On a class of reductions of Manakov-Santini hierarchy connected with the interpolating system
aa r X i v : . [ n li n . S I] D ec On a class of reductions of Manakov-Santinihierarchy connected with the interpolating system
L.V. Bogdanov ∗ September 17, 2018
Abstract
Using Lax-Sato formulation of Manakov-Santini hierarchy, we in-troduce a class of reductions, such that zero order reduction of thisclass corresponds to dKP hierarchy, and the first order reduction givesthe hierarchy associated with the interpolating system introduced byDunajski. We present Lax-Sato form of reduced hierarchy for the inter-polating system and also for the reduction of arbitrary order. Similarto dKP hierarchy, Lax-Sato equations for L (Lax fuction) due to the re-duction split from Lax-Sato equations for M (Orlov function), and thereduced hierarchy for arbitrary order of reduction is defined by Lax-Sato equations for L only. Characterization of the class of reductionsin terms of the dressing data is given. We also consider a waterbagreduction of the interpolating system hierarchy, which defines (1+1)-dimensional systems of hydrodynamic type. In this work we construct a class of reductions of the hierarchy associatedwith the system recently introduced by Manakov and Santini [1] (see also[2], [3]), u xt = u yy + ( uu x ) x + v x u xy − u xx v y ,v xt = v yy + uv xx + v x v xy − v xx v y , (1)whose Lax pair is ∂ y Ψ = (( p − v x ) ∂ x − u x ∂ p ) Ψ ,∂ t Ψ = (( p − v x p + u − v y ) ∂ x − ( u x p + u y ) ∂ p ) Ψ , (2) ∗ L.D. Landau ITP, Kosygin str. 2, Moscow 119334, Russia, e-mail [email protected] p plays a role of a spectral variable. Manakov-Santini system is ageneralization of dispersionless KP (Khohlov-Zabolotskaya) equation to thecase of general (non-Hamiltonian) vector fields in the Lax pair. For v = 0the system reduces to the dKP equation. Respectively, u = 0 reductiongives an equation [4] (see also [5, 6, 7]) v xt = v yy + v x v xy − v xx v y . (3)Using Lax-Sato formulation of the hierarchy [8, 9, 10], we introduce aclass of reductions, such that zero order reduction of this class correspondsto dKP hierarchy, and the first order reduction gives the hierarchy connectedwith the interpolating system, which was introduced in [11], where it wasproved that it is ”the most general symmetry reduction of the second heav-enly equation by a conformal Killing vector with a null self-dual derivative”.In [11] it was also shown that the interpolating system corresponds to sim-ple differential reduction cu = bv x of Manakov-Santini equation. We presentLax-Sato form of reduced hierarchy for interpolating system and also for re-duction of arbitrary order. Similar to dKP hierarchy, Lax-Sato equations for L (Lax fuction) due to the reduction split from Lax-Sato equations for M (Orlov function), and the reduced hierarchy for arbitrary order of reductionis defined by Lax-Sato equations for L only. In terms of Manakov-Santinisystem this class defines differential reductions (not changing the numberof dimensions). Characterization of the class of reductions in terms of thedressing data is given. We also consider waterbag type reductions of re-duced hierarchies (including interpolating equation hierarchy), which define(1+1)-dimensional systems of hydrodynamic type.Reductions of Manakov-Santini system were considered also in the works[12], [13], [14], concentrating mostly on (1+1)-dimensional reductions ofhydrodynamic type. Manakov-Santini hierarchy is defined by Lax-Sato equations [8, 9, 10] ∂∂t n (cid:18) LM (cid:19) = (cid:18)(cid:18) L n L p { L, M } (cid:19) + ∂ x − (cid:18) L n L x { L, M } (cid:19) + ∂ p (cid:19) (cid:18) LM (cid:19) , (4)2here L , M , corresponding to Lax and Orlov functions of dispersionless KPhierarchy, are the series L = p + ∞ X n =1 u n ( t ) p − n , (5) M = M + M , M = ∞ X n =0 t n L n ,M = ∞ X n =1 v n ( t ) L − n = ∞ X n =1 ˜ v n ( t ) p − n , (6)and x = t , ( P ∞−∞ u n p n ) + = P ∞ n =0 u n p n , { L, M } = L p M x − L x M p . A morestandard choice of times for dKP hierarchy corresponds to M = P ∞ n =0 ( n + 1) t n L n ,it is easy to transfer to it by rescaling of times.Lax-Sato equations (4) are equivalent to the generating relation [8, 9, 10] (cid:18) d L ∧ d M { L, M } (cid:19) − = 0 , (7)where differential takes into account all times t and variable p .Equations (4) imply that the dynamics of the Poisson bracket J = { L, M } is described by the equation [12] ∂∂t n ln J = ( A n ∂ x − B n ∂ p ) ln J + ∂ x A n − ∂ p B n , (8) A n = (cid:18) L n L p J (cid:19) + , B n = (cid:18) L n L x J (cid:19) + . This equation together with the first equation of (4) forms a closed systemwhich defines Manakov-Santini hierarchy and can be used as an equivalentto system (4), it is very useful for the description of reductions. Thus, todefine Manakov-Santini hierarchy, it is possible to consider the equations ∂∂t n L = (cid:16)(cid:0) L n L p J − (cid:1) + ∂ x − (cid:0) L n L x J − (cid:1) + ∂ p (cid:17) L,∂∂t n ln J = (cid:16)(cid:0) L n L p J − (cid:1) + ∂ x − (cid:0) L n L x J − (cid:1) + ∂ p (cid:17) ln J + ∂ x (cid:0) L n L p J − (cid:1) + − ∂ p (cid:0) L n L x J − (cid:1) + (9)for the series L ( p ) (5) and J , J = 1 + ∞ X j n ( t ) L − n = 1 + ∞ X ˜ j n ( t ) p − n . (10)3unction M can be found from L and J using the relation [12] J = { L, M } = ( ∂ p L ) ∂ x M | L , where | L means that a partial derivative is taken for fixed L . Then ∂ x M | L = J ( ∂ p L ) − = J ∂ L p ( L ) , (11)and, introducing series for p ( L ) (inverse to L ( p ) (5)), p = L + ∞ X p n ( t ) L − n , (12)it is possible to find coefficients of the series for ∂ x M | L explicitly and definethe function M . For the first coefficient of the series (6) we get ∂ x v ( t ) = j ( t ). In the case of Hamiltonian vector fields J = 1 and ∂ x M | L = ∂ L p ( L ).Lax-Sato equations for the first two flows of the hierarchy (4) ∂ y (cid:18) LM (cid:19) = (( p − v x ) ∂ x − u x ∂ p ) (cid:18) LM (cid:19) , (13) ∂ t (cid:18) LM (cid:19) = (( p − v x p + u − v y ) ∂ x − ( u x p + u y ) ∂ p ) (cid:18) LM (cid:19) , (14)where u = u , v = v , x = t , y = t , t = t , correspond to the Lax pair (2)of Manakov-Santini system (1).Equation (13) gives recursion relations, defining the coefficients of theseries L ( p ), M ( p ) (5), (6) through the functions u , v , ∂ x u n +1 = ∂ y u n + v x u n − ( n − u x u n − , (15) ∂ x ˜ v n +1 − u n = ∂ y ˜ v n + v x ˜ v n − ( n − u x ˜ v n − , n > , ˜ v = v. (16)Using these relations, Manakov-Santini system can be directly obtained fromequation (14) without the application of compatibility conditions for linearequations. It is also possible to use equations for ln J (9), the first two flowsread ∂ y ln J = (( p − v x ) ∂ x − u x ∂ p ) ln J − v xx , (17) ∂ t ln J = (( p − v x p + u − v y ) ∂ x − ( u x p + u y ) ∂ p ) ln J − v xx p − v xy , (18)and recursion relation for ln J = P ∞ n =1 (ln J ) n p − n is similar to recursion for L ( p ), ∂ x (ln J ) n +1 = ∂ y (ln J ) n + v x (ln J ) n − ( n − u x (ln J ) n − , where n >
1, (ln J ) = v x . 4 A class of reductions connected with the inter-polating system
