The dispersionless multi-dimensional integrable systems and related conformal structure generating equations of mathematical physics

Based on the diffeomorphism group vector fields on the complexified torus and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems, describing conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to the spatial dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied to proving complete integrability of some conformal structure generating equations. As examples, we analyzed the Einstein--Weyl metric equation, the modified Einstein--Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations, the inverse first Shabat reduction heavenly equation, the first and modified Pleba\'nski heavenly equations and its multi-dimensional generalizations, the Husain heavenly equation and its multi-dimensional generalizations, the general Monge equation and its multi-dimensional generalizations.

Abstract. Based on the diffeomorphism group vector fields on the complexified torus and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems, describing conformal structure generating equations of mathematical physics. An interesting modification of the devised Liealgebraic approach subject to the spatial dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied to proving complete integrability of some conformal structure generating equations. As examples, we analyzed the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations, the inverse first Shabat reduction heavenly equation, the first and modified Plebański heavenly equations and its multi-dimensional generalizations, the Husain heavenly equation and its multi-dimensional generalizations, the general Monge equation and its multi-dimensional generalizations.

Vector fields on the complexified torus and the related Lie-algebraic properties
Consider the loop Lie groupG := Dif f(T n C ), consisting [14] of the set of smooth mappings {C 1 ⊃ S 1 −→ G : = Dif f (T n }, extended, respectively, holomorphically from the circle S 1 ⊂ C 1 on the set D 1 + of the internal points of the circle S 1 , and on the set D 1 − of the external points λ ∈ C\D 1 + . The corresponding diffeomorphism Lie algebra splittingG :=G + ⊕G − , whereG + := dif f (T n ) + ⊂ Γ(T n C ; T (T n C )) is a Lie subalgebra, consisting of vector fields on the complexified torus T n C ≃ T n × C, suitably holomorphic on the disc D 1 + ,G − := dif f (T n C ) − ⊂ Γ(T n C ; T (T n C )) is a Lie subalgebra, consisting of vector fields on the complexified torus T n C ≃ T n × C, suitably holomorphic on the set D 1 − . The adjoint spaceG * :=G * + ⊕G * − , where the spaceG * + ⊂ Γ(T n C ; T * (T n C )) consists, respectively, from the differntial forms on the complexified torus T n C , suitably holomorphic on the set C\D 1 + , and the adjoint spaceG * − ⊂ Γ(T n C ; T * (T n C )) consists, respectively, from the differntial forms on the complexified torus T n C , suitably holomorphic on the set D 1 + , so that the spaceG * + is dual toG + andG * − is dual toG − with respect to the following convolution form on the productG * ×G : (1.1) (l|ã) := res λ T n < l, a > dx for any vector fieldã :=< a(x), ∂ ∂x >∈G and differential forml :=< l(x), dx > ∈ G * on T n C , depending on the coordinate x := (λ; x) ∈ T n C , where, by definition, < ·, · > is the usual scalar product on the Euclidean space E n+1 and ∂ ∂x := ( ∂ ∂λ , ∂ ∂x1 , ∂ ∂x2 , ..., ∂ ∂xn ) ⊤ is the usual gradient vector. The Lie algebraG allows the direct sum splittingG =G + ⊕G − , causing with respect to the convolution (1.1) the direct sum splittingG * =G * + ⊕G * − . If to define now the set I(G * ) of Casimir invariant smooth functionals h :G * → R on the adjoint spaceG * via the coadjoint Lie algebrã G action (1.2) ad * ∇h(l)l = 0 at a seed elementl ∈G * , by means of the classical Adler-Kostant-Symes scheme [15,7,2,1] one can generate [11,12,16,9] a wide class of multi-dimensional completely integrable dispersionless (heavenly type) commuting to each other Hamiltonian systems for all h ∈ I(G * ), ∇h(l) := ∇h + (l) ⊕ ∇h − (l) ∈G + ⊕G − , on suitable functional manifolds. Moreover, these commuting to each other flows (1.3) can be equivalently represented as a commuting system of Lax-Sato type [9] vector field equations on the functional space C 2 (T n C ; C), generating an complete set of first integrals for them. As it was appeared, amongst them there are present important equations for modern studies in physics, hydrodynamics and, in particular, in Riemannian geometry, being related with such interesting conformal structures on Riemannian metric spaces as Einstein and Einstein-Weyl metrics equations, the first and second Plebański conformal metric equations, Dunajski metric equations etc. What was observed, some of them were generated by seed elementsl ∈G * , meromorphic at some points of the complex plane C, whose analysis needed some modification of the theoretical backgrounds. Moreover, the general differential-geometric structure of seed elements, related with some conformal metric equations, proved to be invariant subject to the spatial dimension of the Riemannian spaces under regard, that made it possible to describe them analytically. We analyzed the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations, the inverse first Shabat reduction heavenly equation, the first and modified Plebański heavenly equations and its multi-dimensional generalizations, the Husain heavenly equation and its multidimensional generalizations, the general Monge equation and its multi-dimensional generalizations. Namely these and related aspects of the integrable multi-dimensional conformal metric equations, mentioned above, are studied and presented in the work.

