Kadomtsev-Petviashvili hierarchies of types B and C
aa r X i v : . [ n li n . S I] F e b Kadomtsev-Petviashvili hierarchies of types B and C
A. Zabrodin ∗ February 2021
Dedicated to the memory of S.M. Natanzon
ITEP-TH-04/21
Abstract
This is a short review of the Kadomtsev-Petviashvili hierarchies of types B andC. The main objects are the L -operator, the wave operator, the auxiliary linearproblems for the wave function, the bilinear identity for the wave function and thetau-function. All of them are discussed in the paper. The connections with theusual Kadomtsev-Petviashvili hierarchy (of the type A) are clarified. Examples ofsoliton solutions and the dispersionless limit of the hierarchies are also considered. Contents ∗ Skolkovo Institute of Science and Technology, 143026, Moscow, Russia and Institute of BiochemicalPhysics, Kosygina str. 4, 119334, Moscow, Russia and ITEP NRC KI, 25 B.Cheremushkinskaya, Moscow117218, Russia; e-mail: [email protected] The CKP hierarchy 13
In the papers [1, 2, 3] infinite integrable hierarchies of partial differential equations withO ( ∞ ) and Sp ( ∞ ) symmetry were introduced. They can be called the Kadomtsev-Petviashvili hierarchies of type B (BKP) and C (CKP). The BKP (respectively, CKP)hierarchy was also discussed in [4, 5, 6, 7]. (respectively, [7, 8, 9, 10, 11]).As is pointed out in [3], the general solutions to the BKP and CKP hierarchies dependon functional parameters in two variables. In a certain sense to be clarified below in themain text these hierarchies can be regarded as restrictions of the well known Kadomtsev-Petviashvili (KP) hierarchy. In a nutshell, this can be made more precise as follows. Let X KP be the moduli space of solutions to the KP hierarchy (according to Segal and Wilson,it is an infinite dimensional Grassmann manifold). The modili spaces of solutions to theBKP and CKP hierarchies are submanifolds of X KP : X BKP ⊂ X KP , X CKP ⊂ X KP , and the“even” time evolution (i.e., the evolution with respect to the times t k , k ≥
1) is frozen.This paper is a short review of the BKP and CKP hierarchies. We discuss the mainobjects and notions related to them: the L -operator, the wave operator, the auxiliarylinear problems for the wave function, the bilinear identity for the wave function and thetau-function. The tau-function satisfies certain equations (the Hirota equations) whichare bilinear in the BKP case and have a more complicated structure in the CKP case.The connection between tau-functions of the KP, BKP and CKP hierarchies is clarified.As examples of solutions, we give explicit formulas for soliton solutions. BKP andCKP N -solitons are specializations of 2 N soliton solutions of the KP hierarchy. Asis known, soliton solutions are degenerations of more general quasi-periodic (algebro-geometric) solutions. According to the Krichever’s construction [12], any smooth al-gebraic curve with some additional data provides a quasi-periodic solution. The quasi-periodic solutions of the BKP hierarchy were constructed in [13], see also [14]. A detailed2iscussion of quasi-periodic solutions to the CKP hierarchy can be found in [11]. Thecorresponding algebraic curves should admit a holomorphic involution with two fixedpoints. Double-periodic in the complex plane (elliptic) solutions were studied in [15, 16]for BKP and [11] for CKP.We also discuss the zero dispersion limit of the BKP and CKP hierarchies whichappears to be the same for both of them. In the dispersionless limit, the operator ∂ x entering the pseudo-differential Lax operator is replaced by a commuting variable p , theLax operator becomes a commuting function (a Laurent series) and the commutator isreplaced by the Poisson bracket { p, x } = 1. Here we briefly recall the main notions related to the KP hierarchy. The set of indepen-dent variables (“times”) is t = { t , t , t , . . . } . It is convenient to set t = x + const, sothat the vector fields ∂ t and ∂ x are identical: ∂ t = ∂ x . The main object is the L -operatorwhich is a pseudo-differential operator of the form L = ∂ x + u ∂ − x + u ∂ − x + . . . (2.1)with no restrictions on the coefficient functions u i . The coefficient functions depend on x and on all the times: u i = u i ( x, t ). Together with the Lax operator, it is convenient tointroduce the wave operator (or dressing operator) W = 1 + ξ ∂ − x + ξ ∂ − x + . . . (2.2)such that L = W ∂ x W − (2.3)(the latter equality is interpreted as “dressing” of the operator ∂ x by W ). Clearly, thereis a freedom in the definition of the wave operator: it can be multiplied from the rightby any pseudo-differential operator with constant coefficients.The functions u i ( x,
0) are initial conditions for the time evolution u i ( x, → u i ( x, t )which is generated by the Lax equations of the KP hierarchy: ∂ t k L = [ B k , L ] , B k = (cid:16) L k (cid:17) + , k = 1 , , , . . . , (2.4)where ( . . . ) + means the differential part of a pseudo-differential operator (i.e. terms withnon-negative powers of ∂ x ). In particular, B = ∂ x and B = ∂ x + 2 u . Since B = ∂ x ,it follows from (2.4) that the evolution in the time t is simply the shift of x , i.e. thesolutions depend on x + t .An equivalent formulation of the hierarchy is through the zero curvature (Zakharov-Shabat) equations ∂ t l B k − ∂ t k B l + [ B k , B l ] = 0 . (2.5)The equivalence of the Lax and Zakharov-Shabat formulations was proved in [20]. Thefamous KP equation for u is obtained from (2.5) at k = 2, l = 3.3he Lax equations and the zero curvature equations are compatibility conditions ofthe auxiliary linear problems ∂ t k ψ = B k ψ, Lψ = zψ (2.6)for the formal wave function ψ = ψ ( x, t , z ) = W e xz + ξ ( t ,z ) , (2.7)where W is the wave operator (2.2), z is the spectral parameter and ξ ( t , z ) = X k ≥ t k z k (2.8)(it is implied that the operator ∂ − x acts to the exponential function as ∂ − x e xz = z − e xz ).One can also introduce the adjoint (dual) wave function ψ † = ψ † ( x, t , z ) = ( W † ) − e − xz − ξ ( t ,z ) , (2.9)where † means the formal adjoint defined by the rule (cid:16) f ( x ) ◦ ∂ nx (cid:17) † = ( − ∂ x ) n ◦ f ( x ). Itcan be shown that the adjoint wave function satisfies the adjoint linear equations − ∂ t k ψ † = B † k ψ † . (2.10)The tau-function τ KP ( x, t ) of the KP hierarchy is consistently introduced by theequations ψ ( x, t , z ) = e xz + ξ ( t ,z ) τ KP ( x, t − [ z − ]) τ KP ( x, t ) , (2.11) ψ † ( x, t , z ) = e − xz − ξ ( t ,z ) τ KP ( x, t + [ z − ]) τ KP ( x, t ) , (2.12)where we have used the standard notation t + j [ z − ] = n t + jz , t + j z , t + j z , . . . o , j ∈ Z . The wave functions satisfy the bilinear equation [4] I C ∞ ψ ( x, t , z ) ψ † ( x, t ′ , z ) dz πi = 0 (2.13)for all t , t ′ . Here C ∞ is a contour surrounding ∞ (a big circle of radius R → ∞ ). Using(2.11), (2.12), one can rewrite (2.13) as the following bilinear relation for the tau-function: I C ∞ e ξ ( t − t ′ ,z ) τ (cid:16) x, t − [ z − ] (cid:17) τ (cid:16) x, t ′ + [ z − ] (cid:17) dz πi = 0 . (2.14)This is the generating equation for all differential equations of the KP hierarchy. Adirect consequence of the bilinear relation (2.13) is the Hirota-Miwa equation for thetau-function of the KP hierarchy( z − z ) τ KP (cid:16) x, t − [ z − ] − [ z − ] (cid:17) τ KP (cid:16) x, t − [ z − ] (cid:17) + ( z − z ) τ KP (cid:16) x, t − [ z − ] − [ z − ] (cid:17) τ KP (cid:16) x, t − [ z − ] (cid:17) + ( z − z ) τ KP (cid:16) x, t − [ z − ] − [ z − ] (cid:17) τ KP (cid:16) x, t − [ z − ] (cid:17) = 0 . (2.15)4t is a generating equation for the differential equations of the hierarchy. In the limit z → ∞ it becomes the equation ∂ x log τ KP (cid:16) x, t + [ z − ] − [ z − ] (cid:17) τ KP ( x, t )= ( z − z ) τ KP (cid:16) x, t + [ z − ] (cid:17) τ KP (cid:16) x, t − [ z − ] (cid:17) τ KP ( x, t ) τ KP (cid:16) x, t + [ z − ] − [ z − ] (cid:17) − . (2.16)The tau-function ˜ τ ( x, t ) = e ℓ ( x, t ) τ ( x, t ), where ℓ ( x, t ) = γ + γ x + X k ≥ γ k t k is a linearfunction of the times, satisfies the same bilinear equations. We say that the tau-functionswhich differ by a factor of the form e ℓ ( x, t ) are equivalent. Here we present the main formulas related to the BKP hierarchy with some details. Themain reference is [2], see also [4, 5, 6, 7].
