Quasi-classical -dressing approach to the weakly dispersive KP hierarchy

Transcription

1 arxiv:nlin/ v3 [nlin.si] 23 Jul 2003 Quasi-classical -dressing approach to the weakly dispersive KP hierarchy Boris Konopelchenko and Antonio Moro Dipartimento di Fisica dell Università di Lecce and INFN, Sezione di Lecce, I Lecce, Italy Abstract Recently proposed quasi-classical -dressing method provides a systematic approach to study the weakly dispersive limit of integrable systems. We apply the quasi-classical -dressing method to describe dispersive corrections of any order. We show how calculate the problems at any order for a rather general class of integrable systems, presenting explicit results for the KP hierarchy case. We demonstrate the stability of the method at each order. We construct an infinite set of commuting flows at first order which allow a description analogous to the zero order purely dispersionless) case, highlighting a Whitham type structure. Obstacles for the construction of the higher order dispersive corrections are also discussed. PACS numbers: 02.30Ik, 02.30Jr 1 Introduction During the last decade a great interest has been focused on the dispersionless limit of integrable nonlinear dispersive systems. They arise in various contexts of physics and mathematical physics: topological field theory and strings, matrix models, interface dynamics, nonlinear optics, conformal mappings theory[1]-[19]. Several methods and approaches have been used in order Supported in part by the COFIN PRIN Sintesi konopel@le.infn.it antonio.moro@le.infn.it 1

2 to study the features of this type of systems, like the quasi-classical Lax pair representation with its close relationship with the Whitham universal hierarchy, highlighting a symplectic structure[4]-[7]; the hodograph transformations to calculate particular and interesting classes of exact solutions rarefaction and shock waves)[9]; the quasi-classical version of the inverse scattering method allowing to analyze various 1+1-dimensional systems[10]. Moreover, the recent formulation of the quasi-classical -dressing method[20]-[22] introduces a more general and systematic approach to multidimensional integrable systems with weak dispersion, preserving the power of the standard method. It allows to build integrable systems and solutions simultaneously; that is a non trivial fact, because of the difficulty to obtain exact explicit solutions[23]. Besides, new interesting interrelations have been revealed, like an intriguing connection with the quasi-conformal mapping theory, a strong similarity with the theory of semi-classical approximation to quantum mechanics[24] and geometric asymptotics methods to calculate wave corrections to geometrical optics[25]. While the method has been rather widely discussed for purely dispersionless limit of some famous integrable hierarchies KP, mkp, 2DTL)[20, 21, 22, 26, 27] the study of the dispersive corrections is open yet. The latter is an interesting and challenging question in the context where one would like to be able to investigate the properties of the full dispersive system through an approximation theory. A main goal of the present paper is to extend the quasi-classical -dressing method to higher order dispersive corrections, starting an investigation of the general properties at the dispersive orders of an integrable hierarchy. We will show that the quasi-classical -dressing method works as effectively as in the pure dispersionless case, and the first order admits a parallel construction to leading order case. We will make all calculations for the weakly dispersive KP hierarchy as illustrative example. In this paper we don t take care of the problem to find bounded solutions. Indeed, by an asymptotic expansion secular terms are appearing that must be suppressed by suitable conditions on the corrective equations[28]. We briefly present the standard -method in the Sec.2 and the quasiclassical -method in the Sec.3. In the Sec.4, we calculate the quasi-classical -problems at any order in the dispersive parameter, giving the explicit formulas at fourth order. In the Sec.5 we recall the method at the first order[20], and prove a theorem allowing us to build an infinite set of flows reproducing the first corrections to the dkp equation. In the Sec.6 we present the generalization of the -dressing method for higher orders. The Sec.7 contains some 2

