The direct algorithm to construct the Davey-Stewartson hierarchy by two scalar pseudo-differential operators
aa r X i v : . [ n li n . S I] J a n The direct algorithm to construct theDavey-Stewartson hierarchy by two scalarpseudo-differential operators
G.Yi ∗ and X. Liao School of Mathematics, Hefei University of Technology, Hefei 230601,China ∗ Corresponding author: [email protected]
January 29, 2021
Abstract
The infinite many symmetries of DS(Davey-Stewartson) systemare closely connected to the integrable deformations of surfaces in R .In this paper, we give a direct algorithm to construct the expression ofthe DS(Davey-Stewartson) hierarchy by two scalar pseudo-differentialoperators involved with ∂ and ˆ ∂ . The KP (Kadomtsev-Petviashvili) equation [1, 2]( u t + 6 uu x + u xxx ) x + 3 σ u yy = 0 , (1)and the DS (Davey-Stewartson) system [2, 3] i q t + 12 ( q xx + σ q yy ) + δqφ = 0 , (2a) σ φ yy − φ xx + ( | q | ) xx + σ ( | q | ) yy = 0 , (2b)as the most important classical integrable models in (2+1) dimensions, havebeen extensively studied with many important results obtained [5–27]. Anintegrable system is usually associated with a hierarchy of nonlinear partialdifferential equations defining infinitely many symmetries. This is one of themost important and valuable properties of integrable systems.1he KP hierarchy plays a fundamental role in the theory of integrablesystems, a key reason is the clear and explicit definition via Lax equationsof pseudo-differential operators [28–36]. Let L = ∂ + u ∂ − + u ∂ − + · · · , ∂ = ∂∂x , (3)be a pseudo-differential operator whose coefficients u i depending on the spa-tial coordinate x . The KP hierarchy is defined as the following set of equa-tions ∂L∂t n = [( L n ) + , L ] , n = 1 , , , · · · . (4)Here the subscript ” + ” means to take the purely differential part (nonnega-tive part) of the pseudo-differential operator, while the subscript ” − ” meansto take the negative part. The well-known GD (Gelfand-Dickey) hierarchyis a reduction of the KP hierarchy with the constraint ( L m ) − = 0 for somenatural number m .The KP hierarchy has been generalized to multicomponent cases withscalar pseudo-differential operators replaced by matrix-value ones [37–40].Wu, Zhou and Lu studied an extension of the KP hierarchy by consider-ing two particular pseudo-differential operators [41, 42]. The infinite manysymmetries of DS system are closely connected to the 2-component KP hier-archy [2, 4]. Konopelchenko, Taimanov studied the infinite many symmetriesof DS system and pointed out that any symmetry induces an infinite familyof geometrically different deformations of tori in R preserving the Willmorefunctional. They defined the DS hierarchy by considering the compatibilityof undetermined differential operators in terms of ∂ z and ˆ ∂ z [43–45] and gaveexamples of t and t flows. But how to characterize the compatibility forthe infinite many equations in the whole hierarchy?In this paper, we give a direct algorithm to construct the expression ofthe flows of the DS hierarchy by two scalar pseudo-differential operatorsinvolved with ∂, ˆ ∂ and proof the compatibility for these infinite flows. The(1+1) dimensional reduction and some examples are discussed in the finalsection. 2 The Davey-Stewartson hierarchy
Firstly we introduce two scalar pseudo-differential operators L = ∂ + ∞ X j =1 u j ∂ − j , (5)ˆ L = ˆ ∂ + ∞ X j =1 ˆ u j ˆ ∂ − j , (6)in which the coefficients u j = u j ( t mn ) and ˆ u j = ˆ u j ( t mn ) depend on complexvariables t mn ( m, n are nonnegative integers and m + n ≥ t ≡ z, t ≡ ˆ z . Here and hereafter we denote ∂ = ∂∂z , ˆ ∂ = ∂∂ ˆ z in this paper.Similar to the definition of the KP hierarchy, we denote A m = ( L m ) + , (7a) B n = ( ˆ L n ) + , (7b)and we introduce ˜ A m = ( ∂ − ◦ q ◦ A m ) + , (8a)˜ B n = ( ˆ ∂ − ◦ p ◦ B n ) + , (8b)here and hereafter the subscript ” + ” means to take the purely differentialpart (nonnegative part) of the pseudo-differential operators both in ∂ and ˆ ∂ ,while the subscript ” − ” means to take the negative part, q = q ( t mn ) and p = p ( t mn ) depend on complex variables t mn are unknown complex-valuedfunctions in this DS hierarchy. The main result of this paper is the followingtheorem. Theorem.