The set of independent variables (“times”) is t o = { t , t , t , . . . } . They are indexed bypositive odd numbers. As in the KP hierarchy, we set t = x + const, so that the vectorfields ∂ t and ∂ x are identical: ∂ t = ∂ x . The main object is the L -operator which is apseudo-differential operator of the form L = ∂ x + u ∂ − x + u ∂ − x + . . . (3.1)with the constraint L † = − ∂ x L∂ − x . (3.2)Unlike in the case of a reduction, when only a finite number of the coefficient functions u i remain independent, the constraint (3.2) implies that there are still infinitely manyindependent coefficients functions. As we shall see soon, the constraint (3.2) is invariantunder the “odd” flows t , t , t , . . . of the KP hierarchy, so the BKP hierarchy can beregarded as a sub-hierarchy (a restriction) of the KP one with the “even” times frozen.It is instructive to reformulate the constraint (3.2) in terms of the wave operator W (2.2) such that L = W ∂ x W − . The constraint (3.2) implies that W † ∂ x W commutes with ∂ x , i.e., it is a pseudo-differential operator with constant coefficients. The freedom in thedefinition of the wave operator can be fixed by demanding that W † ∂ x W = ∂ x , i.e. W † = ∂ x W − ∂ − x . (3.3)The Lax equations of the hierarchy are the same as (2.4) but with odd indices: ∂ t k L = [ B k , L ] , B k = (cid:16) L k (cid:17) + , k = 1 , , , . . . . (3.4)5he constraint (3.2) is equivalent to the condition that the differential operators B k satisfy B k · k ), i.e., that they have the form B k = ∂ kx + k − X j =1 b k,j ∂ jx with b k, = 0. Indeed, if (3.2) is satisfied, then L n ∂ − x = − ∂ − x L † n for odd n . On theother hand, ( L n ∂ − x ) † = − ∂ − x L † n = L n ∂ − x which implies that the coefficient in front of ∂ x in L n vanishes: b n, = 0 for odd n . Conversely, assuming that b n, = 0 for all odd n ,we shall prove that R = ∂ − x L † + L∂ − x = 0. Obviously, R is of the general form R = a∂ − mx + lower order termsand the identity R † = − R implies that if a is not identically zero, then m is odd. Thenwe have (see [1]): L m ∂ − x = ( R∂ x − ∂ − x L † ∂ x ) m ∂ − x = − ( ∂ − x L † ∂ x ) m ∂ − x + mR ( ∂ − x L † ∂ x ) m − + an operator of order less than −
1= ( L n ∂ − x ) † + ma∂ − x + lower order termswhich contradicts the assumption that a = 0.Note that the constraint (3.2) implies ( L † ) k + = − ( ∂ x L k ∂ − x ) + = − ( L k ) + − (( ∂ x L k ) ∂ − x ) + which can be rewritten as B † k = − ∂ x B k ∂ − x , k odd (3.5)(taking into account that b k, = 0). Using this relation, it is straightforward to check,using the Lax equations, that the constraint (3.2) is indeed invariant under odd flows ofthe KP hierarchy: ∂ t k (cid:16) L † + ∂ x L∂ − x (cid:17) = 0 , k odd . (3.6)Therefore, the BKP hierarchy is well-defined as a subhierarchy of the KP hierarchy.The first three differential operators B k are as follows: B = ∂ x ,B = ∂ x + 6 u∂ x , u = u ,B = ∂ x + 10 u∂ x + 10 u ′ ∂ x + v∂ x . (3.7)An equivalent formulation of the hierarchy is through the zero curvature equations ∂ t l B k − ∂ t k B l + [ B k , B l ] = 0 , k, l odd . (3.8)The first equation of the BKP hierarchy follows from the zero curvature equation ∂ t B − ∂ t B + [ B , B ] = 0. The calculations yield the following system of equations for theunknown functions u, v : v ′ = 10 u t + 20 u ′′′ + 120 uu ′ v t − u t = v ′′′ − u ′′′′′ − uu ′′′ − u ′ u ′′ + 6 uv ′ − vu ′ , (3.9)Note that the variable v can be excluded by passing to the unknown function U suchthat U ′ = u . 6 .2 The wave function and the tau-function The Lax equations and the zero curvature equations are compatibility conditions of theauxiliary linear problems (2.6) for the formal wave function ψ = ψ ( x, t o , z ) = W e xz + ξ ( t o ,z ) , (3.10)where ξ ( t o , z ) = X k ≥ , odd t k z k . (3.11)As it follows from (2.2), the wave function ψ = ψ ( x, t o , z ) has the following expansion as z → ∞ : ψ ( x, t o , z ) = e xz + ξ ( t o ,z ) (cid:16) X k ≥ ξ k z − k (cid:17) . (3.12)As is proved in [3], the wave function satisfies the bilinear relation I C ∞ ψ ( x, t o , z ) ψ ( x, t ′ o , − z ) dz πiz = 1 (3.13)valid for all t o , t ′ o . For completeness, we give a sketch of proof here. By virtue of thedifferential equations (3.4), the bilinear relation is equivalent to vanishing of b m = ∂ mx ′ I C ∞ ψ ( x, t o , z ) ψ ( x ′ , t o , − z ) dz πiz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ′ = x for all m ≥ b = I C ∞ ψ ( x, t o , z ) ψ ( x, t o , − z ) dz πiz = 1 . We have: b m = I C ∞ (cid:16)X k ≥ ξ k ( x ) z − k (cid:17) ∂ mx ′ (cid:16)X l ≥ ξ l ( x ′ )( − z ) − l (cid:17) e ( x − x ′ ) z dz πiz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ′ = x = I C ∞ (cid:16)X k ≥ ξ k z − k (cid:17) ( ∂ x − z ) m (cid:16)X l ≥ ξ l ( − z ) − l (cid:17) dz πiz = X j + k + l = m ( − m + j + l mj ! ξ k ∂ jx ξ l . But the last expression is the coefficient of ( − m ∂ − m − x in the operator W ∂ − x W † : W ∂ − x W † = ∂ − x + X m ≥ ( − m b m ∂ − m − x . Since