3 considerations about the possibility to describe the first order introducing a certain generalization of the Whitham hierarchy. Finally, we present some concluding remarks. 2 The standard -dressing method. The standard -dressing method is a powerful procedure allowing to construct multidimensional integrable hierarchies of equations and their solutions. In this section we outline the main ideas of the method[29, 30]. The -dressing method is based on the nonlocal -problem χz, z; t) z = C dµ d µ χµ, µ; t) gµ, t) R 0 µ, µ; z, z) g 1 z, t), 1) where z is the spectral parameter and z its complex conjugate, t = t 1, t 2, t 3,...) the vector of the times, χz, z; t) a complex-valued function on the complex plane C and R 0 µ, µ; z, z) is the -data. The -equation 1) encodes all informations about integrable hierarchies. It is assumed that the problem 1) is uniquely solvable. Usually, one consider its solutions with canonical normalization, i.e. χ 1 + χ 1 z + χ , 2) z2 as z. The particular form of the function gz, t) sets an integrable hierarchy. Specifically, in order to describe the KP hierarchy one has to take gz, t) = e S 0z,t), S 0 z, t) = z k t k. Following a standard procedure[30], it is possible to construct an infinite set of linear operators M i, providing an infinite set of linear equations k=1 M i χ = 0, 3) which are automatically compatible. Equations 3) provide us with the corresponding integrable hierarchy. For example, the pair of operators Lax pair) 3

4 with the notations M 2 = y 2 ux, y, t), x2 M 3 = t 3 x u x 3 2 u x x u y, t 1 = x; t 2 = y; t 3 = t; u ti = u x ; x 1 = dx, t i provides us with the KP equation sometimes called dispersionfull KP) 2 u t x = 1 3 u x 4 x + 3 ) 3 2 u u u x 4 y. 4) 2 -dressing method allow us to construct a wide class of exact solutions of the KP equation and the whole KP hierarchy. 3 Quasi-classical -dressing method. 3.1 Dispersionless limit. In order to illustrate the quasi-classical formalism, we, following Ref.[20], introduce slow variables t i = T i ǫ, ti = ǫ Ti and assume that gz, T; ǫ) = e S 0 z,t) ǫ, where ǫ is a small dispersive parameter. We are looking for solutions of the -problem in the form χ z, z; T ) ) Sz, z; T) = χz, z; T; ǫ) exp, 5) ǫ ǫ where χz, z; T; ǫ) = n=0 ϕ nz, z; T)ǫ n is an asymptotic expansion. The function S has the following expansion S S 1 z + S 2 z + S , z. 6) 2 z3 4

5 Let us consider the -kernel of the following, quite general form: R 0 µ, µ; z, z) = i 2 1) k ǫ k 1 δ k) µ z)γ k z, z). 7) k=0 The derivative δ-functions δ k,n) are defined, in the standard way, as the distributions such that 1 2i C and δ k) := δ k,0). Introducing one, in the KP case, has dµ d µ δ k,n) µ z)fµ, µ) = 1) k+n k+n fz, z) z k z n, 8) Sz, z; T) = S 0 z; T) + Sz, z; T), 9) S = z k T k + k=1 j=1 S j, z. 10) zj A straightforward calculation for the problem 1) with the kernel 7) gives the dispersionless classical) limit of -problem S z = W z, z, S ), 11) z where W = k=0 1) k S z ) k Γ k. 12) In this limit the role of the nonlocal linear -problem in equation 1) is held by the nonlinear -equation 11), that is a local nonlinear equation of Hamilton-Jacobi type. The latter encodes all informations about the integrable systems, like in the dispersionfull case. Nevertheless, there are substantial technical differences in the dressing procedure. In particular, the quasi-classical -method is based crucially on the Beltrami equation s properties. We present them in the next subsection. 5

6 3.2 The Beltrami equation BE). The Beltrami equation BE) has the form Ψ = Ωz, z) Ψ z z, 13) where z C. Under certain conditions, it has the following properties see e.g.[32]): 1. If Ω satisfies Ω < k < 1, then the only solution of BE such that Ψ z is locally L p for some p > 2, and such that Ψ vanishes at some point of the extended plane C is Ψ 0 Vekua s theorem). 2. If Ψ 1, Ψ 2,...,Ψ N are solutions of BE, a differentiable function fψ 1, Ψ 2,..., Ψ N ) with arbitrary N is a solution of BE too. 3. The solutions of the BE under the previous conditions give quasiconformal maps with the complex dilatation Ωz, z)[33]. It is assumed, in all our further constructions, that these properties of BE are satisfied. 3.3 Zero order. The zero order case has been widely discussed, for different integrable hierarchies in various papers[20, 21, 22, 26, 27]. Here, we outline the procedure for the KP hierarchy. The first crucial observation is that the symmetries of the -problem at leading order in ǫ, for any time T j, i.e. δs = S T j δt j, are given by the BE ) ) S = W S. 14) z T j z T j We assume that the complex dilatation W has good properties required for the application of the Vekua s theorem. From 10) it follows that S = z j + 1 S S , z. 15) T j z T j z 2 T j Using the BE properties, one gets the following equations in time variables [20, 21] 6