The compatibility condition of the following linear system Lϕ = σ ϕ, (9a)ˆ Lψ = σ ψ, (9b)3 t mn = A m ϕ + ˜ B n ψ, (9c) ψ t mn = ˜ A m ϕ + B n ψ, (9d)is equivalent to the equations about the complex-valued functions q and p : q t mn = B n ( q ) − A ∗ m ( q ) , (10a) p t mn = A m ( p ) − B ∗ n ( p ) , (10b)in which σ and σ are two parameters, A m , B n , ˜ A m , ˜ B n are defined by (7)(8), F ∗ means the adjoint operator to F . This compatible system (9) is definedas DS (Davey-Stewartson) hierarchy, and the infinite number of (2+1) di-mensional nonlinear partial differential equations (10) are corresponding flowequations of the DS hierarchy. Remark 1.
Without the effect of ˜ A m , ˜ B n , the above system is nothingbut two separated KP hierarchies. In fact, ˜ A m , ˜ B n connect the two KPhierarchies and keep the compatibility. This is the key point for constructingthis DS hierarchy. Remark 2.
Notice that ϕ t = ϕ z , ψ t = ψ ˆ z . This implies ∂∂t = ∂, ∂∂t = ˆ z . This is the reason we identify t with z and t with ˆ z in thispaper. Therefore, one obtains the following Dirac system ϕ ˆ z = ϕ t = ˜ B ψ = pψ, (11a) ψ z = ψ t = ˜ A ϕ = qϕ. (11b)Then the compatibility condition of linear system (9) reads as follows ϕ ˆ z = pψ, (12a) ψ z = qϕ, (12b) ϕ t mn = A m ϕ + ˜ B n ψ, (12c)4 t mn = ˜ A m ϕ + B n ψ. (12d)The above hierarchy is called the DS hierarchy for the reason that its t flowis the well-known DS system (2).To prove this main theorem, we need the following two lemmas. Lemma 1.
The pseudo-differential operators L and ˆ L satisfy the follow-ing structure equations respectively ∂L m ∂ ˆ z + [ L m , R ] = 0 , (13a) ∂ ˆ L n ∂z + [ ˆ L n , ˆ R ] = 0 , (13b)in which R = p ◦ ∂ − ◦ q, ˆ R = q ◦ ˆ ∂ − ◦ p. (14)Correspondingly, A m and B n satisfy ∂A m ∂ ˆ z + [ A m , R ] + = 0 , (15a) ∂B n ∂z + [ B n , ˆ R ] + = 0 . (15b) Proof.
In fact, the equation (9a) yields L m ϕ = σ m ϕ . By taking the deriva-tive of this equation with respect to ˆ z , one obtains ∂L m ∂ ˆ z ϕ + L m ϕ ˆ z − σ m ϕ ˆ z = 0 , which leads to( ∂L m ∂ ˆ z + L m ◦ p ◦ ∂ − ◦ q − p ◦ ∂ − ◦ q ◦ L m ) ϕ = 0 . Hence,(13a) is true. The discussion for (13b) is similar. By taking thedifferential part (nonnegative part) of (13a) and (13b), one obtains (15a)and (15b). (cid:3) emark 3. From the structure equations (13), we can deduce that allthe coefficients u j and ˆ u j of the pseudo-differential operators L and ˆ L dependon the two complex-valued functions p, q and their derivatives or integralswith respect to the independent variables z, ˆ z . The leading terms of L andˆ L can be obtained directly by (13) as follows u = − ∂ − z (( pq ) z ) , u = − ∂ − z (( q z p ) z ) ,u = − ∂ − z (( pq zz ) z ) − ∂ − z (( pq ) z ∂ − z (( pq ) z )) + ∂ − z ( pq∂ − z (( pq ) zz )) , · · · , (16a)ˆ u = − ∂ − z (( pq ) ˆ z ) , ˆ u = − ∂ − z (( p ˆ z q ) ˆ z ) , ˆ u = − ∂ − z (( p ˆ z ˆ z q ) ˆ z ) − ∂ − z (( pq ) ˆ z ∂ − z (( pq ) ˆ z )) + ∂ − z ( pq∂ − z (( pq ) ˆ z ˆ z )) , · · · . (16b) Lemma 2.