W ∂ − x W † = ∂ − x , we conclude that b m = 0 for all m ≥ b = 1.The tau-function τ = τ ( x, t o ) of the BKP hierarchy is consistently introduced by theformula ψ = e xz + ξ ( t o ,z ) τ ( x, t o − z − ] o ) τ ( x, t o ) , (3.14)7here t o + k [ z − ] o ≡ n t + kz , t + k z , t + k z , . . . o , k ∈ Z . (3.15)The proof of the existence of the tau-function is based on the bilinear relation. Let usrepresent the wave function in the form ψ ( x, t o , z ) = e xz + ξ ( t o ,z ) w ( x, t o , z )and set t ′ o = t o − a − ] o in the bilinear relation. We have e ξ ( t o − t ′ o ,z ) = a + za − z and theresidue calculus yields w ( t o , a ) w ( t o − a − ] o , − a ) = 1 , (3.16)where we do not indicate the dependence on x for brevity. Next, we set t ′ o = t o − a − ] o − b − ] o in the bilinear relation, so that e ξ ( t o − t ′ o ,z ) = ( a + z )( b + z )( a − z )( b − z ) . In this casethe residue calculus yields w ( t o , a ) w ( t o − a − ] o − b − ] o , − a ) = w ( t o , b ) w ( t o − a − ] o − b − ] o , − b ) . (3.17)With the help of (3.16) this latter relation can be rewritten as w ( t o , a ) w ( t o − a − ] o , b ) w ( t o , b ) w ( t o − b − ] o , a ) = 1 . (3.18)Now we are going to show that (3.18) implies that there exists a function τ ( t o ) suchthat w ( t o , z ) = τ ( t o − z − ] o ) τ ( t o ) . (3.19)To see this, let us represent (3.19) in an equivalent form. Taking logarithm and z -derivative, we have from (3.19) ∂ z log w = 2 X m ≥ , odd z − m − ∂ t m log τ ( t o − z − ] o ) , or, substituting τ ( t o − z − ] o ) expressed through w ( t o , z ) and τ ( t o ) from (3.19) in theright hand side, ∂ z log w = 2 ∂ t o ( z ) log w + 2 ∂ t o ( z ) log τ, (3.20)where ∂ t o ( z ) is the differential operator ∂ t o ( z ) = X j odd z − j − ∂ t j . In fact (3.20) is equivalent to (3.19). Indeed, writing (3.20) as ( ∂ z − ∂ t o ( z )) log( wτ ) = 0,we conclude that wτ = ρ is a function of t o − z − ] o , and the normalization condition w ( t o , ∞ ) = 1 implies that ρ = τ , so we arrive at (3.19).Equation (3.19) means that Y n := res z = ∞ h z n ( ∂ z − ∂ t o ( z )) log w i = 2 ∂ log τ∂t n , z = ∞ ( z n − ) = δ n . Therefore, the existence of the tau-function will be proved if we prove that ∂ t n Y m ( t o ) = ∂ t m Y n ( t o ). Changing a → z , b → ζ in (3.18), and applying the operator ∂ z − ∂ t o ( z ) to logarithm of this equality, we rewriteit as( ∂ z − ∂ t o ( z )) log w ( t o , z ) − ( ∂ z − ∂ t o ( z )) log w ( t o − ζ − ] o , z ) = − ∂ t o ( z ) log w ( t o , ζ ) , or Y n ( t o ) − Y n ( t o − ζ − ] o ) = − ∂ t n log w ( t o , ζ ) . (3.21)Therefore, denoting F mn = ∂ t m Y n − ∂ t n Y m , we see from (3.21) that F mn ( t o ) = F mn ( t o − ζ − ] o ) . (3.22)This equality is valid identically in ζ . Expanding its right hand side in a power series, F mn ( t o − ζ − ] o ) = F mn ( t o ) − ζ − ∂ t F mn ( t o ) − ζ − ( ∂ t F mn ( t o )+2 ∂ t F mn ( t o )) + . . . , we conclude from compating of the ζ − -terms that F mn does not depend on t . From the ζ − -terms we see that it does not depend on t and so on. In this way we can concludethat it does not depend on t k for all (odd) k , i.e. F mn = 2 a mn , where a mn are someconstants such that a mn = − a nm . Therefore, we can write Y n = X m a mn t m + ∂ t n h, with some function h = h ( t o ). Then from (3.21) we have − ∂ t n log w ( t o , z ) = ∂ t n ( h ( t o ) − h ( t o − z − ] o )) + 2 X m odd a mn m z − m , or, after integration,log w ( t o , z ) = 12 h ( t o − z − ] o ) − h ( t o ) − X m odd a mn m z − m t n + ϕ ( z ) , where ϕ ( z ) is a function of z only. Substituting this into (3.18), we conclude that a mn = 0,and so ∂ t m Y n = ∂ t n Y m .Let us show how to obtain (3.14) up to a common x -independent factor in a very easyway. Apply ∂ t to (3.13) and set t ′ o = t o − a − ] o . The residue calculus yields2 a (cid:16) w ( t o , a ) w ( t o − a − ] o , − a ) − (cid:17) + 2 w ′ ( t o , a ) w ( t o − a − ] o , − a )+ ξ ( t o − a − ] o ) − ξ ( t o ) = 0 . (3.23)Using (3.16), we conclude from (3.23) that ∂ x log w ( t o , a ) = 12 (cid:16) ξ ( t o ) − ξ ( t o − a − ] o ) (cid:17) . (3.24)Now, setting ξ ( x, t o ) = − ∂ x log τ ( x, t o ) and integrating, we arrive at (3.14) up to acommon x -independent factor. 9he function u in (3.7) can be also expressed through the tau-function with the helpof the following argument. It is a matter of direct verification that the result of the actionof the operator ∂ x + 6 u∂ x − ∂ t to the wave function ψ of the form (3.12) is O ( z − ) as z → ∞ , i.e., ( ∂ x + 6 u∂ x − ∂ t ) ψ = O ( z − ) e xz + ξ ( t o ,z ) (3.25)if the conditions u = − ξ ′ , ξ ξ ′ − ξ ′′ − ξ ′ = 0 (3.26)hold true (actually we know that ( ∂ x + 6 u∂ x − ∂ t ) ψ = 0 but here we only need the weakercondition (3.25)). Since from (3.14) it follows that ξ = − ∂ x log τ, ξ = 2( ∂ x log τ ) + 2 ∂ x log τ, we have u = ∂ x log τ (3.27)and the second equality in (3.26) holds identically.The change of dependent variables from u, v to the tau-function as in (3.27) and v = 103 ∂ t ∂ x log τ + 203 ∂ x log τ + 20( ∂ x log τ ) (3.28)makes the first of the equations (3.9) trivial and the other one turns into the bilinearform [2] (cid:16) D − D D − D + 9 D D (cid:17) τ · τ = 0 , (3.29)where D i are the Hirota operators defined by the rule P ( D , D , D , . . . ) τ · τ = P ( ∂ y , ∂ y , ∂ y , . . . ) τ ( x, t + y , t + y , . . . ) τ ( x, t − y , t − y , . . . ) (cid:12)(cid:12)(cid:12) y i =0 for any polynomial P ( D , D , D , . . . ).As it follows from (3.14), the BKP hierarchy is equivalent to the following relationfor the tau-function: I C ∞ e ξ ( t o − t ′ o ,z ) τ (cid:16) x, t o − z − ] o (cid:17) τ (cid:16) x, t ′ o + 2[ z − ] o (cid:17) dz πiz = τ ( x, t o ) τ ( x, t ′ o ) (3.30)valid for all t o , t ′ o . Set t ′ o = t o − a − ] o − b − ] o − c − ] o , then e ξ ( t o − t ′ o ,z ) = ( a + z )( b + z )( c + z )( a − z )( b − z )( c − z )and the residue calculus in (3.30) gives the following equation:( a + b )( a + c )( b − c ) τ [ a ] τ [ bc ] + ( b + a )( b + c )( c − a ) τ [ b ] τ [ ac ] +( c + a )( c + b )( a − b ) τ [ c ] τ [ ab ] + ( a − b )( b − c )( c − a ) τ τ [ abc ] = 0 , (3.31)10here τ [ a ] = τ ( x, t o + 2[ a − ] o ), τ [ ab ] = τ ( x, t o + 2[ a − ] o + 2[ b − ] o ), and so on. Equation(3.31) should be valid for all a, b, c . Taking the limit c → ∞ , we get the equation τ τ [ ab ] (cid:18) a + b ∂ t log ττ [ ab ] (cid:19) = τ [ a ] τ [ b ] a − b ∂ t log τ [ b ] τ [ a ] ! (3.32)This is the equation for the tau-function of the BKP hierarchy. It should hold for all a, b . The differential equations of the hierarchy are obtained by expanding it in inversepowers of a, b . Here we compare the spaces of solutions to the BKP and KP hierarchies by an explicitembedding of the former into the latter on the level of tau-functions.We begin with the adjoint wave function: ψ † ( x, t o , z ) = ( W † ) − e − xz − ξ ( t o ,z ) = ∂ x W ∂ − x e − xz − ξ ( t o ,z ) = − z − ∂ x ψ ( x, t o , − z ) , (3.33)so that the bilinear relation (2.13) reads I C ∞ ψ ( x, t o , z ) ∂ x ψ ( x, t ′ o , − z ) dz πiz = 0 (3.34)which is a consequence of (3.13). Using (2.11), (2.12), one can write (3.33) in terms ofthe KP tau-function: e − xz τ KP ( x, ˙t + [ z − ]) τ KP ( x, ˙t ) = − z − ∂ x e − xz τ KP ( x, ˙t − [ − z − ]) τ KP ( x, ˙t ) ! or ∂ x log τ KP ( x, ˙t − [ z − ]) τ KP ( x, ˙t ) = − z − τ KP ( x, ˙t + [ − z − ]) τ KP ( x, ˙t − [ z − ]) ! , (3.35)where we use the short-hand notation ˙t = { t , , t , , . . . } . Shifting the times t o , we canrewrite this as ∂ x log τ KP ( t , − z − , t , − z − , . . . ) τ KP ( t + z − , , t + z − , , . . . ) = − z + z τ KP ( t , z − , t , z − , . . . ) τ KP ( t , − z − , t , − z − , . . . ) , or, subtracting these equalities with z and − z , ∂ x log τ KP ( x, t − z − , , t − z − , , . . . ) τ KP ( x, t + z − , , t + z − , , . . . ) = − z + 2 z τ KP ( x, t , z − , t , z − , . . . ) τ KP ( x, t , − z − , t , − z − , . . . ) . (3.36)Comparing this with the KP hierarchy in the form (2.16) at z = − z = z , we concludethat (cid:16) τ KP ( x, ˙t − [ z − ]) (cid:17) = τ KP ( x, ˙t ) τ KP ( x, ˙t − z − ] o ) (3.37)(in the second tau-function in the right hand side, the even times are equal to 0). Thisis the constraint which distinguishes solutions to the BKP hierarchy among all solutionsto the KP hierarchy. 11nother way to come to (3.37) is to notice that we have two different expressions forthe wave function ψ (one in terms of the KP tau-function τ KP and the other in terms ofthe BKP tau-function τ ) from which it follows that τ KP ( x, ˙t − [ z − ]) τ KP ( x, ˙t ) = τ ( x, t o − z − ] o ) τ ( x, t o )or, after a shift of the times t o ,log τ KP ( x, t , − z − , t , − z − , . . . ) τ KP ( x, t + z − , , t + z − , , . . . ) = log τ ( x, t o − [ z − ] o ) τ ( x, t o − [ − z − ] o ) . (3.38)The right hand side is and odd function of z , therefore, we havelog τ KP ( x, t , − z − , t , − z − , . . . ) τ KP ( x, t + z − , , t + z − , , . . . ) + log τ KP ( x, t , − z − , t , − z − , . . . ) τ KP ( x, t − z − , , t − z − , , . . . ) = 0which is (3.37).The constraint (3.37) should be valid for all values of t , t , . . . and z . Expanding itin powers of z , one can represent it as an infinite number of differential constraints thefirst of which is ( ∂ t + ∂ t ) log τ KP (cid:12)(cid:12)(cid:12) t k =0 = 0 , k ≥ . (3.39)This constraint was mentioned in [14].Let us represent (3.37) in the form τ KP ( x, ˙t − z − ] o ) τ KP ( x, ˙t ) = τ KP ( x, ˙t − [ z − ]) τ KP ( x, ˙t ) ! = τ ( x, t o − z − ] o ) τ ( x, t o ) . (3.40)It follows from here that τ ( x, t o ) = C q τ KP ( x, ˙t ) (3.41)and that τ KP ( x, ˙t ) is a full square, i.e., q τ KP ( x, ˙t ) does not have square root singularitiesin all the times. N -soliton solutions of the BKP hierarchy are obtained by imposing certain constraintson the parameters of 2 N -soliton solutions to the KP hierarchy. The tau-function of theBKP hierarchy is related to the KP tau-function as τ = √ τ KP , with “even” times t k putequal to zero and it is implied that the parameters of the KP tau-function τ KP are chosenin a special way. With this choice, τ KP is a full square, i.e., τ does not have square rootsingularities. The tau-function for one BKP soliton is the square root of a specialization of 2-solitontau-function of the KP hierarchy: τ KP (cid:12)(cid:12)(cid:12) t k =0 = 1 + α ( p − q ) w + α p − q ) w = (cid:16) α p − q ) w (cid:17) , (3.42)12here w = e ( p + q ) x + ξ ( t o ,p )+ ξ ( t o ,q ) , ξ ( t o , z ) is given by (3.11) (3.43)and α, p, q are arbitrary parameters. Therefore, the tau-function of the BKP hierarchy is τ = 1 + α p − q ) w. (3.44)It is an entire function of x .Note that the extension of the tau-function τ KP to the modified KP hierarchy (mKP)reads τ mKP n = 1 + α (cid:16) − q ( − p/q ) n + p ( − q/p ) n (cid:17) w + α p − q ) w , (3.45)where n is the integer-valued “zeroth time” (clearly, τ KP = τ mKP0 = τ mKP1 ). Then we seethat the parameters of the soliton solutions