7 S y where the notation is adopted and S t ) 2 S u 0 x, y, t) = 0, 16) x ) 3 S 3 x 2 u S 0 x V 0x, y, t) = 0, 17) T 1 = x; T 2 = y; T 3 = t, 18) u 0 = 2 S 1 x ; V 0 x = 3 u 0 4 y. 19) Indeed, due to the property 2 of BE, the left hand sides of equations 16) and 17) are solutions of BE and since they vanish at z, then, by virtue of the Vekua s theorem, they vanish identically on whole complex plane. Equations 16) and 17) are automatically compatible and by expansion in 1/z-power, according to the standard -dressing procedure, one obtains the well known dispersionless KP dkp) equation or Zabolotskaya-Khokhlov equation ) 2 u 0 t x = 3 ) u 0 u u 0 2 x x 4 y. 20) 2 Usually the dkp equation and its dispersive corrections are obtained, by a slow times limit, from dispersionfull KP equation 4), inserting the following asymptotic expansion in terms of a small dispersive parameter ) T u = ǫ u n T)ǫ n. 21) n=0 We stress that the expansion 21) is a formal one since it implies secular terms are arising from. In order to ensuring the uniform validity of the expansion, additive conditions associated to the corrective equations to 16) and 17) are necessary. Actually, as it is well known, equations 16) and 17) are the first ones of an infinite set of equations in all time variables T n, with n N \ {0}, 7

8 reproducing the whole KP hierarchy. Assuming W z, z, T) = θr z )V z, z, S ), z with r > 0, the function S is an analytic function outside the circle D = {z C z < r}. That is in agreement with the asymptotic behavior 10). At last, the dkp hierarchy can be written in compact form as follows where z C \ D, and Taking into account that S T n z, T) = Ω n pz, T), T); n 1, 22) p := S x. 23) p = z + j=1 S T n = z n + O 1 z j S j x, 24) ) 1 ; z z, 25) then Ω n p, T) Z n 0 p, T)) + = O 1 z ) ; z, 26) where Z 0 denotes the expansion for z obtained by inversion of equation 24), and the symbol ) + means the polynomial part of the expansion. The left hand sides of equations 26) are solutions of BE, so they vanish identically[22], i.e. Ω n p, T) = Z n 0 p, T)) +. 27) Then, Ω n can be connected to a suitable expansion series in terms of the p variable. Assuming Z 0 p, T) = p + j=1 a j T) p j, 28) 8

9 one has Ω 1 = p; Ω 2 = p 2 + 2a 1 ; Ω 3 = p 3 + 3a 1 p + 3a 2, 29) that reproduces equations 16) and 17), by identifications u 0 = 2a 1 ; V 0 = 3a 2. 30) The set of flows Ω n represents a set of quasi-conformal maps for which the dispersionless integrable hierarchy describes a class of integrable deformations[21]. 4 Quasi-classical -dressing method at any order. In this section we calculate the corrections at any order for the kernel 7). R 0 µ, µ; z, z) = i 2 1) k ǫ k 1 δ k) µ z)γ k z, z). k=0 Let us observe that the independence of Γ k,n on the integration variables µ and µ, do not reduce the generality of the kernel 7). Indeed, by the delta function properties, it provides the same result, up to a redefinition, as for the kernels where Γ k,n depends on integration variables µ and µ. Introducing the expansion 5) in equation 1), it is not too difficult obtain, in direct way, the quasiclassical -problem at any order introducing the Faà de Bruno polynomials[31] defined as follows h n [gx)] = x h n 1 [gx)] + h[gx)]h n 1 [gx)], 31) with h 0 [gx)] = 1 and h 1 [gx)] = h[gx)] = ) x gx), x. Let us note x that h n [gx)] = h n [gx)] x g, x 2) g,..., x n) g, in other words, it depends on the derivatives of g until n th order. In our case we consider h l [ k log Sz, z)], 32) 9