The Lie-algebraic structures and integrable Hamiltonian systems
Consider the loop Lie algebraG, determined above. This Lie algebra has elements representable as a(x; λ) .., ∂ ∂xn ) ⊤ is the generalized Euclidean vector gradient with respect to the vector variable x := (λ, x) ∈ T n C . As it was mentioned above, the Lie algebraG naturally splits into the direct sum of two subalgebras: allowing to introduce on it the classical R-structure: The spaceG * ≃Λ 1 (T n C ), adjoint to the Lie algebraG of vector fields on T n C , is functionally identified withG subject to the metric (1.1). Now for arbitrary f, g ∈ D(G * ), one can determine two Lie-Poisson type brackets where at any seed elementl ∈G * the gradient element ∇f (l) and ∇g(l) ∈G are calculated with respect to the metric (1.1). Now let us assume that a smooth function γ ∈ I(G * ) is a Casimir invariant, that is ad * ∇γ(l)l = 0 for a chosen seed elementl ∈G * . As the coadjoint mapping ad * ∇f (l) :G * →G * for any f ∈ D(G * ) can be rewritten in the reduced form as where, by definition, ∇f (l) :=< ∇f (l), ∂ ∂x > . For the Casimir function γ ∈ D(G * ) the condition (2.7) is then equivalent to the equation which should be solved analytically. In the case when an elementl ∈G * is singular as |λ| → ∞, one can consider the general asymptotic expansion )| + are, respectively, defined for special integers p y , p t ∈ Z + . These invariants generate, owing to the Lie-Poisson bracket (2.6), the following commuting flows: where y, t ∈ R are the corresponding evolution parameters. Since the invariants h (y) , h (t) ∈ I(G * ) commute with respect to the Lie-Poisson bracket (2.6), the flows (2.12) and (2.13) also commute, implying that the corresponding Hamiltonian vector field generators (2.14) A ∇h (t) satisfy the Lax compatibility condition for all y, t ∈ R. On the other hand, the condition (2.15) is equivalent to the compatibility condition of two linear equations for a function ψ ∈ C 2 (R 2 × T n C ; C) for all y, t ∈ R and any λ ∈ C. The above can be formulated as the following key result: Proposition 2.1. Let a seed vector field bel ∈G * and h (y) , h (t) ∈ I(G * ) be Casimir functions subject to the metric (·|·) on the loop Lie algebraG and the natural coadjoint action on the loop co-algebraG * . Then the following dynamical systems are commuting Hamiltonian flows for all y, t ∈ R. Moreover, the compatibility condition of these flows is equivalent to the vector fields representation (2.16), where ψ ∈ C 2 (R 2 × T n C ; C) and the vector fields A ∇h (y) ∈G are given by the expressions (2.14) and (2.11).
Remark 2.2. As mentioned above, the expansion (2.10) is effective if a chosen seed elementl ∈G * is singular as |λ| → ∞. In the case when it is singular as |λ| → 0, the expression (2.10) should be replaced by the expansion for suitably chosen integers p ∈ Z + , and the reduced Casimir function gradients then are given by the Hamiltonian vector field generators for suitably chosen positive integers p y , p t ∈ Z + and the corresponding Hamiltonian flows are, respectively, written as ∂l/∂t = ad * It is also worth of mentioning that, following Ovsienko's scheme [11,12], one can consider a slightly wider class of integrable heavenly equations, realized as compatible Hamiltonian flows on the semidirect product of the holomorphic loop Lie algebraG of vector fields on the torus T n C and its regular co-adjoint spaceG * , supplemented with naturally related cocycles.
3. The Lax-Sato type integrable multi-diemnsional heavenly systems and related conformal structure generating equations x dλ, which generates with respect to the metric (1.1) the gradient of the Casimir invariants h (pt) , h (py) ∈ I(G * ) in the form as |λ| → ∞ at p t = 2, p y = 1. For the gradients of the Casimir functions h (t) , h (y) ∈ I(G * ), determined by (2.11) one can easily obtain the corresponding Hamiltonian vector field generators satisfying the compatibility condition (2.15), which is equivalent to the set of equations describing general integrable Einstein-Weyl metric equations [6].
As is well known [10], the invariant reduction of (3.3) at v = 0 gives rise to the famous dispersionless Kadomtsev-Petviashvili equation for which the reduced vector field representation (2.16) follows from (3.2) and is given by the vector fields satisfying the compatibility condition (2.15), equivalent to the equation (3.4). In particular, one derives from (2.16) and (3.5) the vector field compatibility relationships