In this section we consider a class of reductions of Manakov-Santini hier-archy, characterized by existense of order k polynomial (with respect to p )solution of non-homogeneous linear equation (8). For k = 0 this reductioncorresponds to Hamiltonian vector fields and dKP hierarchy. For k = 1we obtain the interpolating system [11] hierarchy. For general k J can beexplicitely expressed through L , and the reduced hierarchy is defined byLax-Sato equations for L only (similar to dKP hierarchy).Let ln J satisfy non-homogeneous equations (8) and L satisfy homoge-neous equations (4), than the function ln J + F ( L ) also satisfies equations(8). We define a class of reductions of Manakov-Santini hierarchy by thecondition (ln J − αL k ) − = 0 , (19)where α is a constant, that means that equations (8) have an analytic so-lution (ln J − αL k ). This condition defines a reduction because A n , B n inequations (8) are polynomials, and the dynamics, defined by these equa-tions, preserves analitycity of the functions, so analytic solutions form aninvariant manifold. Thus, if (ln J − αL k )( x, p ) is polynomial with respect to p at initial point in higher times, then it is polynomial for arbitrary valuesof higher times.Reduction (19) is completely characterized by the existence of polyno-mial solution of equations (8). Proposition 1
Existence of polynomial solution f = − αp k + i = k − X f i ( t ) p i , (where coefficients f i don’t contain constants, see below) of equations (8), ∂∂t n f = ( A n ∂ x − B n ∂ p ) f + ∂ x A n − ∂ p B n , (20) is equivalent to the reduction condition (19). Proof
First, reduction condition (19) directly implies that f = (ln J − αL k )is a polinomial solution of equations (20) of required form, that proves thatexistence of polynomial solution is necessary.5o prove that it is sufficient, we note that F = ln J − f solves homoge-neous equations (20) (equations (4)). Let us expand p into the powers of L (12), and represent F in the form F = αL k + i = k − X −∞ F i ( t ) L i , where F i ( t ) can be expressed through f i ( t ) and coefficients of expansion of J and L (respectively, j n ( t ) and u n ( t )). It is easy to check that F solveshomogeneous equations (20) iff all the coefficients F i ( t ) are constants. Sug-gesting that coefficients f i of the polynomial f ( p ) don’t contain constants(in the sense that they are equal to zero if all the coefficients j n = u n = 0),we come to the conclusion that ln J − f = αL k . (cid:3) The simplest case k = 0 corresponds to Hamiltonian vector fields. In-deed, in this case J = 1, and from equations (8) we have ∂ x A n − ∂ p B n = 0 . In the case k = 1(ln J − αL ) − = 0 ⇒ (ln J − αL ) = (ln J − αL ) + = − αp,J = exp α ( L − p ) . (21)So, similar to the case of Hamiltonian vector fields, equation for L splits andthe reduced hierarchy is defined by Lax-Sato equations ∂∂t n L = ( e α ( p − L ) L n L p ) + ∂ x L − ( e α ( p − L ) L n L x ) + ∂ p L. (22)Generating relation for the reduced hierarchy reads (cid:16) e α ( p − L ) d L ∧ d M (cid:17) − = 0 , or, equivalently, (cid:0) e − αL d L ∧ d M (cid:1) − = 0 . Representing relation (21) as a series in p − , in the first nontrivial order weget (see (11)) αu = j = v x , (23)6hat is exactly the condition used in [11] to reduce Manakov-Santini systemto interpolating equation ( α = cb in the notations of [11]). Manakov-Santinisystem (1) with reduction (23) is equivalent to interpolating equation upto a simple transformation, and we will call hierarchy (22) the interpolatingequation hierarchy .Reduction condition (21) implies that ( − αp ) is a solution of equations(8) (in fact, these conditions are equivalent ), and, substituting it, we getreduction equations in term of vector fields components, ∂ x A n − ∂ p B n − B n = 0 . (24)It is easy to check that for n = 1 we obtain a reduction condition (23). General k In the general case,(ln J − αL k ) − = 0 ⇒ (ln J − αL k ) = (ln J − αL k ) + = − α ( L k ) + ,J = exp α ( L k − ( L k + )) = exp α ( L k − ) , (25)and Lax-Sato equations of reduced hierarchy read ∂∂t n L = ( e − α ( L k − ) L n L p ) + ∂ x L − ( e − α ( L k − ) L n L x ) + ∂ p L. (26)These equations imply equations (9) for J (25), function M is defined byrelation (11), ∂ x M | L = J ( ∂ p L ) − = e α ( L k − ( L k + )) ( ∂ p L ) − . Generating relation (7) in this case takes the form (cid:16) e − αL k d L ∧ d M (cid:17) − = 0 . (27)Reduction (25) is equivalent to the condition that ( − αL k + ) is a solution toequations (8), that gives a differential characterization of reduction in termsof Manakov-Santini hierarchy, ∂∂t n ( αL k + ) = ( A n ∂ x − B n ∂ p ) ( αL k + ) − ∂ x A n + ∂ p B n , (28) A n = (cid:18) L n L p J (cid:19) + , B n = (cid:18) L n L x J (cid:19) + . n = 1 we obtain a condition (compare (17)) ∂ y ( αL k + ) = (( p − v x ) ∂ x − u x ∂ p )( αL k + ) + v xx . (29)This condition defines a differential reduction of Manakov-Santini system.Let us consider in more detail the case k = 2. Reduction is defined byrelation (25), J = e α ( L − ) . (30)Taking an expansion into powers of p − , in the first nontrivial order we get j = 2 αu . Using recursion formula (15), we obtain ∂ x u = u y + v x u x . Thus we come to the conclusion that in terms of Manakov-Santini system(1) reduction (30) leads to a condition2 α ( u y + v x u x ) = v xx (31)This condition defines a differential reduction of Manakov-Santini system.Another way to obtain differential reduction is to use relation (29). In-deed, ( L ) = p + 2 u , and, substituting this expression to relation (29), weget 2 αu y = 2 α (( p − v x ) u x − u x p ) + v xx ⇒ α ( u y + 2 v x u x ) = v xx . Relation (29) explicitly gives differential reductions of arbitrary order k forManakov-Santini system.For illustration we will also calculate differential reduction of Manakov-Santini system of the order k = 3. In this case ( L ) = p + 3 pu + 3 u , and,substituting this expression to (29), we get3 α (cid:0) ∂ y ( u y + u x v x ) + ∂ x ( u y v x + u x v x + uu x ) (cid:1) = v xxx . (32) A pair of reductions with different k – reduction to (1+1) If we consider a pair of reductions with different k , we obtain a closed (1+1)-dimensional system of equations for the functions u , v . First let us considerreductions of interpolating system, i.e., reduction with k = 1, which leads tothe condition (23), together with reduction (19) of some order k = 1 (witha constant β ). 8or k = 2, using (19) and (31), we obtain a system u y + v x u x = (2 β ) − v xx ,v x = αu, which implies hydrodynamic type equation (Hopf type equation) for u , u y + αuu x = α β u x . The system for k = 3 reads (see (32)) ∂ y ( u y + u x v x ) + ∂ x ( u y v x + u x v x + uu x ) = 3 β − v xxx ,v x = αu, it implies an equation for u , u yy + ∂ x (2 αu y u + α u x u + uu x − α β u x ) = 0 , which can be rewritten as a system of hydrodynamic type for two functions u , w , w y = ( α β − α u − u ) u x − αuw x ,u y = w x . A system of equations of hydrodynamic type corresponding to the reductionof interpolating system of arbitrary order k > f = βL k + − αp is a solution of linear equation ∂ y f = ( p − αu ) ∂ x f − u x ∂ p f, which provides a system of hydrodynamic type for the coefficients of thepolynomial f = βp k + kβup