Commutator representations of nonlinear evolution equations: Harry-Dym and Kaup-Newell cases

Transcription

1 Nonlinear Mathematical Physics 995, V.2, N 2, Commutator representations of nonlinear evolution equations: Harry-Dym and Kaup-Newell cases Zhijun QIAO Department of Mathematics, Liaoning University, Shenyang, Liaoning 36, People s Republic of China Submitted by A. NIKITIN Received September, 994 Abstract A general structure of commutator representations for the hierarchy of nonlinear evolution equations (NLEEs) is proposed. As two concrete examples, the Harry-Dym and Kaup-Newell cases are discused. Recently, the commutator representations of the hierarchy of nonlinear evolution integrable equations (NLEEs) and the related Lax operator algebra properties have been intensively discussed [ 6]. It is well-known for the spectral problem Lψ L (u) ψ = λψ (u = (u,..., u N ) T is a potential vector, λ is a constant parameter) that if its hierarchy of evolution equations possesses commutator representations, then its key lies in solving an operator equations of the differential operator V = V (G) [, 3, 5] [V, L] = L (KG) L (JG) L () where K, J are the pair of Lenard s operators corresponding to the spectral problem L (u) ψ = λψ, G = (G (),..., G (N) ) T is an arbitrary given vector function, L (ξ) = d dε ε= L (u + εξ). Now, consider the spectral problem ψ x = U (u, λ) ψ (2) where ψ = (ψ,..., ψ n ) T, each element of the n n matrix U (u, λ) is the polynomial of λ, λ and the coefficients of its every term depend on u. Copyright c 995 by Mathematical Ukraina Publisher. All rights of reproduction in any form reserved.

2 52 ZHIJUN QIAO According to the methods proposed in ref. [7, 8], we can always acquire the spectral gradient u λ = (δλ/δu,..., δλ/δu N ) T of the spectral parameter λ with respect to the potential vector u. Generally, u λ is related to λ, u, and the special function ψ. The integro-differential operators K = K (u), J = J (u) depending on the potential vector an satisfying the following linear relation K u λ = λ θ J u λ (θ is a fixed constant) (3) are called the pair of Lenard s operators of (2). The operators K, J can be obtained with (2) and the concrete expression of u λ after some delicate calculations. As U (u, λ) is linear on λ, (2) can always lead to L ψ L (u) ψ = λ ψ. (4) Otherwise, (2) can t read the form like (4). Nevertheless, because each element of the n n matrix U (u, λ) is the polynomial in λ, λ and the coefficients of its every term depend on u, the spectral problem (2) can be usually rewritten as L ψ L (u, λ) ψ = λ γ ψ (5) where γ is the highest order of λ in U (u, λ), L = L (u, λ) is a differential operator related to u and λ. A basic problem is: what is conditions under which the isospectral hierarchy of evolution equations (5) posesses the commutator representations? For the spectral problem of its form like (5), here we construct a wider operator equation with the differential operator V = V (G) than () [V, L] = L (KG) L β L (JG) L α (6) where [, ] stands for the commutator; L = L (u, λ); K, J are the pair of Lenard s operators determined by (3); G = (G (),, G (N) ) T is an arbitrary given vector function, L (ξ) = d dε ε= L (u + εξ, λ), ξ = (ξ,..., ξ N ) T ; α, β are two fixed constants associated with (5) and β < α. Let η = α β, choose G η Ker J = {G JG = }, and define Lenard s recursive secuence {G jη }: KG (j )η = JG jη, j =,, 2,.... (7) The NLEEs u t = X m (u) (m =,, 2,...) produced by the vector field X m = JGmη (m =,, 2,...) and called the hierarchy of evolution equations (5). The following two theorems give a simple and clear approach that the hierarchy of isospectral evolution equations u t = X m (u) (m =,, 2,...) of (5) owns the commutator representations. Theorem Let {G jη } j= be the Lenard s recursive sequence of (5). For any G jη, the operator equation (6) has the commutator solution V j = V (G jη ). Then the operator W m = m V j L (m j)η β is the Lax operator (4) of the vector field X m (u), that is, W m satisfies j= [W m, L] = L (X m ), m =,, 2,.... (8)

