Dispersionless BKP equation, the Manakov-Santini system and Einstein-Weyl structures

We construct a map of solutions of the dispersionless BKP (dBKP) equation to solutions of the Manakov-Santini (MS) system. This map defines an Einstein-Weyl structure corresponding to the dBKP equation through the general Lorentzian Einstein-Weyl structure corresponding to the MS system. We give a spectral characterisation of reduction of the MS system which singles out the image of the dBKP equation solutions and also consider more general reductions of this class. We define the BMS system and extend the map defined above to the map (Miura transformation) of solutions of the BMS system to solutions of the MS system, thus obtaining an Einstein-Weyl structure for the BMS system.


Introduction
Dispersionless BKP hierarchy is a reduction of the dispersionless KP hierarchy by a special symmetry, which is compatible only with odd times of the hierarchy [1], [2]. Equations of the hierarchy can be represented as compatibility conditions for the Hamilton-Jacobi equations. For the first equation of the hierarchy (dispersionless BKP equation) corresponding Hamilton-Jacobi equations are The symmetry characterising the reduction is a simple condition for Hamiltonians H(−p) = −H(p), x = t 1 , y = t 3 , t = t 5 (in terms of dispersionless KP hierarchy times). Compatibility condition of Hamilton-Jacobi equations (1) can be written in the form where the Poisson bracket is {f, g} = f p g x − f x g p , it gives the dispersionless BKP equation (see [1], [2]) In what follows we will rescale the times to simplify the coefficients of equation and use the Hamiltonians In potential form, u = f x , we have The Lax pair in terms of Hamilton-Jacobi equations (pseudopotentials) can be represented as commutation relations for Hamiltonian vector fields where In this Lax pair p plays a role of 'spectral parameter', and commutation relation [V 1 , V 2 ] = 0 gives exactly equation (4). The formalism of integration of equations arising as commutation relations for vector fields is not restricted to Hamiltonian vector fields. Moreover, several interesting examples corresponding to general vector fields were discussed, e.g., the Manakov-Santini (MS) system [3], [4], which was recently demonstrated to describe a general local form of the Einstein-Weyl equations [5].

From the dBKP equation to the MS system
In explicit form vector fields (6) read Symmetry of vector fields V (−p) = V (p) characterises dispersionless BKP hierarchy in the framework of dKP hierarchy.
Let us transform the spectral variable p 2 = µ, commutation relations evidently remain the same, Vector fields are still Hamiltonian (the bracket should be changed). Now let us make a change λ = µ + 2f x (which also preserves commutation relations) Lax pair V 1 , V 2 has the structure of the Manakov-Santini (MS) system Lax pair the MS system reads A comparison of Lax pairs (9), (8) gives a map from solution of dBKP equation (5) to solution of MS system (10), corresponding solutions of the MS system satisfy a reduction This map defines the Einstein-Weyl structure corresponding to dBKP equation.

Einstein-Weyl structure for the dBKP equation
Let us remind (see [6], [7], for more detail), that a Weyl space is a manifold with a conformal structure [g] and a symmetric connection D consistent with [g] in a sense that for every g ∈ [g] for some differential form ω (the connection preserves a conformal class). Einstein-Weyl spaces are defined by the condition that the trace-free part of the symmetrised Ricci tensor of the connection D vanishes (Einstein equations), which together with relation (13) constitute Einstein-Weyl equations system, in a coordinate form here Λ is some function. Einstein-Weyl equations are correctly defined for arbitrary manifold dimension not less than three, but the most interesting case is three-dimensional when they are integrable [8]. The Manakov-Santini system (10) defines a general local form of the (2+1)-dimensional Lorentzian Einstein-Weyl structure (modulo coordinate transformations) with the metric g and oneform (covector) ω defined as [5] where u, v satisfy the MS system. Using the map (11), we obtain the Einstein-Weyl structure corresponding to solutions of the potential dispersionless BKP equation (5), It could be possible to construct this Einstein-Weyl structure by the methods of the work [7], starting from the symbol of linearisation of equation (5). Here we do it directly, using the map (11).