are such that τ mKP1 − n = τ mKP n , which is theconstraint necessary for the BKP hierarchy [4, 21].The bilinear identity (3.13), which in the present case has the explicit form I C ∞ e ξ ( t o ,z ) − ξ ( t ′ o ,z ) α z − p )( z − q )( z + p )( z + q ) ( p − q ) w ! α z + p )( z + q )( z − p )( z − q ) ( p − q ) w ′ ! dz πiz = (cid:18) α p − q ) w (cid:19) (cid:18) α p − q ) w ′ (cid:19) can be checked directly by the residue calculus. (Here w ′ = e ( p + q ) x + ξ ( t ′ o ,p )+ ξ ( t ′ o ,q ) .) The general KP tau-function for 2 N -soliton solution has 6 N arbitrary parameters α i , p i , q i ( i = 1 , . . . , N ) and is given by τ KP " α p , q ; α p , q ; α p , q ; α p , q ; · · · ; α N − p N − , q N − ; α N p N , q N = det ≤ i,k ≤ N δ ik + α i p i − q i p i − q k e ( p i − q i ) x + ξ ( t ,p i ) − ξ ( t ,q i ) ! . (3.46)The N -soliton tau-function of the BKP hierarchy is the square root of the τ KP specializedas τ KP " − q α p , − q ; p α q , − p ; − q α p , − q ; p α q , − p ; · · · ; − q N α N p N , − q N ; p N α N q N , − p N , (3.47)where it is implied that even times evolution is frozen ( t k = 0 for all k ≥ τ = √ τ KP is an entire function ofthe times (no square root singularities!). In this section we present the main formulas related to the CKP hierarchy with somedetails. The main references are [3, 7, 8]. 13 .1 The CKP equation and the hierarchy
The set of independent variables (“times”) is t o = { t , t , t , . . . } . Like in the BKPhierarchy, they are indexed by positive odd numbers. We set t = x + const. The mainobject is the L -operator (2.1) with the constraint L † = − L. (4.1)It is instructive to reformulate this constraint in terms of the wave operator W (2.2) suchthat L = W ∂ x W − . The constraint implies that W † W commutes with ∂ x , i.e., it is apseudo-differential operator with constant coefficients. The freedom in the definition ofthe wave operator can be fixed by demanding that W † W = 1, i.e. W † = W − . (4.2)The evolution equations (the Lax equations) and the zero curvature equations have thesame form (3.4) and (3.8) as in the BKP hierarchy. By applying the † -operation tothe evolution equations (3.4) it is not difficult to see that they are consistent with theconstraint (4.1), i.e., ∂ t k ( L + L † ) = 0 for odd k , so the CKP hierarchy is well-defined.Clearly, the differential operators B k satisfy B † k = − B k (for odd k ). In particular, B = ∂ x ,B = ∂ x + 6 u∂ x + 3 u ′ ,B = ∂ x + 10 u∂ x + 15 u ′ ∂ x + v∂ x + ( v ′ − u ′′′ ) . (4.3)where u ′ ≡ ∂ x u , u = u . Since B = ∂ x , it follows from (3.4), like in the KP hierarchy,that the evolution in the time t is simply the shift of x , i.e. the solutions depend on x + t .The first equation of the CKP hierarchy follows from the zero curvature equation ∂ t B − ∂ t B + [ B , B ] = 0 with B , B as in (4.3). The calculations yield the followingsystem of equations for the unknown functions u, v : ∂ t u = 3 v ′ − u ′′′ − uu ′ ∂ t u − ∂ t v = u ′′′′′ + 150 uu ′′′ + 180 u ′ u ′′ − v ′′′ + 6 vu ′ − uv ′ . (4.4)Note that the variable v can be excluded by passing to the unknown function U suchthat U ′ = u . Like in the KP and BKP hierarchies, the Lax equations and the zero curvature equationsare compatibility conditions of the auxiliary linear problems ∂ t k Ψ = B k Ψ , L Ψ = z Ψ (4.5)14or the formal wave function Ψ = Ψ( x, t o , z ) = W e xz + ξ ( t o ,z ) , (4.6)where z is the spectral parameter and ξ ( t o , z ) is defined in (3.11). The wave function hasthe following expansion as z → ∞ :Ψ( x, t o , z ) = e xz + ξ ( t o ,z ) (cid:16) X k ≥ ξ k z − k (cid:17) . (4.7)As is proved in [3], it satisfies the bilinear relation I C ∞ Ψ( x, t o , z )Ψ( x, t ′ o , − z ) dz πi = 0 (4.8)valid for all t o , t ′ o . Here is a sketch of proof. By virtue of the differential equations (4.5),the bilinear relation is equivalent to vanishing of b m = ∂ mx ′ I C ∞ Ψ( x, t o , z )Ψ( x ′ , t o , − z ) dz πi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ′ = x for all m ≥ b m = I C ∞ (cid:16)X k ≥ ξ k ( x ) z − k (cid:17) ∂ mx ′ (cid:16)X l ≥ ξ l ( x ′ )( − z ) − l (cid:17) e ( x − x ′ ) z dz πi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ′ = x = I C ∞ (cid:16)X k ≥ ξ k z − k (cid:17) ( ∂ x − z ) m (cid:16)X l ≥ ξ l ( − z ) − l (cid:17) dz πi = X j + k + l = m +1 ( − m + j + l mj ! ξ k ∂ jx ξ l . It remains to notice that this expression is the coefficient of ( − m ∂ − m − x in the operator W W † : W W † = 1 + X m ≥ ( − m b m ∂ − m − x . Since
W W † = 1 (see (4.2)), we conclude that b m = 0 for all m ≥ τ = τ ( x, t o ) of the CKP hierarchy is consistently introduced by theformula [7, 8] Ψ = e xz + ξ ( t o ,z ) G ( x, t o , z ) τ ( x, t o − z − ] o ) τ ( x, t o ) , (4.9)where G ( x, t o , z ) = z − ∂ t log τ ( x, t o − z − ] o ) τ ( x, t o ) ! / (4.10)(cf. (3.14), where there is no factor G ( x, t o , z )). The formula (4.9) (and the very existenceof the tau-function) is based on the bilinear relation (4.8) The proof can be found in [11].For completeness, we give it here. Let us represent the wave function in the formΨ( x, t o , z ) = e xz + ξ ( t o ,z ) w ( x, t o , z )15nd set t ′ o = t o − a − ] o in the bilinear relation. We have e ξ ( t o − t ′ o ,z ) = a + za − z . The residuecalculus in (4.8) yields w ( t o , a ) w ( t o − a − ] o , − a ) = f ( t , a ) , (4.11)where f ( t o , z ) = 1 + 12 z (cid:16) ξ ( t o ) − ξ ( t o − z − ] o ) (cid:17) (4.12)(here and below we do not indicate the x -dependence for brevity). Next, we set t ′ o = t o − a − ] o − b − ] o in the bilinear relation and the residue calculus yields a + ba − b (cid:16) aw ( t o , a ) w ( t o − a − ] o − b − ] o , − a ) − bw ( t o , b ) w ( t o − a − ] o − b − ] o , − b ) (cid:17) = a + b + 12 (cid:16) ξ ( t o ) − ξ ( t o − a − ] o − b − ] o (cid:17) . (4.13)Expressing w ( . . . , − a ), w ( . . . , − b ) through w ( . . . , a ), w ( . . . , b ) by means of the relation(4.11), we can represent this equation in the form1 a − b af ( t o − b − ] o , a ) w ( t o , a ) w ( t o − b − ] o , a ) − bf ( t o − a − ] o , b ) w ( t o , b ) w ( t o − a − ] o , b ) ! = 1 + ξ ( t o ) − ξ ( t o − a − ] o − b − ] o )2( a + b ) . (4.14)Shifting here t o → t o + 2[ b − ] o , changing the sign of b (i.e, substituting b → − b ) andusing (4.11) again after that (in the second term in the left hand side), we arrive at theequivalent equation1 a + b af ( t o , a ) w ( t o − b − ] o , a ) w ( t o , a ) − bf ( t o , b ) w ( t o − a − ] o , b ) w ( t o , b ) ! = 1 + ξ ( t o − b − ] o ) − ξ ( t o − a − ] o )2( a − b ) . (4.15)Together equations (4.14), (4.15) form the system of two equations a − b (cid:16) af ( t o − b − ] o , a ) X − − bf ( t o − a − ] o , b ) Y − (cid:17) = af ( t o , a )+ bf ( t o − a − ] o , b ) a + b a + b (cid:16) af ( t o , a ) X − bf ( t o , b ) Y (cid:17) = af ( t o , a ) − bf ( t o , b ) a − b (4.16)for the two “unknown quantities” X = w ( t o − b − ] o , a ) w ( t o , a ) , Y = w ( t o − a − ] o , b ) w ( t o , b ) . (4.17)16he next step is to multiply the two equations (4.16). After some algebra, one obtainsthe following simple relation: YX = w ( t o , a ) w ( t o − a − ] o , b ) w ( t o , b ) w ( t o − b − ] o , a ) = f ( t o , a ) f ( t o − a − ] o , b ) f ( t o , b ) f ( t o − b − ] o , a ) ! / . (4.18)In the calculations, the identity af ( t o , a ) − af ( t o − b − ] o , a ) − bf ( t o , b ) + bf ( t o − a − ] o , b ) = 0 (4.19)has been used. Introducing w ( t o , z ) = w ( t o , z ) f − / ( t o , z ), we get from it the relation w ( t o , a ) w ( t o − a − ] o , b ) w ( t o , b ) w ( t o − b − ] o , a ) = 1 (4.20)which has the same form as (3.18) for the BKP hierarchy, with the change in the notation w → w .As soon as the relation of the form (4.20) is established, the rest of the argument isthe same as for the BKP hierarchy. In the same way as in section 3.2 we can prove thatthere exists a function τ ( t o ) such that w ( t o , z ) = τ ( t o − z − ] o ) τ ( t o ) . (4.21)This function is called the tau-function of the CKP hierarchy. Finally, writing w ( t o , z ) = f / ( t o , z ) w ( t o , z ) and noting that f ( t o , z ) = 1 + O ( z − ), we see that ξ ( t o ) = − ∂ t log τ ( t o ) (4.22)and, recalling (4.12), we arrive at (4.9) with G = f / .Let us show that equation (4.9) can be obtained up to a common x -independent factorin the following easy way [7, 8]. Apply ∂ t to (4.8) and set t ′ o = t o − a − ] o . The residuecalculus yields2 a (cid:16) − w ( t o , a ) w ( t o − a − ] o , − a ) (cid:17) − aw ′ ( t o , a ) w ( t o − a − ] o , − a )+2 a (cid:16) ξ ( t o ) − ξ ( t o − a − ] o ) (cid:17) + ξ ( t o − a − ] o ) + ξ ( t o ) + ξ ′ ( t o ) − ξ ( t o ) ξ ( t o − a − ] o ) = 0 , (4.23)where prime means the x -derivative and we again do not indicate the dependence on x explicitly. Tending a → ∞ in (4.23), we get the relation2 ξ ( t o ) = ξ ( t o ) − ξ ′ ( t o ) (4.24)(it also directly follows from the constraint W W † = 1 for the dressing operator). Pluggingit back to (4.23), we can rewrite this equation in the form w ′ ( t o , a ) w ( t o − a − ] o , − a ) = af ( t o , a )( f ( t o , a ) −
1) + f ′ ( t o , a ) . (4.25)17sing (4.11), we conclude that ∂ x log w ( t o , a ) = a ( f ( t o , a ) −
1) + ∂ x log f ( t o , a )= (cid:16) ξ ( t o ) − ξ ( t o − a − ] o ) (cid:17) + ∂ x log f ( t o , a ) . (4.26)Now, setting ξ ( x, t o ) = − ∂ x log τ ( x, t o ) with some function τ ( x, t o ) and integrating, wearrive at (4.9) with G = f / up to a common multiplier which does not depend on x .The function u in (4.3) can be also expressed through the tau-function. It is a matterof direct verification that the result of the action of the operator ∂ x + 6 u∂ x + 3 u ′ − ∂ t tothe wave function Ψ of the form (4.7) is O ( z − ) as z → ∞ , i.e.,( ∂ x + 6 u∂ x + 3 u ′ − ∂ t )Ψ = O ( z − ) e xz + ξ ( t o ,z ) (4.27)if the conditions u = − ξ ′ , u ′ = ξ ξ ′ − ξ ′′ − ξ ′ (4.28)hold true. Since from (4.9) it follows that ξ = − ∂ x log τ, ξ = 2( ∂ x log τ ) + ∂ x log τ, we have u = ∂ x log τ (4.29)and the second equality in (4.28) holds identically. As it follows from (4.8), (4.9), the CKP hierarchy is equivalent to the following relationfor the tau-function: I C ∞ e ξ ( t o − t ′ o ,z ) τ (cid:16) x, t o − z − ] o (cid:17) τ (cid:16) x, t ′ o + 2[ z − ] o (cid:17) G ( x, t o , z ) G ( x, t ′ o , − z ) dz πi = 0 (4.30)valid for all t o , t ′ o . Set t ′ o = t o − a − ] o − b − ] o − c − ] o , then e ξ ( t o − t ′ o ,z ) = ( a + z )( b + z )( c + z )( a − z )( b − z )( c − z )and the residue calculus in (4.30) gives the following equation: a ( a + b )( a + c )( b − c ) τ [ a ] o τ [ bc ] o G ( − a ) G [ abc ] o ( a )+ b ( b + a )( b + c )( c − a ) τ [ b ] o τ [ ac ] o G ( − b ) G [ abc ] o ( b )+ c ( c + a )( c + b )( a − b ) τ [ c ] o τ [ ab ] o G ( − c ) G [ abc ] o ( c )+( a + b + c )( a − b )( b − c )( c − a ) τ τ [ abc ] o +( a − b )( b − c )( c − a ) (cid:16) τ [ abc ] o ∂ t τ − τ ∂ t τ [ abc ] o (cid:17) = 0 , (4.31)18here τ [ a ] o = τ ( x, t o + 2[ a − ] o ), τ [ ab ] o = τ ( x, t o + 2[ a − ] o + 2[ b − ] o ), G [ a ] o ( z ) = G ( x, t o +2[ a − ] o , z ) and so on. Equation (4.31) should be valid for all a, b, c . Let us tend c toinfinity. The highest terms proportional to c vanish identically. The terms of order c give the equation a + ba − b (cid:16) aG ( − a ) G [ ab ] o ( a ) − bG ( − b ) G [ ab ] o ( b ) (cid:17) = a + b − ∂ t log τ [ ab ] o τ ! τ τ [ ab ] o τ [ a ] o τ [ b ] o , (4.32)or, in the more detailed notation, z + z z − z "(cid:16) z − ∂ t log τ [ z z ] o τ [ z ] o (cid:17) / (cid:16) z − ∂ t log τ [ z ] o τ (cid:17) / − (cid:16) z − ∂ t log τ [ z z ] o τ [ z ] o (cid:17) / (cid:16) z − ∂ t log τ [ z ] o τ (cid:17) / = z + z − ∂ t log τ [ z z ] o τ ! τ τ [ z z ] o τ [ z ] o τ [ z ] o . (4.33)This is the equation for the tau-function of the CKP hierarchy. It should hold for all z , z . In contrast to the cases of the KP and BKP hierarchies, this equation is notbilinear. Here we give a characterization of those KP tau-functions which correspond to solutionsof the CKP hierarchy and show that the CKP tau-function is the square root of the KPone.Comparing (2.13) and (4.8), we conclude that the wave functions of the CKP and KPhierarchies are related asΨ( x, t o , z ) = e χ ( z ) Ψ KP ( x, t , , t , , . . . , z ) , Ψ( x, t o , − z ) = e − χ ( z ) Ψ † KP ( x, t , , t , , . . . , z ) . Here χ ( z ) is some function such that χ ( ∞ ) = 0, i.e.Ψ † KP ( x, t , , t , , . . . , z ) = e χ e ( z ) Ψ KP ( x, t , , t , , . . . , − z ) , (4.34)where χ e ( z ) = ( χ ( z ) + χ ( − z )) is the even part of the function χ ( z ). From (2.11), (2.12)and (4.34) it follows that the KP tau-function is the extension of a solution of the CKPhierarchy if and only if the equation τ KP (cid:16) x, t + z − , z − , t + z − , z − , . . . (cid:17) = e χ e ( z ) τ KP (cid:16) x, t + z − , − z − , t + z − , − z − , . . . (cid:17) (4.35)holds identically for all z, x, t , t , t , . . . . Shifting the odd times, we can rewrite thiscondition aslog τ KP (cid:16) x, t , z − , t , z − , . . . (cid:17) − log τ KP (cid:16) x, t , − z − , t , − z − , . . . (cid:17) = 2 χ e ( z ) . (4.36)19omparing the coefficients at z − of the expansions of the left and right hand sides of(4.36) and passing to an equivalent tau-function if necessary, we get the condition ∂ t log τ KP (cid:12)(cid:12)(cid:12) t k =0 = 0 , k ≥ . (4.37)It is the CKP counterpart of the condition (3.39) specific for the BKP hierarchy. In [11]it is proved that this is also a sufficient condition that the tau-function τ KP generatesa solution to the CKP hierarchy. The proof is based on the technique developed in[17, 18, 19].Let us introduce the auxiliary wave function ψ in the same way as for the BKPhierarchy: ψ = e xz + ξ ( t o ,z ) τ ( x, t o − z − ] o ) τ ( x, t o ) , (4.38)then the wave function (4.9) isΨ = z − / q ∂ x log ψ · ψ = (2 z ) − / q ∂ x ψ . (4.39)We will prove that the CKP and KP tau-functions are related as τ = q τ KP . (4.40)The tau-function τ ( x, t o ) has square root singularities in x . However, it appears thatthe expression ∂ x ψ is a full square for all z , i.e., q ∂ x ψ and, therefore, Ψ is an entirefunction of x and t o (no square root singularities!). This is similar to the fact that forthe BKP hierarchy the function τ KP is a full square. The bilinear identity (4.8) acquiresthe form I C ∞ q ∂ x ψ ( x, t o , z ) q ∂ x ψ ( x, t ′ o , − z ) dz πiz = 0 . (4.41)In order to prove that τ = √ τ KP (see [9]) we compare two expressions for the wavefunction Ψ of the CKP hierarchy. The first one is in terms of the KP tau-function,Ψ KP = e xz + ξ ( t o ,z ) τ KP ( x, t − z − , − z − , t − z − , − z − , . . . ) τ KP ( t , , t , , . . . ) , (4.42)and the second one (4.9) is in terms of the CKP tau-function τ . Comparing (4.39) and(4.42), we get the equation12 z ∂ x e xz τ (cid:16) x, t o − z − ] o (cid:17) τ ( x, t o ) = e xz τ KP ( x, ˙t − [ z − ]) τ KP ( x, ˙t ) ! , (4.43)where we again use the short-hand notation ˙t = { t , , t , , . . . } . Then we get that (4.43)is equivalent to the differential equation ∂ x ϕ = − zϕ, where ϕ = τ (cid:16) x, t o − z − ] o (cid:17) τ ( x, t o ) − τ KP (cid:16) x, ˙t − z − ] o (cid:17) τ KP ( x, ˙t ) . (4.44)In (4.44), ˙t − z − ] o = { t − z − , , t − z − , , . . . } . The general solution of thedifferential equation is ϕ = c ( z, t , t , . . . ) e − x + t ) z but from (4.44) it follows that ϕ is20xpanded in a power series as ϕ = ϕ z − + ϕ z − + . . . as z → ∞ , and this means that c must be equal to 0. Therefore, ϕ = 0, i.e. τ (cid:16) x, t o − z − ] o (cid:17) τ ( x, t o ) = τ KP (cid:16) x, ˙t − z − ] o (cid:17) τ KP ( x, ˙t ) (4.45)for all z . This is an identity on solutions to the KP/CKP hierarchies. It follows from(4.45) that τ KP = const · τ , i.e. τ ( x, t o ) = q τ KP ( x, ˙t ) is a tau-function of the CKPhierarchy. We start from the simplest example of one-soliton solution. The tau-function for oneCKP soliton is the square root of a specialization