homogeneous(1+1,nh)u1t...unt=v11(u)· · ·v1n(u)·· · ··vn1 (u)· · ·vnn(u)u1x...unx+f1(u)...fn(u),ui = ui(x, t),i = 1, . .

. , nsystems.Some systems (1+1,h) of physical importance such as Whitham equations (theaveraged 1-phase KdV equation) and Benney equations have the notable property ofbeing diagonalizable: under a suitable choice of field variables ui (Riemann invariants)the equations becomeuit(x) = vi(u)uix(1)(there is no summation over i!).

As we have proved in [32], these properties (hamilto-nian property and diagonalizability) imply integrability. Deeper insight into this typeof integrability is given by the theory of orthogonal curvilinear coordinate systems.This chapter of classical differential geometry was being intensively developed at thebeginning of the XX century ([6], [8], [20]).

In fact this theory gives the geometricbackground for integrability of systems (1+1,h), (N+1,h), (1+1,nh). These forgottencorners of differential geometry seem to be worth revisiting.An example.

The well-known Bullough-Dodd-Jiber-Shabat equation uxx −utt =eu −e−2u (in the form (ln h)uv = h −1/h2) was introduced for the first time in [35]where the respective linear problem was given as well as a proper B¨acklund trans-formation for it! It is much simpler and ”geometric” than B¨acklund transformationsdiscussed recently [1], [31] in the context of integrable systems.In this paper we will sketch some applications of methods originating from classicaldifferential geometry to equations of types (1+1,h), (N+1,h), (1+1,nh).1.Diagonal systems of hydrodynamic type and orthogonal curvilinearcoordinate systems in RnLet us recall briefly the main results of [10],[32].

A (generally nondiagonal) systemuit =Pnj=1 vij(u)ujx is hamiltonian if there exist a hamiltonian H =R h(u) dx and a2

hamiltonian operatorˆAij = gij(u) ddx + bijk (u) ukxwhich define a skew-symmetric Poisson bracket on functionals{I, J} =ZδIδui(x)ˆAijδJδuj(x)dxsatisfying the Jacobi identity and generate the systemuit(x) = {ui(x), H} = ˆAij∂H∂uj(x) = (gij∂k∂jh + bijk ∂jh)ukx = vik(u)ukx(2)where ∂s = ∂/∂us. B.A.Dubrovin and S.P.Novikov [10] proved that the necessary andsufficient conditions for ˆAij to be a hamiltonian operator in the case of non-degeneracyof the matrix gij are:a) gij = gji, i.e.

the inverse matrix g−1 defines a Riemannian metric.b) bijk = −gisΓjsk for the standard Christoffel symbols Γjsk generated bygij.c) the metric gij has identically vanishing curvature tensor.In such case we have vij(u) = ∇i∇jh = gis∇s∇jh with the covariant derivativesdefined by gij.Lemma [32], [33]. In order that a matrix vij(u) be a matrix of a hamiltoniansystem (1+1,h) with a nondegenerate metric in ˆAij it is necessary and sufficient thatthere exists a nondegenerate zero curvature metric gij such thata) gikvkj = gjkvki andb) ∇jvik = ∇kvij, where ∇is the covariant differentiation generated by the metricgij.For a diagonal matrix vij(u) = vj(u)δij this implies that (see [32], [33]) gij is alsodiagonal and∂ivk(vi −vk) = Γkki = 12∂i ln gkk,∂i = ∂/∂ui(3)(hereafter we do not imply the summation on repeated indices!).

From (3) we deduce∂j∂ivkvi −vk= ∂i∂jvkvj −vk, i ̸= j ̸= k.(4)From a differential geometric point of view, to give a zero curvature nondegeneratediagonal metric is equivalent to giving an orthogonal curvilinear coordinate system ona flat (possibly pseudo-Euclidean) space (see [6]). Locally these coordinate systemsare determined by n(n −1)/2 functions of two variables (L.Bianchi).

A striking fact3

can be discovered: formula (3) was found in [6] (p. 353)! This formula is crucial forthe integrability property of diagonal hamiltonian systems (1): if we interpret it asan overdetermined (compatible in view of zero curvature property of g) system on nunknown functions vj(u) (gii given) we can generate from every its solution ¯vj(u) asymmetry (commuting flow)uit = ¯vi(u)uix, i = 1, ..., n,of (1) and a solution of (1) (the generalized hodograph method, see [33] for thedetails).