The modified Einstein-Weyl metric equation. This equation system is
where a x := u x w x − w xy , and was recently derived in [17]. In this case we take alsõ which with respect to the metric (1.1) generates two Casimir invariants γ (j) ∈ I(G * ), j = 1, 2, whose gradients are as |λ| → ∞ at p y = 1, p t = 2. The corresponding gradients of the Casimir functions h (t) , h (y) ∈ I(G * ), determined by (2.11), generate the Hamiltonian vector field expressions ∇h (y) Now one easily obtains from (3.10) the compatible Lax system of linear equations ∂ψ ∂y 3.3. The Dunajski heavenly equation system. This equation, suggested in [5], generalizes the corresponding anti-self-dual vacuum Einstein equation, which is related to the Plebański metric and the celebrated Plebański [13,8] second heavenly equation. To study the integrability of the Dunajski equations C ) * and take the following as a seed elementl ∈G * With respect to the metric (1.1), the gradients of two functionally independent Casimir invariants h (py) , h (py) ∈ I(G * ) can be obtained as |λ| → ∞ in the asymptotic form as at p t = 1 = p y . Upon calculating the Hamiltonian vector field generators ∇h (y) following from the Casimir functions gradients (3.14), one easily obtains the following vector fields satisfying the Lax compatibility condition (2.15), which is equivalent to the vector field compatibility relationships ∂ψ ∂t satisfied for ψ ∈ C 2 (R 2 × T 2 C ; C), any (y, t) ∈ R 2 and all (λ; x 1 , x 2 ) ∈ T 2 C . As was mentioned in [3], the Dunajski equations (3.12) generalize both the dispersionless Kadomtsev-Petviashvili and Plebański second heavenly equations, and is also a Lax integrable Hamiltonian system.

3.5.
Second conformal structure generating equation: u xt +u x u yy −u y u xy = 0. For a seed elementl ∈G ,and α, β ∈ R, there is one independent Casimir functional γ (1) ∈ I(G * ) with the following asymptotic as |λ| → 0 expansion of its functional gradient: , where c r ∈ R, r = 1, 2. If one assumes that c 0 = 1, c 1 = 0 and c 2 = 0, then we obtain two functionally independent gradient elements The corresponding commutativity condition (3.19) of the vector fields (3.20) give rise to the following heavenly type equation: (3.25) u xt + u x u yy − u y u xy = 0, whose linearized Lax-Sato representation is given by the first order system of linear vector field equations on a function ψ ∈ C 2 (R 2 × T 1 C ; R).

Inverse first Shabat reduction heavenly equation. A seed elementl
, and a 0 , a 1 ∈ R, generates two independent Casimir functionals γ (1) and γ (2) ∈ I(G * ), whose gradients have the following asymptotic expansions: , as |µ| → 0, µ := λ + 1, and as |λ| → ∞. If we put, by definition, the commutativity condition (3.19) of the vector fields (3.20) leads to the heavenly equation which can be obtained as a result of the simultaneous changing of independent variables R ∋x → t ∈ R, R ∋y → x ∈ R and R ∋t → y ∈ R in the first Shabat reduction heavenly equation. The corrersponding Lax-Sato representation is given by the compatibility condition for the first order vector field equations equations

First Plebański heavenly equation and its generalizations. The seed el-
where u ∈ C 2 (T 2 × R 2 ; R), (x 1 , x 2 ) ∈ T 2 , λ ∈ C\{0} and "d" designates a full differential, generates two independent Casimir functionals γ (1) and γ (2) ∈ I(G * ), whose gradients have the following asymptotic expansions: as |λ| → 0. The commutativity condition (3.19) of the vector fields (3.20), where leads to the first Plebański heavenly equation [4]: Its Lax-Sato representation entails the compatibility condition for the first order partial differential equations where ψ ∈ C ∞ (R 2 × T 2 C ; C). Remark 3.1. Taking into account that the determining condition for Casimir invariants is symmetric and equivalent to the system of nonhomogeneous linear first order partial differential equations for the covector function l = (l 1 , l 2 ) ⊤ , the corresponding seed element can be also chosen in another forms. Moreover, the form (3.33) is invariant subject to the spatial dimension of the underlying torus T n , what makes it possible to describe the related generalized conformal metric equations for arbitrary dimension.
In particular, one easily observes that the asymptotic expansions (3.34) are also true for such invariant seed elements as The above described Lie-algebraic scheme can be easily generalized for any dimension n = 2k, where k ∈ N, and n > 2. In this case one has 2k independent Casimir functionals γ (j) ∈ I(G * ), whereG * = dif f (T 2k ) * , j = 1, 2k, with the following asymptotic expansions for their gradients: . . . , If we put , , the commutativity condition (3.19) of the vector fields (3.20) leads to the following multi-dimensional analogs of the first Plebański heavenly equation:

Conclusion
We succeeded in applying the Lie-algebraic approach to studying vector fields on the complexified n-dimensional torus and the related Lie-algebraic structures, which made it possible to construct a wide class of multi-dimensional dispersionless integrable systems, describing conformal structure generating equations of modern mathematical physics. There was described a modification of the approach subject to the spatial dimensional invariance and meromorphicity of the related differentialgeometric structures, giving rise to new generalized multi-dimensional conformal metric equations. There have been analyzed in detail the related differential-geometric structures of the Einstein-Weyl conformal metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations, the inverse first Shabat reduction heavenly equation, the first and its multi-dimensional generalizations, the modified Plebański and Husain heavenly equations, the general Monge equation and its multi-dimensional generalizations.