On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation

Different gauge copies of the Ablowitz-Kaup-Newell-Segur (AKNS) model labeled by an angle θ are constructed and then reduced to the two-component Camassa-Holm model. Only three different independent classes of reductions are encountered corresponding to the angle θ being 0, π/2 or taking any value i...

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Дата:	2006
Автори:	Aratyn, H., Gomes, J.F., Zimerman, A.H.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2006
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/146097
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Цитувати:	On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation / H. Aratyn, J.F. Gomes, A.H. Zimerman // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 21 назв. — англ.

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spelling	irk-123456789-1460972019-02-08T01:23:48Z On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation Aratyn, H. Gomes, J.F. Zimerman, A.H. Different gauge copies of the Ablowitz-Kaup-Newell-Segur (AKNS) model labeled by an angle θ are constructed and then reduced to the two-component Camassa-Holm model. Only three different independent classes of reductions are encountered corresponding to the angle θ being 0, π/2 or taking any value in the interval 0 < θ < π/2. This construction induces Bäcklund transformations between solutions of the two-component Camassa-Holm model associated with different classes of reduction. 2006 Article On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation / H. Aratyn, J.F. Gomes, A.H. Zimerman // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 21 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K10; 35Q53; 53A07; 53B50 http://dspace.nbuv.gov.ua/handle/123456789/146097 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Different gauge copies of the Ablowitz-Kaup-Newell-Segur (AKNS) model labeled by an angle θ are constructed and then reduced to the two-component Camassa-Holm model. Only three different independent classes of reductions are encountered corresponding to the angle θ being 0, π/2 or taking any value in the interval 0 < θ < π/2. This construction induces Bäcklund transformations between solutions of the two-component Camassa-Holm model associated with different classes of reduction.
format	Article
author	Aratyn, H. Gomes, J.F. Zimerman, A.H.
spellingShingle	Aratyn, H. Gomes, J.F. Zimerman, A.H. On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Aratyn, H. Gomes, J.F. Zimerman, A.H.
author_sort	Aratyn, H.
title	On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation
title_short	On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation
title_full	On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation
title_fullStr	On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation
title_full_unstemmed	On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation
title_sort	on a negative flow of the akns hierarchy and its relation to a two-component camassa-holm equation
publisher	Інститут математики НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/146097
citation_txt	On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation / H. Aratyn, J.F. Gomes, A.H. Zimerman // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 21 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed	2025-07-10T23:09:25Z
last_indexed	2025-07-10T23:09:25Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 070, 12 pages On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa–Holm Equation? Henrik ARATYN † and Jose Francisco GOMES ‡ and Abraham H. ZIMERMAN ‡ † Department of Physics, University of Illinois at Chicago, 845 W. Taylor St., Chicago, Illinois 60607-7059 E-mail: aratyn@uic.edu ‡ Instituto de F́ısica Teórica-UNESP, Rua Pamplona 145, 01405-900 São Paulo, Brazil E-mail: jfg@ift.unesp.br, zimerman@ift.unesp.br Received September 13, 2006, in final form October 05, 2006; Published online October 17, 2006 Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper070/ Abstract. Different