One can prove ([33]) the completeness property for this class of symmetriesand solutions parameterized by n functions of 1 variable - the generic Cauchy datafor our diagonal system (1).The corresponding geometric notion used in the theory of orthogonal curvilinearcoordinate systems corresponding to (3) is the so called Combescure transformation(see [6]).Definition. Two orthogonal curvilinear coordinate systems xi = xi(u1, .

. .

, un)and ˆxi = ˆxi(u1, . .

. , un) in the same flat (pseudo)Euclidean space Rn = {(x1, .

. .

, xn)}are said to be related by a Combescure transformation (or simply parallel) ifftheirtangent frames ⃗ei = ∂⃗x/∂ui and ˆ⃗ei = ∂ˆ⃗x/∂ui are parallel in points corresponding tothe same values of curvilinear coordinates ui.Let us take the quantities Hi(u) = |⃗ei| = √gii, ˆHi(u) = |ˆ⃗ei| (Lam´e coefficients).Proposition The quantities ¯vi(u) = ˆHi(u)/Hi(u) satisfy (3) with Γkki = ∂iHk/Hk,the connection coefficients for the metric gii = H2i . Conversely, for any solution ¯vi of(3) ˆgii = (¯viHi)2 will give an orthogonal curvilinear coordinate system related to thecoordinate system with the metric gii = H2i by a Combescure transformation.The theory of Combescure transformations coincides with the theory of integrablediagonal systems of hydrodynamic type.Physical examples of such systems ( Whitham equations, Benney equations) havehamiltonian structures (2) with diagonal metrics gii possessing the so called Egorovproperty: ∂igkk = ∂kgii.

As we have demonstrated earlier ([34]) this is a consequenceof Galilei invariance of the original systems. See also [12] for the algebro-geometricbackground of this property for averaged integrable systems.

Using this property andhomogeneity of coefficients one can find explicit formulas for solutions of (3) for thesystems in question [33], [34].The class of Egorov orthogonal curvilinear coordinate systems is interesting initself and merits our special attention.2. Egorov coordinate systems, the N-wave problem and its generalizationsIntroducing βik(u) = ∂iHk/Hi,i ̸= k, βii(u) = 0 (rotation coefficients of a given4

orthogonal curvilinear coordinate system with gii = H2i , see [6]) one can easily checkthe following:a) vanishing of the curvature tensor is equivalent to∂jβik = βijβjk,i ̸= j ̸= k,(5)∂iβik + ∂kβki +Xs̸=i,kβsiβsk = 0,i ̸= k.(6)b) the Egorov property ∂igkk = ∂kgii reduces toβik = βki. (7)In the Egorov case condition (6) is equivalent to ˆTβik = 0, ˆT = ∂1 + .

. .

+ ∂n. Con-sequently the problem of classification of Egorov coordinate systems is reduced todescription of all off-diagonal symmetric matrices (βik) satisfying (5) and ˆTβik = 0.B.A.Dubrovin [12] have recently observed that this problem coincides with thepurely imaginary reduction of the well-known integrable system describing resonantN-wave interactions.

Namely, restriction of βik on any (x, t) plane ui = aix+bit gives(compare, for example, [28])[A, Γt] −[B, Γx] = [[A, Γ], [B, Γ]],A = diag(a1, . .

. , an), B = diag(b1, .

. .

, bn), Γ = (βik) with additional reductionIm Γ = 0, ΓT = Γ. For the case N = 3 this reduces tob1t + c1b1x=κb2b3,b2t + c2b2x=κb1b3,b3t + c3b3x=κb1b2.

(8)This is a system of type (1+1,nh),integrable by the IST method ([28]).Now we can compare the progress achieved in the modern integrability theoryfor (8) and the results obtained more than 70 years ago in the theory of Egorovcoordinate systems initiated by G.Darboux in 1866 and continued by D.Th.Egorov in1901 in his thesis (see [13]). It was Darboux [6] who proposed to call this special typeof coordinate systems Egorov type systems.

From the point of view of integrabilityproperties remarkable progress was achieved by L.Bianchi in 1915 (see [2]). He found aB¨acklund transformation for this problem and established the permutability propertyas well as the superposition formula for it.