dBKP equation as a reduction of the MS system
It is possible to get a condition (12) starting from the reduction of the MS hierarchy, which is characterised by the existence of wave function of adjoint linear operators of the hierarchy with special analytic properties (with respect to the spectral variable). The technique to construct this type of reductions was developed in [9]. Here we will do an elementary derivation on the level of Lax operator for the MS system. First we introduce formally adjoint linear operators, defined by the rule (u∂) * = −∂u (for all partial derivatives), −X * = X + div X, for the Lax operator of the MS system (9) we get We should emphasize that adjoint vector fields in general are not vector fields and contain an extra term without derivative, equal to divergence of vector field, for zero divergence vector fields are (anti) self-adjoint. However, the commutation of adjoint vector fields gives the same compatibility conditions. Let us suggest that adjoint Lax operator (17) possesses a wave function of the form where g is a function of times. This condition is compatible with the dynamics of the hierarchy and defines a reduction, see [9]. The form of this wave function can be found considering the map from the dispersionless BKP equation to the MS system, we will skip the details. For the logarithm of the wave function we have an equation and, substituting ψ = (λ − g) α , we get implying a condition For α = − 1 2 this condition coincides with condition (12) and MS system (10) reduces to potential dispersionless BKP equation (5) (f = v). For α = 0 relations (19) imply that v = 0, and we obtain the dKP equation for the function u, u x = g y − gg x . For general α the MS system reduces to the equation An interesting special case corresponds to α = −1, then condition (20) takes the form u = v y and the MS system reduces to the equation The Einstein-Weyl structure for equation (21) is obtained by the substitution of expression for u (20) to the MS system Einstein-Weyl structure (15), It is natural to suggest that, similar to dBKP case, equation (21) for arbitrary α could be obtained from some Hamiltonian Lax pair. And it is indeed so! Let us consider dBKP type Lax pair (1) with the Hamiltonians β = 2 corresponds to the dBKP equation case. In terms of Hamiltonian vector fields we have Similar to dBKP equation Lax pair, we will transform this Lax pair to obtain the Lax pair of the MS type. The first step is to perform a transformation µ = p β , The second step is a transformation λ = µ + βf x , Comparing with the MS Lax operator (9), we get After the identification g = βf x , β −1 = α+1, this transformation coincides with expressions (19). Thus, equation (21) can be obtained from the Hamiltonian Lax pair (23), (24), substituting β −1 = α + 1, v = αf .

BMS system
The symmetry of vector fields V (−p) = V (p) (7) characterising dispersionless BKP hierarchy in the framework of dKP hierarchy can be extended to the Manakov-Santini hierarchy. The Lax pair for the first equation of the hierarchy (the BMS system) reads Compatibility conditions imply that w 2 = u y + (u 2 ) x , and the BMS system can be written in the form For u = 0, corresponding to linearly degenerate case, when vector fields in the Lax pair do not include the derivative over spectral parameter, BMS system (26) reduces to the equation coinciding with the linearly degenerate reduction of MS system (10). It is not unexpected, because in linearly degenerate case we have a freedom of arbitrary change of the spectral variable (independent of times).
Hamiltonian reduction corresponds to u = −v x , and system (26) reduces to potential dBKP equation (5) for the function f = −v.
The condition v = 0 is evidently also a reduction of system (26), it leads to the equation which coincides with equation (22) after the identification v x = 2u.
It is rather suprising that, following the steps of transformation of the dBKP Lax pair to the MS type Lax pair described above, we are able to define a map (Miura transformation) from solutions of the BMS system to solutions of the MS system, thus defining the Einstein-Weyl structure corresponding to the BMS system.
Performing a transformation λ = p 2 + 2u in the Lax pair (25), we obtain a Lax pair of MS type with a Lax operator Comparing this Lax operator to the MS Lax operator (9), we obtain a transformation of solutions of the BMS system to the solutions of MS system, Substituting this transformation to Einstein-Weyl structure (15), we obtain the Einstein-Weyl structure corresponding to BMS system (26),

Hydrodynamic type reductions of the MS system
Let us consider a multicomponent generalisation of reduction (18) ψ = (λ − g i ) αi and complement it with a standard waterbag ansatz [10] for the wave functions of MS linear operators (9) ψ = λ + γ j ln(λ − f j ). Some special examples of related reductions were considered in [11]. These reductions lead to (1+1)-dimensional systems of hydrodynamic type defining the dynamics with respect to y, and with respect to t, These (1+1)-dimensional systems are compatible and their common solution represents a solution of the MS system.