gauge copies of the Ablowitz–Kaup–Newell–Segur (AKNS) model la- beled by an angle θ are constructed and then reduced to the two-component Camassa–Holm model. Only three different independent classes of reductions are encountered corresponding to the angle θ being 0, π/2 or taking any value in the interval 0 < θ < π/2. This construction induces Bäcklund transformations between solutions of the two-component Camassa–Holm model associated with different classes of reduction. Key words: integrable hierarchies; Camassa–Holm equation; Bäcklund transformation 2000 Mathematics Subject Classification: 37K10; 35Q53; 53A07; 53B50 1 Introduction It is widely known that the standard integrable hierarchies can be supplemented by a set of commuting flows of a negative order in a spectral parameter [1]. A standard example is pro- vided by the modified KdV-hierarchy, which can be embedded in a new extended hierarchy. This extended hierarchy contains in addition to the original modified KdV equation also the differential equation of the sine-Gordon model realized as the first negative flow [2, 3, 4, 5, 6, 7]. Quite often the negative flows can only be realized in a form of non-local integral differential equations. The cases where the negative flow can be cast in form of local differential equation which has physical application are therefore of special interest. Recently in [11], a negative flow of the extended AKNS hierarchy [8] was identified with a two-component generalization of the Camassa–Holm equation. The standard Camassa–Holm equation [9, 10] ut − utxx = −3uux + 2uxuxx + uuxxx − κux, κ = const (1.1) enjoys a long history of serving as a model of long waves in shallow water. The two-component extension [11, 13] differs from equation (1.1) by presence on the right hand side of a new term ρρx, with the new variable ρ obeying the continuity equation ρt +(uρ)x = 0. Such generalization was first encountered in a study of deformations of the bihamiltonian structure of hydrodynamic type [12]. Various multi-component generalizations of the Camassa–Holm model have been subject of intense investigations in recent literature [14, 15, 16, 17, 18]. ?This paper is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). The full collection is available at http://www.emis.de/journals/SIGMA/LOR2006.html mailto:aratyn@uic.edu mailto:jfg@ift.unesp.br mailto:zimerman@ift.unesp.br http://www.emis.de/journals/SIGMA/2006/Paper070/ http://www.emis.de/journals/SIGMA/LOR2006.html 2 H. Aratyn, J.F. Gomes and A.H. Zimerman A particular connection between extended AKNS model and a two-component generalization of the Camassa–Holm equation was found in [11] and in [13]. It was pointed out in [19] that the second order spectral equation for a two-component Camassa–Holm model can be cast in form of the first order spectral equation which after appropriate gauge transformations fits into an sl(2) setup of linear spectral problem and associated zero-curvature equations. The goal of this article is to formulate a general scheme for connecting an extended AKNS model to a two-component Camassa–Holm model which would encompass all known ways of connecting the solution f of the latter model to variables r and q of the former model. Our approach is built on making gauge copies of an extended AKNS model labeled by angle θ belonging to an interval 0 ≤ θ ≤ π/2 and then by elimination of one of two components of the sl(2) wave function reach a second order non-linear partial differential equation which governs the two-component Camassa–Holm model. We found that the construction naturally decomposes into three different classes depending on whether angle θ belongs to an interior of interval 0 ≤ θ ≤ π/2 or is equal to one of two boundary values unifying therefore the results of [11] and [20]. The map between these three cases induces a Bäcklund like transformations between different solutions f of the two-component Camassa–Holm equation. 