k − − αp + P k − i =0 f i p i , namely ∂ y u = ( kβ ) − ∂ x f k − − αu∂ x u,∂ y f k − = ∂ x f k − − αu∂ x f k − − k ( k − ∂ x u,∂ y f i = ∂ x f i − − αu∂ x f i − ( i + 1) f i +1 ∂ x u, < i < k − ,∂ y f = − αu∂ x f − ( f − α ) ∂ x u. Let us also consider a simple example of a system defined by two reduc-tions of higher order, taking reductions of the order 2 (31) and of the order3 (32), u y + v x u x = (2 α ) − v xx , (cid:0) ∂ y ( u y + u x v x ) + ∂ x ( u y v x + u x v x + uu x ) (cid:1) = (3 β ) − v xxx . u , w = v x , u y + wu x = (2 α ) − w x ,w y = 2 α β w x − ww x − αuu x . For the class of reduced hierarchies defined by Lax-Sato equations (26) it ispossible to consider manifold of solutions of the form L ( p, x ) = p − N X i =1 c i ln( p − w i ( x )) , N X i =1 c i = 0 , (33)where c i are some constants. Due to the fact that coefficients of vectorfields in equations (26) are polynomial, and ’plus’ projection of equations isidentically zero by construction, it is straightforward to demonstrate thatthis manifold is invariant under dynamics, so it defines a reduction (thistype of reduction is known for dKP hierarchy as a waterbag reduction).Each of Lax-Sato equations (26) in this case is equivalent to the closed(1+1)-dimensional system of equations for the functions u i .Let us study in more detail the waterbag reduction for interpolatingequation hierarchy (22). First two Lax-Sato equations of the hierarchy read ∂ y L = ( p − αu ) ∂ x L − u x ∂ p L,∂ t L = ( p − αup − αu + u ) ∂ x L − ( u x p − αuu x + ∂ x u ) ∂ p L. (34)For Lax-Sato function of the form (33) the coefficients of expansion u n areexpressed through the functions w i as u n = N X i =1 c i n w ni , (35)Substituting ansatz (33) to Lax-Sato equations (34) and using formula (35),we obtain two closed (1+1)-dimensional systems of equations for the func-10ions w i , ∂ y w i = w i − α N X i =1 c i w i ! ∂ x w i + ∂ x N X i =1 c i w i ,∂ t w i = w i − αw i N X i =1 c i w i − α N X i =1 c i w i + N X i =1 c i w i ! ∂ x w i + w i − α N X i =1 c i w i ! ∂ x N X i =1 c i w i + ∂ x N X i =1 c i w i . (36)These systems (as well as higher flows) are compatible, because they areconstructed as a reduction of the flows of Manakov-Santini hierarchy to theinvariant manifold (33). On the invariant manifold equations (36) are equiv-alent to Lax-Sato equations of Manakov-Santini hierarchy. Equations (36)are (1+1)-dimensional systems of hydrodynamic type, their common solu-tion gives a solution of interpolating equation (Mananakov-Santini system(1) with the reduction αu = v x ) by the formula u = N X i =1 c i w i . In the case α = 0 formulae (36) give the waterbag reduction of the dKPhierarchy [15] (to match (36) to the formulae of the work [15], it is necessaryto rescale the times).Minimal number of components w i in equations (36) is two, and forthe simplest case N = 2, L ( p, x ) = p − c ln p − w ( x ) p − w ( x ) , an explicit form ofhydrodynamic type system corresponding to the first flow of (36) is ∂ y w = ∂ x (cid:18) w + c ( w − w ) (cid:19) − αc ( w − w ) ∂ x w ,∂ y w = ∂ x (cid:18) w + c ( w − w ) (cid:19) − αc ( w − w ) ∂ x w , ∂ t w = ∂ x (cid:18) w + cw ( w − w ) + c w − w ) (cid:19) − α (cid:18) cw ( w − w ) ∂ x w + c ∂ x ( w − w ) (cid:19) ,∂ t w = ∂ x (cid:18) w + cw ( w − w ) + c w − w ) (cid:19) − α (cid:18) cw ( w − w ) ∂ x w + c ∂ x ( w − w ) (cid:19) . Zakharov reduction, corresponding to rational L with simple poles, can beconsidered as a degenerate case of the waterbag reduction, when pairs offunctions w i coincide. In the two-component case, considering