10 for some k N. Observing that we denote l h l [ k log Sz, z)] = C l,l S l, 33) l =1 H s,l = C s,l ), 34) l! k l where k is an arbitrary integer larger than l. We note that the quantity 34), on the whole, do not depend on k. Now, we introduce the operator Ŵ p,q W p,q = In particular, we have Ŵ p,q = W p,q z p, ) k 1) k H p k p,k q Γ k,n. k=0 W = Ŵ0,0 = H k,k = S z ) k. 1) k H k,k Γ k,n, k=0 So, we have all ingredients to write the quasiclassical -problem at n th order S z, z) = W z, z, S ), z z 35) ϕ 0 z = W ϕ 0 z W 2 S z ϕ 0, ) ϕ n z = W ϕ n z W 2 S z ϕ n+1 2 n + Ŵ p,q ϕ n q+1, 37) 10 q=1

11 where W i) = i ξ iw z, z, ξ). It is interesting to observe that equation 35) is a Hamilton-Jacobi type equation, the first order correction is a homogeneous linear equation, while the next orders are linear, but nonhomogeneous ones. In agreement with the normalization condition 2) we have ϕ ϕ 0,1 z + ϕ 0,2 z 2 + ϕ 0,3 z , ϕ 1 ϕ 1,1 + ϕ 1,2 + ϕ 1,3 +..., z z 2 z 3... ϕ n ϕ n,1 + ϕ n,2 + ϕ n,3 +...; n N \ {0}. z z 2 z 3 The explicit problem at fourth order, i.e. n = 3 is where S = W z, z, S ), 38) z z D 0 ϕ 0 = 0, 39) D 0 ϕ 1 = D 1 ϕ 0, 40) D 0 ϕ 2 = D 1 ϕ 1 + D 2 ϕ 0, 41) D 0 ϕ 3 = D 1 ϕ 2 + D 2 ϕ 1 + D 3 ϕ 0, 42) 11

12 D 0 = z W z 1 2 W 2 S z 2, D 1 = 1 2 W 2 D 2 = 1 6 W 3 z W 2 S z 2 z W 3 S z W 4) z W 4) 2 S z 2 2 z W 4) 3 S z W 5) W 4) 4 S z W 5) 2 S z 2 3 S z W 6) D 3 = 1 4) 4 W 24 z W 5) 2 S 3 z 2 z W 5) 4 S z W 6) 2 S z 2 3 S z W 7) W 5) 5 S z W 6) + 1 ) 384 W 8) 2 4 S. z 2 3 S z 3 ) 2 3 S, z 2 ) 2 2 S, z 2 ) ) 2 2 S z W 5) 3 S z W 6) ) ) 2 3 S z 2 z + ) ) 2 2 S z + z 2 ) W 6) 2 S z 2 4 S z W 7) 2 S z 2 2 z 2 + ) 2 3 S z 3 + We stress the strong analogy between equations38)-42) and the Hamilton- Jacobi and higher transport equations arising in the semi-classical approximation[24] and theory of geometric asymptotics[25]. Anyway, the main difference with the cited approaches is that in our case the role of the phase function is played by the function S that is a complex valued one rather than a real valued one. 5 First order contribution. The BE s properties are very useful for the study of higher order corrections too[20]. This analysis has been initiated in the paper[20], which results are presented here for completeness. Fixed an arbitrary solution Sz, z, T) of equation 38), let ϕ 0 and Lϕ 0 be two solutions of equation 39), where L is a suitable linear operator depending on time variables. The ratio Lϕ 0 /ϕ 0 12

13 satisfies the BE z ) Lϕ0 ϕ 0 Then, choosing L such that = W z Lϕ0 as a result of the Vekua s theorem, one gets ϕ 0 ). 43) Lϕ 0 0; z 44) Lϕ 0 = 0; z C. 45) In particular, for the first equation of the KP hierarchy we have L 1 ϕ 0 L 2 ϕ 0 y 2 S x x 2 S [ S t 3 x ) x u 1x, y, t) 2 ) ) u 0 x 3 S 2 S x 3 4 ϕ 0 = 0, 46) x 3 S 2 2 x u 1+ ] u 0 x V 1 ϕ 0 = 0, 47) where u 1 = 2 ϕ 0,1 x ; V 1 x = 3 u 1 4 y, 48) are extracted from the condition 44) and u 0 is an arbitrary solution of equation 20). Setting to zero the 1/z-power expansion s coefficients in 46) and 47), we find the first dispersive correction to dkp equation 2 u 1 t x = x u 0u 2 1 ) u 1 y 2. 49) Let us note that u 1 satisfies the equation defining the symmetries of dkp equation 20). Indeed, considering equation 20) for u 0 +δu 0, one, obviously, gets the equation 2 δu) t x = S 0 δu), 50) 13