3 COMMUTATOR REPRESENTATIONS OF NONLINEAR EVOLUTION EQUATIONS 53 Proof. [W m, L] = m [V j, L] L (m j)η β j= = m (L (KG (j )η ) L (m j)η L (JG (j )η ) L (m j+)η ) j= = L (JG mη ) = L (X m ). From this theorem, we can also further discuss the Lax operator algebra generated by the Lax operator W m which is left to a later paper. Theorem 2 Let the conditions in Theorem be satisfied, and the Gateaux derivative mapping L : ξ L (ξ) of the spectral operator L in the direction ξ is an injective homomorphism. Then the isospectral hierarchy of evolution equations u t = X m (u) of (5) possesses the commutator representations L t = [W m, L], m =,, 2,.... (9) Proof. L t = L (u t ), L t [W m, L] = L (u t ) L (X m (u)) = L (u t X m (u)). The above equality implies Theorem 2 holds because L is injective. By Theorem and Theorem 2, we can evidently see that in order to secure the commutator representations (9) of NLEEs u t = X m (u), its key lies in constructing the corresponding operator equation (6) according to the form of (5) and finding an operator solution of (6). Corollary The potential u = (u,..., u N ) T satisfies a stationary nonlinear equation α k X k if and only if [ α k W k, L] =, where α k (k =,, 2,..., l) are some k= k= constants, l Z +. In the following, as two concrete examples of the above approach, we shall discuss the Harry-Dym and Kaup-Newell hierarchies, present the corresponding operator equation (6), solve it, and finally give the commutator representations of these two hierarchies.. Consider the spectral problem iλ (u )λ ψ x = U (u, λ) ψ, U (u, λ) =, ψ = λ iλ () is equivalent to the famous Sturm-Liouville equation ψ ψ 2, i 2 =. () 2 y = µuy, () via the transformations ψ = iy λ y x, ψ 2 = y, µ = λ 2. The isospectral property of the Harry-Dym hierarchy was studided in [9], and the nonlinearization of the Lax pair for the Harry-Dym equation u t = (u (/2) ) xxx was discussed in []. In the present paper, using

4 54 ZHIJUN QIAO the above skeleton, we further give the commutator representations of each equation in the Harry-Dym hierarchy including the Harry-Dym equation u t = (u (/2) ) xxx. The spectral gradient u λ of () with regard to u is (. u λ = λψ2 2 (2iψ ψ 2 uψ2 2 ψ) dx) 2 (2) Ω and (9), only choosing Lenard s ope- Noticing the relation u ψ2 2 = 2iψ ψ 2 + ψ2 2 ψ2 rators we have K = 3, J = 2 ( u + u ), (3) K u λ = λ 2 J u λ. (4) Let G 2 = u (/2) Ker J, define the Lenard recursive sequence {G 2j } of (): KG 2(j ) = JG 2j, j =,, 2,.... The Harry-Dym vector fields X j (u) = JG 2j yield the isospectral hierarchy of NLEEs (): u t = X j (u) (j =,, 2,...), in which the first system is the well-known Harry-Dym equation u t = KG 2 = (u (/2) ) xxx. () can be rewritten as Lψ = λψ, L = L (u) = u i i The Gateaux derivative mapping L of L in the direction ξ is L (ξ) = ξ u 2 i = ξ u i u, = / x. (5) i L (6) and L is an injective homomorphism. Let G (x) be an arbitrary smooth function. For the spectral problem (5), we establish the corresponding operator equation of V = V (G) as follows [ V, L ] = L (KG) L L (JG) L (7) which is equivalent to (6) with α =, β =. Theorem 3 The operator equation (7) has the operator solution V = V (G) = G xx 2i i u + G x L + ( 2G) L 2. (8) i Proof. Let i u W =, i V = G xx,

5 COMMUTATOR REPRESENTATIONS OF NONLINEAR EVOLUTION EQUATIONS 55 2i V = G x, i u V 2 = 2G. (9) i Then the commutator [V, L] of V = V + V L + V 2 L 2 and L is (L = W ): [V, L] = W V x + (V W V W W V x ) L+ (V W V W W V 2x ) L 2 + (V 2 W V 2 W ) L 3. (2) Substituting every expressions of (9) into (2), through lengthy calculations we can find that the right-hand side of (2) is equal to L (KG) L L (JG) L. Thus, the conditions of both Theorem and Theorem 2 hold. So, the Harry-Dym hierarchy of NLEEs u t = X m (u) (m =,, 2,...) possesses the following commutator representations L t = [W m, L], m =,, 2,..., m { ( ) ( ) 2i W m = G 2(j ),xx + G 2(j ),x L j= ( ) } i u 2G 2(j ) L i 2 L 2(m j)+. Particularly, as m =, the Harry-Dym equation u t = X (u) = (u) xxx has the commutator representation L t = [W, L], ( ) W = (u (/2) ) xx L+ (u (/2) ) x ( 2i ) ( ) i u L 2 2u (/2) L i Consider the spectral problem proposed by Kaup and Newell [] iλ 2 λu ψ ψ x = U (u, v, λ) ψ, U (u, v, λ) =, ψ =, i 2 =. (2) λv iλ 2 It isn t difficult to get the spectral gradient (u,v) λ δλ / δu λ ψ 2 2 (u,v) λ = = (vψ 2 δλ / δv λ ψ 2 + 4iψ ψ 2 uψ2) 2 dx Ω which satisfies where K (u,v) λ = λ 2 J (u,v) λ, (23) K = 2 u u 2 i u v 2 i v u 2 v v ψ 2, J = (22)