of 2-soliton tau-function of the KPhierarchy. The latter has the form τ KP ( x, t ) = 1 + α exp (cid:16) ( p + q ) x + X k ≥ ( p k − ( − q ) k ) t k (cid:17) + α exp (cid:16) ( p + q ) x + X k ≥ ( q k − ( − p ) k ) t k (cid:17) − α ( p − q ) pq exp (cid:16) p + q ) x + X k ≥ ( p k + q k − ( − p ) k − ( − q ) k ) t k (cid:17) , (4.46)where α, p, q are arbitrary parameters. When all even times are put equal to zero, t = t = . . . = 0, this expression simplifies: τ KP = 1 + 2 αw − α ( p − q ) pq w , (4.47)where w = e ( p + q ) x + ξ ( t o ,p )+ ξ ( t o ,q ) . (4.48)We see that the tau-function τ = √ τ KP has two square root singularities at the points w = w ± , w ± = ± √ pqα ( √ p ∓√ q ) . A direct calculation shows that ∂ x ψ (where ψ is given by (4.38)) for the solution(4.47) is a full square for all z : ∂ x ψ z = e xz +2 ξ ( t o ,z ) α (2 z − p − q )( z + p )( z + q ) w − α ( p − q ) pq ( z − p )( z − q )( z + p )( z + q ) w αw − α ( p − q ) pq w (cid:16) Ψ KP ( x, t , , t , , . . . , z ) (cid:17) , where Ψ KP is constructed from the KP tau-function τ KP (4.46) according to formula(2.11). Hence (2 z ) − / q ∂ x ψ and, therefore, Ψ does not have square root singularitiesin the variable z . The bilinear identity (4.8), which in the present case has the explicitform I C ∞ e ξ ( t o ,z ) − ξ ( t ′ o ,z ) α (2 z − p − q )( z + p )( z + q ) w − α ( p − q ) pq ( z − p )( z − q )( z + p )( z + q ) w ! × α (2 z − p − q )( z − p )( z − q ) w ′ − α ( p − q ) pq ( z + p )( z + q )( z − p )( z − q ) w ′ ! dz πi = 0 , can be checked directly by the residue calculus. (Here w ′ = e ( p + q ) x + ξ ( t ′ o ,p )+ ξ ( t ′ o ,q ) .) The general KP tau-function for M -soliton solution has 3 M arbitrary parameters α i , p i , q i ( i = 1 , . . . , M ) and is given by equation (3.46). We denote this tau-function as τ KP " α p , q ; α p , q ; α p , q ; α p , q ; · · · ; α N − p M − , q M − ; α N p M , q M . The parameters p i , q i are sometimes called momenta of solitons.The multi-soliton tau-function of the CKP hierarchy is the square root of the τ KP specialized as τ KP " α p , − p ; α p , − q ; α q , − p ; α p , − q ; α q , − p ; · · · ; α N p N , − q N ; α N q N , − p N , (4.49)where it is implied that even times evolution is suppressed ( t k = 0 for all k ≥ N + 2. If α = 0, the tau-function (3.47)reduces to τ KP " α p , − q ; α q , − p ; α p , − q ; α q , − p ; · · · ; α N p N , − q N ; α N q N , − p N , (4.50)and it is this tau-function which is usually called the N -soliton CKP tau-function in theliterature (see, e.g. [3]). It is a specialization of 2 N -soliton KP tau-function and has 3 N free parameters. The dispersionless limit is the limit ¯ h → t k → t k / ¯ h, τ = exp( F/ ¯ h ) . p ( z ) − p ( z ) z − z = p ( z ) + p ( z ) z + z e D o ( z ) D o ( z ) F , (5.1)where p ( z ) = z − ∂ t D o ( z ) F (5.2)and D o ( z ) is the differential operator D o ( z ) = X k ≥ , odd z − k k ∂ t k . (5.3)The (odd) function p ( z ) has the expansion p ( z ) = z − uz + X k ≥ , odd u k z − k , (5.4)where u = 2 ∂ t F. (5.5)Equation (5.1) is the generating equation for the dispersionless BKP (dBKP) hierarchyin the Hirota form [22, 23, 24]. It is remarkable that the dispersionless limit of the CKPequation (4.33) is the same, so the dispersionless limits of the BKP and CKP hierarchiescoincide.Let us show how to represent the dispersionless hierarchy in the Lax form. Takinglogarithm of equation (5.1), differentiating with respect to t and using the definition(5.4), we obtain the equation2 D o ( z ) p ( z ) = ∂ t log p ( z ) + p ( z ) p ( z ) − p ( z ) (5.6)from which it follows that D o ( z ) p ( z ) = D o ( z ) p ( z ) (this follows also from the definition(5.4)). Tending z → ∞ , we get ∂ t p ( z ) = − D o ( z ) u. (5.7)The next step is to rewrite equation (3.11) in terms of the function z ( p ), inverse to the p ( z ) (like p ( z ), it is an odd function with the Laurent series of the form z ( p ) = p + O ( p − )).Using the relation ∂ t k p ( z ) = − ∂ t k z ( p ) ∂ p z ( p ) , k ≥ , (5.8)we get, after simple transformations:2 D o ( z ) z ( p ) = ( z ( p ) , log p ( z ) − pp ( z ) + p ) , (5.9)where { f, g } := ∂f∂t ∂g∂p − ∂g∂t ∂f∂p (5.10)23s the Poisson bracket. This is the generating Lax equation for the dBKP hierarchy, z ( p )being the Lax function (the dispersionless limit of the Lax operator (3.1)). Expandingequation (5.9) in powers of z , one obtains the hierarchy of Lax equations through theFaber polynomials B k ( p ) introduced by the expansion − log p ( z ) − pz = X k ≥ z − k k B k ( p ) . (5.11)For example, B ( p ) = p . It is easy to see that B k ( p ) = (cid:16) z k ( p ) (cid:17) ≥ , (5.12)where ( . . . ) ≥ is the polynomial part of the Laurent series in p (containing only non-negative powers of the variable p ). The fact that p ( z ) is an odd function implies that B k ( − p ) = ( − k B k ( p ) and we have the expansionlog p ( z ) + pp ( z ) − p = 2 X k ≥ , odd z − k k B k ( p ) . (5.13)The Lax equations are of the form ∂ t k z ( p ) = n B k ( p ) , z ( p ) o = n ( z k ( p )) ≥ , z ( p ) o , k odd . (5.14) Acknowledgments
The author thanks V. Akhmedova, I. Krichever, S. Natanzon and D. Rudneva for dis-cussions. This work was supported by the Russian Science Foundation under grant19-11-00275.
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