14 where 2 2 S 0 ) = 3 2 x 2u 0 ) y2 ), 51) that coincide with 49). Now we will present some new results concerning the first order corrections. Theorem 1 Let A n and C n be the differentiable functions defined by ) d Z n A n = A n p, T) = 0 ), dp + C n = C n p, x p, T) = 1 da n p, T), 2 dx where Z 0 is given by 28), B n = B n p, T) an arbitrary differentiable function, ϕ 0 some solution of equation 39) and the linear operator L n) is given by L n) = A n T n x B n C n. 52) Then L n) ϕ 0 is also the solution of equation 39). Proof. By the use of equation 38) and p = S z, T) it s immediate to x verify that A n z B n z C n z = W A n p = W B n p = W C n p S z x = W A n z, 53) S z x = W B n z, 54) x + W C n 2 S x p) x + W C n 2 x p) S z z S x ) 2.55) Differentiating equation 22) with respect to z, we get ) ) S S = A n. 56) z T n z x 14

15 Moreover, the definition of the function C n implies that C n = 1 da n 2 dx = 1 A n p 2 p x + 1 A n 2 x, 57) C n x p) = 1 A n 2 p. 58) Calculating D 0 L n) ϕ 0 ), where D0 is given in equation 43), and exploiting equations 38),39) and the set of equalities 53)-56),58), one finds D 0 L n) ϕ 0 ) = 0. 59) This completes the proof. Based on the theorem 1), we choose, in particular, B n = ηp, T) d ) Zn 0 ) dp where η is a series defined by ηp, T) = j=1 + 60) b j T) p j. 61) A choice of the coefficients in equation 28) and 61) in such way that Lϕ 0 0 for z, together with the previous arguments on the BE, gives us the following set of linear problems in time variables L n) ϕ 0 = 0. 62) These equations can be rearranged by analogy with the leading order case, in the form log ϕ 0 T n z, T) = Λ n qz, T), pz, T), x pz, T), T); z C \ D, 63) where 15

16 Λ n = L 1 Taking n = 1, 2, 3 L 1 d Z n 0 ) dp := q + ηp, T), q := log ϕ 0. x + 1 d 2 dx d Z0 n) ), 64) dp + Λ 1 = q, Λ 2 = 2pq + 2b 1 + x p, Λ 3 = 3p 2 q + 3a 1 q + 3b 1 p + 3p x p xa 1 + 3b 2, we reproduce equations 46) and 47) and, consequently, the first order dispersive correction to the dkp equation by the identifications u 0 = 2a 1 ; V 0 = 3a 2 ; u 1 = 2b 1 ; V 1 = 3b 2. 6 Higher order contributions. Unlike the first order, the higher order corrections are characterized by nonhomogeneous equations. Here we will show how the -dressing procedure works in these cases, discussing explicitly the KP equation. Let us consider a set of functions ϕ 0, ϕ 1,...,ϕ n satisfying the problem at n+1) th order and n+1 linear operators in time variables K 0), K 1),...,K n) such that the quantity n m=0 Km) ϕ n m satisfies equation 39). Then the ratio n m=0 Km) ϕ n m 65) ϕ 0 is a solution of BE with complex dilatation W. 16