We shall remark here that the pioneeringresults on B¨acklund transformations and their permutability in the well-known theoryof constant curvature surfaces in R3 are due to Bianchi also.Let us take an orthogonal curvilinear coordinate system (not necessary of Egorovtype) with Lam´e coefficients Hi(u) and rotation coefficients βik(u). Bianchi appliedto it a generalization of Ribaucour transformations known in the theory of transfor-mations of surfaces.

We recall that two surfaces ⃗x(u, v) and ˆ⃗x(u, v) in R3 are related5

by a Ribaucour transformation iffthere exists a two-parametric family of spheresS(u, v) such that each sphere S(u0, v0) is tangent to both surfaces in correspondingpoints ⃗x(u0, v0), ˆ⃗x(u0, v0) and this correspondence preserves the curvature lines onthe surfaces. For the case of a pair of orthogonal curvilinear coordinate systems inR3 we need a three-parametric family of spheres (or an n-parametric family for then-dimensional case) tangent in the corresponding points to one of three families of co-ordinate surfaces as well as to a coordinate surface of the other curvilinear coordinatesystem.

Since due to the classical Dupin theorem coordinate lines in any orthogonalcurvilinear coordinate system are curvature lines their correspondence is guaranteed.In terms of the rotation coefficients βik one shall find a solution γi(u) of∂iγk = βkiγi, i ̸= k,(9)to define the corresponding Ribaucour transformation ([2])ˆβik = βik −2γiA (∂kγk +Xs̸=kβskγs),A =Xp(γp)2. (10)For the case of Egorov systems we shall complete (9) and restrict γi to satisfy∂iγi = cγi −Xs̸=iβsiγs,(c = const)orˆTγi = cγi,ˆT = ∂1 + .

. .

+ ∂n(11)Then the B¨acklund transformation in question isˆβik = βik −2cγiγkA. (12)The permutability property for any orthogonal coordinate system requires a quadra-ture, but for Egorov systems it may be found explicitly and provides the followingformulas for the fourth Egorov system ¯βik related to β′ik, β′′ik obtained from a givenEgorov system βik with constants c′, c′′ (c′ + c′′ ̸= 0) and potentials γ′i, γ′′i in (10):¯γ′i = γ′′i −2c′γiPs(γ′sγ′′s)(c′ + c′′) Ps(γ′s)2, ¯γ′′i = γ′i −2c′′γiPs(γ′sγ′′s)(c′ + c′′) Ps(γ′′s )2.One can enjoy reading [4], [23] where these formulas were rediscovered in the contextof 3-wave system.

So the basic integrability results for (8) were established long ago byDarboux, Egorov and Bianchi certainly with the exception of the IST transformation.An unexpected result (hidden in [6]) consists in existence of a homogeneous system(1+1,h) of three equations related to (8) by a nonlocal transformation. Geometricallythis is trivial.

Given an orthogonal curvilinear coordinate system in R3 we have ineach its point P(x0, y0, z0) the orthogonal 3-frame of tangent planesz = pk(x −x0) + qk(y −y0) + z0,k = 1, 2, 3. (13)6

Let us parameterize it by 3 functions A(x, y, z), B(x, y, z), C(x, y, z),coefficients pk, qkof tangent planes being three solutions of(pq + Ap + Bq = 0,p2 −q2 + 2(Cp + Hq) = 0,2(BC −AH) + 1 = 0,different from the trivial solution p = q = 0 ([3]). Then the Frobenius compati-bility conditions for these three families of distributions (13) give a system of threehomogeneous first-order equations of type (2+1,h):2(ACz −CAz) = 2Cy + By −Ax2(BHz −HBz) = 2Hx + By −AxAHz −HAz + BCz −CBz = Ay −Bx(14)where H = (2BC −1)/2A .For this system one can reformulate the B¨acklund-like transformation (10) given interms of βik(u).

A number of different transformations producing (with quadratures)solutions of (14) parameterized by arbitrary many functions of one variable may befound in [6]. Thus (14) is integrable in a sense to be discussed elsewhere.If we will search for solutions of (14) which do not depend on z then a remarkableintegrable (1+1,h) system of three equation appears.