the limit c → ∞ , w − w = c − u , we get L = p + up − w , and the equations of reducedhierarchy can be obtained as a limit of equations for the waterbag reduction.For the first two flows ∂ y w = ∂ x (cid:18) w + u (cid:19) − αu∂ x w,∂ y u = ∂ x ( wu ) − αu∂ x u, and ∂ t w = ∂ x (cid:18) w + 2 wu (cid:19) − α (cid:18) wu∂ x w + 12 ∂ x u (cid:19) ,∂ t u = ∂ x (cid:0) w u + u (cid:1) − αu∂ x ( wu ) . A common solution of these systems gives a solution u of interpolating equa-tion. A dressing scheme for Manakov-Santini hierarchy can be formulated in termsof two-component nonlinear Riemann problem on the unit circle S in thecomplex plane of the variable p , L in = F ( L out , M out ) ,M in = F ( L out , M out ) , (37)12here the functions L in ( p, t ), M in ( p, t ) are analytic inside the unit circle,the functions L out ( p, t ), M out ( p, t ) are analytic outside the unit circle andhave an expansion of the form (5), (6). The functions F , F are suggestedto define (at least locally) diffeomorphism of the plane, F ∈ Diff(2), andwe call them dressing data. It is straightforward to demonstrate that theproblem (37) implies analyticity of the differential formΩ = d L ∧ d M { L, M } (where differential takes into account all times t and p ) in the complex planeand generating relation (7), thus defining a solution of Manakov-Santini hier-archy. Considering a reduction to area-preserving diffeomorphisms SDiff(2),we obtain the dKP hierarchy.To obtain interpolating system, it is necessary to consider a more gen-eral class of reductions. Let G ( λ, µ ), G ( λ, µ ) define an area-preservingdiffeomorphism, G ∈ SDiff(2), (cid:12)(cid:12)(cid:12)(cid:12) D ( G , G ) D ( λ, µ ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 . Let us fix a pair of analytic functions f ( λ, µ ), f ( λ, µ ) (reduction data) andconsider a problem f ( L in , M in ) = G ( f ( L out , M out ) , f ( L out , M out )) ,f ( L in , M in ) = G ( f ( L out , M out ) , f ( L out , M out )) , (38)which defines a reduction of MS hierarchy. In terms of initial Riemannproblem for MS hierarchy (37), which can be written in the form( L in , M in ) = F ( L out , M out ) , (39)the reduction condition for the dressing data reads f ◦ F ◦ f − ∈ SDiff(2) . (40)In terms of equations of MS hierarchy the reduction is characterized by thecondition(d f ( L, M ) ∧ d f ( L, M )) out = (d f ( L, M ) ∧ d f ( L, M )) in , thus the form Ω red = d f ( L, M ) ∧ d f ( L, M )13s analytic in the complex plane, and reduced hierarchy is defined by thegenerating relation (d f ( L, M ) ∧ d f ( L, M )) − = 0 . Taking f ( L, M ) =
L,f ( L, M ) = e − αL n M, (41)we obtain the generating relation (cid:16) e − αL k d L ∧ d M (cid:17) − = 0 , coinciding with (27). Thus we come to the following conclusion: Proposition 2
A class of reductions (19) is characterized in terms of thedressing data for the problem (39) by the condition (40), where f is definedby the formulae (41). For interpolating equation f = L , f = e − αL M , and the Riemann problem(38) can be written in the form L in = G ( L out , e − αL out M out ) ,M in = e αG ( L out ,e − αL out M out ) G ( L out , e − αL out M out ) , where G ∈ SDiff(2).
Acknowledgments
The author is grateful to S.V. Manakov and M.V. Pavlov for useful dis-cussions. This research was partially supported by the Russian Foundationfor Basic Research under grants no. 06-01-89507 (Russian-Taiwanese grant95WFE0300007), 08-01-90104, 07-01-00446, 09-01-92439, and by the Presi-dent of Russia grant 4887.2008.2 (scientific schools).
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