17 Using the same arguments as in the first order case, choosing K j), j = 0,..., n, in such a way that n m=0 Km) ϕ n m 0 for z, we get the linear equations n K m) ϕ n m = 0; z C. 66) m=0 For instance, the second order dispersive corrections to dkp equation are associated with the following pair K 0) 1 ϕ 1 + K 1) 1 ϕ 0 = 0, 67) K 0) 2 ϕ 1 + K 1) 2 ϕ 0 = 0, 68) where K 0) 1 = L 1 and K 0) 2 = L 2 are given by 46) and 47), and K 1) 1 = 2 K 1) 2 = 3 S x x u 2, 69) 2 2 x 3 2 S 2 x + 3 ) 2 2 u 1 x 3 S x 3 S 3 2 x u u 1 4 x V 2, 70) u 2 = 2 ϕ 1,1 x ; V 2 x = 3 u 2 4 y, 71) are deducted similarly to 48). Equations 67) and 68) are compatible if and only if u 2 satisfies the second order dispersive correction to dkp equation 2 u 2 t x = x u 0u 2 2 ) u 2 y u x u x. 72) 4 The third order correction corresponds to the compatibility condition for the pair of the following linear problems K 0) 1 ϕ 2 + K 1) 1 ϕ 1 + K 2) 1 ϕ 0 = 0, 73) K 0) 2 ϕ 2 + K 1) 2 ϕ 1 + K 2) 2 ϕ 0 = 0, 74) 17

18 where K 2) 1 = u 3, 75) K 2) 2 = 3 x u S 3 x 3 2 u 2 x 3 u 2 4 x V 3, 76) with the definitions The equation for u 3 is of the form u 3 = 2 ϕ 2,1 x ; V 3 x = 3 u 3 4 y. 77) u 3 t x = 3 2 x u 0u 2 3 ) x u 1u 2 2 ) u 3 y u x. 78) 4 A simple observation from equations 67) and 68) allows us to rewrite them in a compact form, suggesting the generalization to any order. In order to realize that, we introduce the following operator { k θs k [S; θ] = e e θs if k 0 x k 79) 0 if k < 0 Noting that for k 0 d r dθ r k [S; θ] = [ [ ] ] k... x, S,..., S k the transport equations can be written as follows r brackets 80) ϕ n T k = + 1 d r r! dθ r k [S; 0]ϕ n k+r+1 + k n k 2 d r u j+r 2 dθ r k 2 [S; 0]ϕ n j + j=0 r=0 n ) 3 u j 4 x + V j+1 k 3 [S; 0]ϕ n j, 81) k 1 r=0 j=0 for k = 1, 2, 3. It would be nice to get a similar formula for higher times too. 18

19 At last, as we noted above, a solution u 1 of the first dispersive equation is defined up to a symmetry of the dkp equation. Using this freedom one can fix a gauge putting u 1 = 0. Equations for higher corrections imply that the gauge u n = 0 for each odd n is an admissible one. In such a gauge the even order corrective equations take the form 2 u n ) t x = S 0 u n ) + f n, n even. 82) That is a nonhomogeneous equation with the corresponding homogeneous one given by the symmetry equation. By virtue of the Fredholm s theorem, the nonhomogeneous term must be orthogonal to the solutions of homogeneous equation, i.e. δup)f n P)dP = 0, P = x, y, t,...). 83) One can check that for regular solutions of the KP equation, the condition 83) is satisfied. Quite different situation takes place for the singular sector of the dkp equation. For instance, for breaking waves solutions for which δu 0 u 0 x, the condition 83) breaks. So, there are obstacles to construct the higher order quasi-classical corrections for dkp equation originated from its own singular sector. Such an obstacle is analogous to a typical one appearing in the construction of global asymptotics in the semi-classical approximation to the quantum mechanics[24] and in the study of the wave corrections to geometrical optics[25], because of existence of caustics. In the dispersionless KdV case this kind of problem induces a stratification of the affine space of times providing a classification of the singularities[34]. The problem of obstacles for the construction of the higher order corrections to dkp equation will be considered elsewhere. 7 A Whitham type structure. It is well known that the dispersionless limit of some integrable hierarchies admits a symplectic structure[4]-[7]. In particular, one can see that starting with the function Sz, T) outside the circle D it is possible to introduce a 2-form closed ω 0 in terms of the p-variable such that ω 0 = dω n p, T) dt n = dz 0 p, T) d M 0 p, T), 84) 19