Since one can easily prove theequivalence of z-independence in (14) and the Egorov property (7) we have received ahomogeneous system related to (8) by a nonlocal change of variables. In Euler ϕ, ψ, θparameterization of orthogonal 3-frames it readsψtθtϕt=−cos2 ϕ−sin ϕ cos ϕ/ sin θ0−sin θ sin ϕ cos ϕ−sin2 ϕ0−cos θ(1 + cos2 ϕ)−sin ϕ cos ϕ cos θ/ sin θ1ψxθxϕx(15)This nonlocal change does not affect the existence of higher order conserved densities.Recently Ferapontov [17] have proved the uniqueness result for such 3×3 homoge-neous systems possessing higher-order conserved densities: they may be transformedto (15) by reciprocal and point transformations.

Also another nonlocal transitionfrom (8) to (15) as given there.The matrix of (15) has constant eigenvalues −1, 0, +1 but its eigenvector fields(properly normalized) form so(3) Lie algebra, consequently (15) is a non-diagonalizable(1+1,h) integrable system.The complete system (14) certainly may be called a (2+1)-dimensional generaliza-tion of the (1+1)-dimensional 3-wave system (8). Orthogonal curvilinear coordinatesystems in Rn provide also only a (2+1)-dimensional generalization of the (1+1)-dimensional N-wave system since they are parameterized by n(n −1)/2 functions oftwo variables (L.Bianchi).3.

Semihamiltonian diagonal systems and coordinate systems with conju-gate lines7

The class of integrable diagonal systems (1) is wider than the class of hamiltoniansystems of this type. Namely, the property (4) which is a weaker consequence ofthe hamiltonian property is sufficient ([33]).Let us call a diagonal system semi-hamiltonian if n = 2 or if n > 2 and vi(u) satisfy (4).

As a physical example of asemihamiltonian (but non-hamiltonian for n > 3) system one can mention the idealLangmuir chromatography and electrophoresis systems ([33]).To every semihamiltonian system we can relate a diagonal metric gii via∂i ln gkk/2 = ∂ivk/(vi −vk). This metric is not flat in general though some coefficients of the curvature tensorvanish as a consequence of (4).

Namely introducing Hi = √gii , βik(u) = ∂iHk/Hi,we can find out that (4) is equivalent to the set (5) of equations on βik. Solutions of(5) may be parameterized by n(n−1) functions of 2 variables.

This system coincideswith the compatibility conditions for a linear system∂iψk = βikψi, i ̸= k.Restricting (5) on 3-dimensional planes ui = aix + biy + ciz in Rn we obtain (forgeneral nonvanishing constants ai, bi, ci) a (2+1,nh) system on n(n −1) quantitiesβik(x, y, z).As we have seen earlier the theory of hamiltonian diagonal systems (1) is closelyrelated to the theory of orthogonal curvilinear coordinate systems in Rn. The geo-metric background for the theory of semihamiltonian systems is given by the theoryof coordinate systems with conjugate coordinate lines (see [6] and [7], t. 4, ch.

12).A general (non-orthogonal) coordinate system ⃗x(u1, u2, u3) in R3 is called a systemwith conjugate coordinate lines (or simply a conjugate coordinate system) if on ev-ery coordinate surface Si0 = {ui0 = const} in every point P(x0, y0, z0) the lines ofintersection of this surface with two other coordinate surfaces belonging to other one-parametric families of coordinate surfaces and containing P(x0, y0, z0) are conjugateon Si0 (with respect to its second fundamental form). Every orthogonal curvilinearcoordinate system is conjugate due to Dupin theorem mentioned above.

The theoryof conjugate coordinate systems was developed by Darboux and others and borroweda lot of results from the classical theory of conjugate coordinate nets on surfaces in R3(known as ”nets” or ”r´eseaux”, see [14], [21], [22], [36]). A number of B¨acklund-liketransformations for these coordinate systems was given with permutability properties(though the superposition formulas therein require quadratures).Every conjugate coordinate system xi = xi(u1, .

. .

, un) in Rn = {(x1, . .

. , xn) ischaracterized by the conditions of conjugacy of coordinate lines:∂i∂k⃗x = Γkki(u)∂k⃗x + Γiik(u)∂i⃗x,i ̸= k.(16)This system of equations coincides with the system describing hydrodynamic typeconserved quantities of a semihamiltonian system with ([33]).