The two pairs of differential operators A m , ˜ A m and B n , ˜ B n satisfy ∂ ◦ ˜ A m − q ◦ A m = − A ∗ m ( q ) , (17a)ˆ ∂ ◦ ˜ B n − p ◦ B n = − B ∗ n ( p ) . (17b) Proof.
In fact, A m defined in (7) reads as A m = m X k =0 a k ∂ k , where a m = 1 , a m − = 0 and a k ( k = 0 , · · · , m −
2) are differential polyno-mials in u j ( j = 1 , , · · · , m − ∂ ◦ ˜ A m − q ◦ A m = ∂ ◦ ( ∂ − ◦ q ◦ A m ) + − ∂ ◦ ∂ − ◦ q ◦ A m = − ∂ ◦ ( ∂ − ◦ q ◦ A m ) − = − ∂ ◦ ( ∂ − ◦ ( m X k =0 ( − k ( qa k ) ( k ) ))= − A ∗ m ( q ) , where ( qa k ) ( k ) = ∂ k ( qa k ) ∂z k . The proof of (17b) is similar. (cid:3) Proof of the theorem.
In fact, the compatibility condition of (12a)and (12c) reads as ∂ ψ∂z∂t mn = ∂ ψ∂t mn ∂z . By direct calculation, one obtains ∂ ψ∂z∂t mn = q t mn ϕ + qA m ϕ + q ˜ B n ψ,∂ ψ∂t mn ∂z = ∂ ( ˜ A m ϕ ) + ∂ ( B n ψ ) . Therefore, ∂ ψ∂z∂t mn − ∂ ψ∂t mn ∂z = q t mn ϕ + ( qA m ϕ − ∂ ( ˜ A m ϕ )) + ( q ˜ B n ψ − ∂ ( B n ψ )) = 0 . (18)By the virtue of (17a) in Lemma 2, one obtains qA m ϕ − ∂ ( ˜ A m ϕ ) = ( q ◦ A m − ∂ ◦ ˜ A m )( ϕ )= A ∗ m ( q ) ϕ. (19)The other part in (18) can be simplified by direct calculation as follows q ˜ B n ψ − ∂ ( B n ψ ) = q ˜ B n ψ − ∂B n ∂z ψ − B n ( qϕ )= q ˜ B n ψ − ∂B n ∂z ψ − B n ◦ q ◦ ˆ ∂ − ◦ p ( ψ )= ( q ◦ ˜ B n − ( B n ◦ q ◦ ˆ ∂ − ◦ p ) + − ∂B n ∂z )( ψ ) − ( B n ◦ q ◦ ˆ ∂ − ◦ p ) − ( ψ )= ( ∂B n ∂z + [ B n , ˆ R ] + )( ψ ) − ( B n ( q ) ◦ ˆ ∂ − ◦ p )( ψ )= ( ∂B n ∂z + [ B n , ˆ R ] + )( ψ ) − B n ( q ) ϕ. (20)7hen the structure equation (15b) in Lemma 1 gives q ˜ B n ψ − ∂ ( B n ψ ) = − B n ( q ) ϕ. (21)Therefore the compatibility equation (18) can be simplified as ∂ ψ∂z∂t mn − ∂ ψ∂t mn ∂z = ( q t mn + A ∗ m ( q ) − B n ( q )) ϕ = 0 , (22)which leads to the flow equation (10a).Similarly, by considering the compatibility condition ∂ ϕ∂ ˆ z∂t mn = ∂ ϕ∂t mn ∂ ˆ z , one obtains the flow equation (10b). (cid:3) Some examples from the DS hierarchy and the (1+1) dimensional reductionare presented below.