20 where the sum over index n is assumed, while the last member contains the pair of Darboux coordinates Z 0 and M 0 p, T) = M 0 Z 0 p, T), T) = M 0 z, T) and the function M 0 z, T) := S z, T) 85) z is usually called Orlov s function[35]. From 84), it follows that 1 = {Z 0, M 0 } p,x, 86) Z 0 = {Ω n, Z 0 } T p,x, 87) n M } 0 = {Ω n, M 0. 88) T n p,x where the Poisson bracket is defined as follows {fα, β), gα, β)} α,β = f g α β g f α β. 89) Equation 86) is usually referred to as string equation[6] and equation 87) is the dispersionless Lax pair. Moreover, equation 84) implies the zero curvature condition ω 0 ω 0 = 0. 90) That is another way to write the compatibility condition between equations 87) or 88) ; indeed, expanding the total differentials, one gets from equation 90) the set of equations Ω n Ω m + {Ω n, Ω m } T m T p,x = 0, 91) n that are the universal Whitham hierarchy equations[4]. In particular, for n = 2 and m = 3 one obtains equation 20). The building of the infinite set of flows in equation 63), allows us to describe the first correction by an analoguos of the zero curvature condition 90). Let us start with the total differential of log ϕ 0 d log ϕ 0 z, T) = Λ n qz, T), pz, T), x pz, T), T)dT n + M 1 z, T)dz. 92) 20

21 The function M 1 z, T) = log ϕ 0 z, T) 93) z is an analogous of the Orlov s function. Differentiating equation 92), one can introduces the 2-form ω 1 := dλ n qz, T), pz, T), x pz, T), T) dt n = dz dm 1 z, T). 94) The equation 84) allows us to identify a Whitham type structure defined by the equation which can be given explicitly ω 1 ω 1 = 0, 95) Λ n Λ m p + {Λ n, Λ m } T m T q,x + {Λ n, Λ m } q,p n x + {Λ 2 p n, Λ m } q, xp x = 2 = G n Λ m ) G m Λ n ), 96) where G n is defined as G n f) = dωn dx p + d2 Ω n dx 2 ) f. 97) x p) Let us note that for n = 2 and m = 3, equation 96) gives, of course equation 49). 8 Concluding remarks and perspectives. In this paper we have seen how construct sistematically an integrable hierarchy by the quasiclassical -dressing method generalized at any order. Besides, a deeper investigation of the first dispersive contribution reveals a regular structure, by the formula 63), allowing us to handle the whole herarchy. There are also a lot of very interesting facts arising from our description, that will be inspiring our future work. From a general point of view it would be useful to go into the connection with the standard theory of asymptotics 21

22 approximation[24, 25] to generalize the results holding in that context. Moreover, there is the question if the obstacles arising in the construction of the corrections, discussed in the section 6) are connected to the caustics problem arising in the geometrical optics approximation. As regards possible applications it would be interesting consider the KdV limit of the KP hierarchy to establish a connection with the Dubrovin-Zhang theory[3]. Another intriguing matter of study and application is associated with the possibility of a physically meaningful interpretation of the solutions of dispersionless systems as describing integrable dynamics of interfaces[17]. In this context, one should investigate a quasiconformal dynamics described by the quasiclassical problem and the corrections 35-37), analyzing, in particular, the singular behavior of interfaces evolution, by introducing, for instance, a small dispersive parameter. In order to do that, a deeper study of the singular sector order by order will be also necessary. References [1] Witten E., 1990, Nucl. Phys. B, 123, 281; 1991, Surv. Diff. Geom., 1, 243. [2] Kontsevich M., 1991, Functs. Anal. Priloz., 25, 123; 1992, Commun. Math. Phys., 143, 1. [3] Dubrovin B.A., 1992, Commun. Math. Phys., 145, 195; 1992, Nucl. Phys. B, 379, 627; Dubrovin B.,Zhang Y., 1998, Commun. Math. Phys., 198, 311. [4] Krichever I.M., 1992, Comm. Math. Phys., 143, 415. [5] Krichever I.M., 1994, Comm. Pure. Appl. Math, 47, 437. [6] Takasaki K., Takebe T., 1992, Int. Jour. Mod. Phys., 7 Suppl.1B, 889. [7] Takasaki K., Takebe T., 1995, Rev. Mod. Phys., 7 n o 5, 743. [8] Dijkgraaf R., 1991, Intersection theory, integrable hierarchies and topological field theory, New Symmetry Principles in Quantum Field Theory, NATO ASI, Cargese1991), Plenum Press, London; A. Morozov, 22

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