Quantities Γkki in (16)8

satisfy its compatibility conditions∂jΓkki = ∂iΓkkj,∂jΓkki = ΓkkjΓjji + ΓkkiΓiij −ΓkkiΓkkj,i ̸= j ̸= k.equivalent to the semihamiltonian property (4). Introducing Hi(u) as solutions of∂iHk(u) = ΓkkiHk(u) and βik = ∂iHk/Hi,i ̸= k, we receive a set of βik satisfying(5).

The converse is also true: given a solution βik of (5) one can find (a numberof)semihamiltonian systems related to it. Any semihamiltonian system also may berelated to a Combescure transformation of conjugate coordinate systems ([6]).This geometric interpretation provides another example of integrable (2+1,h) sys-tem.

Namely, given a conjugate coordinate system in R3 one can take the field ofits (non-orthogonal) tangent 3-frames (⃗e1,⃗e2,⃗e3) and parameterize it by 6 indepen-dent functions eik(x, y, z), i = 1, 2, k = 1, 2, 3 (the coefficients e3k may be set to 1due to normalization). Then the Frobenius compatibility conditions give 3 homo-geneous first-order PDE on eik(x, y, z), i = 1, 2, k = 1, 2, 3.

Another 3 equations aregiven by the conjugacy condition det((⃗ei · ∇)⃗ek,⃗ei,⃗ek) = 0, i < k. This system of6 equations is a homogeneous (2+1,h) system in question. Its z-independent solu-tions satisfy a (1+1,h) system enjoying properties analogous to those of (15): it hasconstant eigenvalues −1, 0, +1 (all doubly degenerate) and 6 linearly independentfields of eigenvectors forming (if properly normalized) a nontrivial Lie algebra.

Thisremarkable system will be studied in subsequent publications.4. Additional topicsRecently O.I.Mokhov and E.V.Ferapontov [27], [16] found a nonlocal generalization ofthe hamiltonian formalism of hydrodynamic type (2).

V.E.Ferapontov communicatedto the author about the further generalization resulting in the following beautifultheorem: any semihamiltonian system (1) has a nonlocal hamiltonian structure witha hydrodynamic hamiltonian and a hamiltonian operator with (possibly infinitelymany) nonlocal terms similar to those in [16].Grinevich [19] derived a series ofnonlocal symmetries for Whitham equations as well as original KdV equations.Weakly nonlinear semihamiltonian systems (i.e. systems (6) with ∂ivi = 0 withoutsummation on i, such systems are also called ”linearly degenerate”) were studied in[15], [30].The theory of such systems is connected to the theory of n-webs on Euclideanplane, Dupin cyclids and St¨ackel metrics (E.V.Ferapontov).

Among the results are:quasiperiodic behavior of their solutions ([30]), complete description of such systemsand complete sets of their hydrodynamic symmetries ([15]). E.V.Ferapontov com-municated to the author the following fact: any n-phase (n-zone) quasiperiodic (ora n-soliton) solution of the KdV equation can be represented with a solution of aweakly nonlinear semihamiltonian system Rit = (Pk̸=i Rk)Rix, i = 1, .

. .

, n. Theseresults shall be compared with Curro and Fusco’s results [5] in the soliton-like inter-actions of Riemann simple waves for some 2 × 2 systems.9

IST-like methods were developed in [18], [24], [25] for some diagonal hamiltoniansystems of physical importance.Certainly this approach shall be related to ourgeometric methods.In a series of preprints (see [9],[26] and references therein) I.M.Krichever andB.A.Dubrovin exposed a remarkable link between the theory of Egorov coordinatesystems and Witten-Dijgraagh-Verlinder-Verlinder equations for the correlation func-tions of topological conformal field theories proving their integrability.Acknowledgement. The author enjoys the occasion to thank Dr. E.V.Ferapontovand Professor J.Gibbons for many useful discussions and hospitality in Imperial Col-lege, London, where a part of this paper was written as well as Professor P. Clarksonand our colleagues in Exeter for their hospitality during the Workshop.References[1] V.A.Andreev, B¨acklund transformation for the Bullough-Dodd-Jiber-Shabatequation and symmetries of integrable equations, Theor.

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[11] B.A.Dubrovin, S.P.Novikov, Hydrodynamics of weakly deformed soliton lattices.Differential geometry and hamiltonian theory, Russ. Math.

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[16] E.V.Ferapontov, Differential geometry of nonlocal hamiltonian operators of hy-drodynamic type, Func. Anal.

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