6 56 ZHIJUN QIAO are the pair of Lenard s operators of (2). Let G = (, ) T Ker J, G = J KG = (v, u) T. The Lenard recursive sequences G j (j =,, 2,...) are determined by KG j = JG j, j =,, 2,..., (24) which produces the Kaup-Newell hierarchy of NLEEs (u, v) T t = X j (u, v) = JG j, j =,, 2,..., (25) with the representative equation (u, v) T t ( = X (u, v) 2 iu xx + 2 (u2 v) x, 2 iv xx + ) T 2 (v2 u) x. (26) As j = and v = u, (26) reduces to the famous derivative Schrödinger equation (DSE): u t = 2 iu xx + 2 (u u 2 ) x. (27) (2) is equivalent to i Lψ = λ 2 ψ, L = iλv iλu. (28) i The Gateaux derivative L of L is iξ L (ξ) = L /2, ξ = (ξ, ξ 2 ) T, L is injective. (29) iξ 2 Let G (x) =(G () (x), G (2) (x)) T be any given smooth vector field. For the spectral problem (28), we construct the related operator equation with V = V (G) as follows [V, L] = L (KG) L (/2) L (JG) L /2 (3) which is exactly (6) with α = 2, β = 2. Theorem 4 The operator equation (3) possesses the operator solution 2 ig(2) x + 2 u (ug () x +vg (2) x ) V =V (G)= + 2 ig() x + 2 v (ug () x +vg (2) x ) 2 i (ug () x + vg (2) x ) 2 i (ug () x + vg (2) L /2. (3) x ) Proof. The method of prooving this Theorem is similar to that used in Theorem 3. The process is omitted.

7 COMMUTATOR REPRESENTATIONS OF NONLINEAR EVOLUTION EQUATIONS 57 So, the Kaup-Newell hierarchy of NLEEs (u, v) T t commutator representations L t = [W m, L], m =,, 2,..., W m = m j= = X m (u, v) (m =,, 2,...) has the 2 ig(2) j,x + 2 u (ug () j,x + vg(2) 2 ig() j,x + 2 v (ug () j,x + vg(2) j,x ) 2 i (ug () j,x + vg(2) j,x ) j,x ) + L /2 Lm j+(/2). 2 i (ug () j,x + vg(2) j,x ) Especially, if one lets m =, v = u, then the DSE (27) has the commutator representation L t = [W, L], iu x + u u 2 i u 2 W = 2 L /2 + iu x + u u 2 2 L+ i u 2 u i L 3/2 + L 2. u i References [] Cao C., Commutator representation of isospectral equation, Chin. Sci. Bull., 989, V.34, N, [2] Cheng Y. and Li Y., Lax operator algebra of the AKNS hierarchy, Chin. Sci. Bull., 99, V.35, N 9, [3] Ma W., Commutator representations of the Yang hierarchy of integrable evolution equations, Chin. Sci. Bull., 99, V.36, N 6, [4] Ma W., The algebraic structure of isospectral Lax operator and applications to integrable equations, J. Phys. A: Math. Gen., 992, V.25, N 2, [5] Qiao Z., Lax representations for the Levi hierarchy, Chin. Sci. Bull., 99, V.36, N 2, [6] Qiao Z., Commutator representations of the D-AKNS hierarchy of evolution equations, Mathematica Applicata, 99, V.4, N 4, [7] Cao C., Nonlinerization of Lax pair for the AKNS hierarchy, Sci. Sin. A, 99, V.33, N 5, [8] Tu G., An extension of a theorem on gradient of conserved densities of integrable systems, Northeastern Math. J., 99, V.6, N, [9] Li Y., Chen D. and Zeng Y., Some equivalent classes of soliton equations, Proc. 983 Beijing Symp. on Diff. Geom. and Diff. Equ s, Science Press, Beijing, 986, [] Cao C., Stationary Harry-Dym equation and its relation with geodesics ellipsoid, Acta Math. Sin., New Series, 99, V.6, N, [] Kaup J.D. and Newell A.C., An exact solution for a derivate nonlinear Schrödinger equation, J. Math. Phys., 978, V.9, N 4,