Example 1:
By taking m = n = 2, then A = ∂ + 2 u = ∂ − ∂ − z (( pq ) z ) , (23a) B = ˆ ∂ + 2ˆ u = ˆ ∂ − ∂ − z (( pq ) ˆ z ) , (23b)˜ A = q∂ − q z , (23c)˜ B = p ˆ ∂ − p ˆ z . (23d)Therefore, linear system (12) reads as ϕ ˆ z = pψ, (24a)8 z = qϕ, (24b) ϕ t = ( ∂ − ∂ − z (( pq ) z )) ϕ + ( p ˆ ∂ − p ˆ z ) ψ, (24c) ψ t = ( q∂ − q z ) ϕ + ( ˆ ∂ − ∂ − z (( pq ) ˆ z )) ψ. (24d)Then the following system q t = B ( q ) − A ∗ ( q ) = q ˆ z ˆ z − q zz − φq, (25a) p t = A ( p ) − B ∗ ( p ) = p zz − p ˆ z ˆ z + φp, (25b) φ z ˆ z = 2( pq ) ˆ z ˆ z − pq ) zz , (25c)in the form (10) arising from (24) is nothing but the well-known DS systemwith p = ¯ q . Example 2:
By taking m = n = 3,then A = ∂ + 3 u ∂ + 3 u + 3 u z = ∂ − ∂ − z (( pq ) z ) ∂ − ∂ − z (( q z p ) z ) − ∂ − z (( pq ) zz ) , (26a) B = ˆ ∂ + 3ˆ u ˆ ∂ + 3ˆ u + 3ˆ u z = ˆ ∂ − ∂ − z (( pq ) ˆ z ) ˆ ∂ − ∂ − z (( p ˆ z q ) ˆ z ) − ∂ − z (( pq ) ˆ z ˆ z ) , (26b)˜ A = q∂ − q z ∂ + q zz + 3 qu , (26c)˜ B = p ˆ ∂ − p ˆ z ˆ ∂ + p ˆ z ˆ z + 3 p ˆ u . (26d)Correspondingly, the adjoint operators read as A ∗ = − ∂ − u ∂ + 3 u + 3 u z = − ∂ + 3 ∂ − z (( pq ) z ) ∂ − ∂ − z (( q z p ) z ) − ∂ − z (( pq ) zz ) , (27a)9 ∗ = − ˆ ∂ − u ˆ ∂ + 3ˆ u + 3ˆ u z = − ˆ ∂ + 3 ∂ − z (( pq ) ˆ z ) ˆ ∂ − ∂ − z (( p ˆ z q ) ˆ z ) − ∂ − z (( pq ) ˆ z ˆ z ) . (27b)Then the communication of linear system (12) is equivalent to the followingintegrable system q t = B ( q ) − A ∗ ( q ) , (28a) p t = A ( p ) − B ∗ ( p ) . (28b) Example 3:
By taking m = 2 , n = 3, then the communication condition of the linearsystem (12) reads as q t = B ( q ) − A ∗ ( q ) , (29a) p t = A ( p ) − B ∗ ( p ) . (29b)Now, we consider the (1+1) dimensional reduction of the DS hierarchywhich includes the important integrable model NLS (nonlinear Schr¨ o dinger)equation. By considering the conjugate independent variables z = x + i y, ˆ z = x − i y and the reduced condition ∂ = i ˆ ∂ , i.e., x = − y , one obtains thefollowing (1+1) reduction of the linear system (12) under the transform p → − i p, q → i q ϕ x = pψ, (30a) ψ x = qϕ, (30b) ϕ t mn = α m ϕ + ˜ β n ψ, (30c) ψ t mn = ˜ α m ϕ + β n ψ, (30d)10here α m , β n , ˜ α m and ˜ β n are differential operators in terms of ∂ x . Thefollowing nontrivial example of this reduced hierarchy is the NLS equation. Example 4:
By taking m = n = 2, the linear system (30) reads as ϕ x = pψ, (31a) ψ x = qϕ, (31b) ϕ t = ( i ∂ x − i pq ) ϕ + ( − i p∂ x + i p x ) ψ, (31c) ψ t = ( i q∂ x − i q x ) ϕ + ( − i ∂ x + i pq ) ψ. (31d)The compatibility condition of (31) is equivalent to i q t − q xx + 2 pq = 0 , (32a) i p t + p xx − pq = 0 , (32b)which is exactly the classical NLS equation with p = ± ¯ q . Research on the dispersionless integrable systems (integrable systems of hy-drodynamic type) which arise from the commutation condition of vector fieldsLax pairs, is an important subject. One important kind of dispersionless in-tegrable systems comes from the semiclassical limit (dispersionless limit) ofthe classical integrable systems. In [46], we discussed the semiclassical limitof the DS system (2) and the relevant nonlinear Riemann-Hilbert problem.In [47], we defined a new class of dispersionless integrable systems called dDS(dispersionless Davey-Stewartson) hierarchy. In fact, the semiclassical limit11dispersionless limit) of the DS hierarchy (9)(12) is closely connected to thedDS hierarchy. We will show the details in the next separated paper.
Acknowledgements:
This work has been supported by the Key Labo-ratory Foundation (No.6142209180306) and National Natural Science Foun-dation of China (No. 11501222).
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