2 A simple derivation of a relation between AKNS and two-component Camassa–Holm models Our starting point is a standard first-order linear spectral problem of the AKNS model: Ψy = (λσ3 +A0) Ψ = λ [ 1 0 0 −1 ] Ψ + [ 0 q r 0 ] Ψ, (2.1) where λ is a spectral parameter, y a space variable and Ψ a two-component object: Ψ = [ ψ1 ψ2 ] . (2.2) In addition, the system is augmented by a negative flow defined in terms of a matrix, which is inverse proportional to λ: Ψs = D(−1)Ψ = 1 λ [ A B C −A ] Ψ. (2.3) The compatibility condition arising from equations (2.1) and (2.3): (A0)s −D(−1) y + [ λσ3 +A0, D (−1) ] = 0. (2.4) has a general solution: D(−1) = 1 4βλ M0σ3M −1 0 , A0 = M0 yM −1 0 , (2.5) in terms of the zero-grade group element, M0, of SL(2). Note that the solution, D(−1), of the compatibility condition is connected to (1/λ)σ3-matrix by a similarity transformation. The factor 1/4β in (2.5) is a general proportionality factor which implies a determinant formula: A2 +BC = 1 16β2 (2.6) for the matrix elements of D(−1) . On a Negative Flow of the AKNS Hierarchy 3 From (2.4) we find that( Tr(A2 0) ) s = 2Tr(A0A0 s) = −2 Tr ( A0 [ λσ3, D (−1) ]) = 2Tr ( λσ3 [ A0, D (−1) ]) = 2Tr ( λσ3D (−1) y ) = 4Ay or Ay = 1 2 (rq)s. (2.7) When projected on the zero and the first powers of λ the compatibility condition (2.4) yields qs = −2B, rs = 2C, (2.8) and Ay = qC − rB, By = −2Aq, Cy = 2Ar, (2.9) respectively. Note that the first of equations (2.9) together with equations (2.8) reproduces formula (2.7). Combining the above equations we find that A = −By 2q = qsy 4q = Cy 2r = rsy 4r . (2.10) The spectral equation (2.1) reads in components: ψ1 y = λψ1 + qψ2, ψ2 y = −λψ2 + rψ1. (2.11) Now we eliminate the wave-function component ψ2 by substituting ψ2 = 1 q (ψ1y − λψ1) into the remaining second equation of (2.11). In this way we obtain for ψ1 ψ1yy − qy q ψ1y + λqy q ψ1 − λ2ψ1 − rqψ1 = 0. Introducing ψ = e− ∫ p dyψ1 (2.12) with the integrating factor p(y) = 1 2 (ln q)y allows to eliminate the term with ψ1y and obtain ψyy = ( λ2 − λ (ln q)y −Q ) ψ (2.13) with Q = 1 2 (ln q)yy − 1 4 (ln q)2y − rq = qyy 2q − 3 4 ( qy q )2 − rq (2.14) as in [20]. 4 H. Aratyn, J.F. Gomes and A.H. Zimerman Eliminating ψ2 from equation (2.3) yields for ψ the following equation: ψs = 1 4λ ( qs q ) y ψ − 1 2λ qs q ψy. (2.15) Compatibility equation ψyys − ψsyy = 0 yields( qsy 4q ) y = 1 2 (rq)s (2.16) in total agreement with (2.7). To eliminate r from (2.16) we use that r = −Ay + qC B (2.17) as follows from the first equation from (2.9). Replacing C by 1/(B16β2) − A2/B as follows from the determinant relation (2.6) and recalling that B = −qs/2 according to equation (2.8) we obtain after substituting r from (2.17) into (2.16):( qsy q ) y = ( qsyy qs − qsyqy qqs + 1 2β2 q2 q2s − q2sy 2q2s ) s . (2.18) Note that alternatively we could have eliminated q from equation(rsy 4r ) y = 1 2 (rq)s and obtained an equation for r only. It turns out that the equation for r follows from equa- tion (2.18) by simply substituting r for q. For brevity we introduce, as in [20], f = ln q. Then expression (2.18) becomes: (fsfy)y = − ( f2 y 2 + f2 sy 2f2 s − 1 2β2f2 s − fsyy fs ) s . (2.19) The above relation can be cast in an equivalent form: fss 2β2f3 s + fsyfy + 1 2 fsfyy − fssyy 2fs + fssyfsy 2f2 s + fssfsyy 2f2 s − fssf 2 sy 2f3 s = 0, (2.20) which first appeared in [11]. The relation (2.20) is also equivalent to the following condition( 1 fs ) s = β2 ( f2 s fy − fssy + fssfsy fs ) y . (2.21) For a quantity u defined as: u = β2 ( f2 s fy − fssy + fssfsy fs ) − 1 2 κ, (2.22) with κ being an integration constant, it holds from relation (2.21) that uy = ( 1 fs ) s . (2.23) On a Negative Flow of the AKNS Hierarchy 5 Next, as in [21], we define a quantity m as β2f2 s fy and derive from relations (2.22) and (2.23) that m = β2f2 s fy = u+ β2 ( fssy − fssfsy fs ) + 1 2 κ = u− β2fs ( fs ( 1 fs ) s ) y + 1 2 κ = u− β2fs(fsuy)y + 1 2 κ. (2.24) Taking a derivative of m with respect to s yields ms = β2 ( 2fyfsfss + f2 s fsy ) = 2m fss fs + β2f2 s fsy = −2mfs ( 1 fs ) s + β2f2 s fsy = −2mfsuy + β2f2 s fsy. (2.25) In terms of quantities u and ρ = fs equations (2.23) and (2.25) take the following form ρs = −ρ2uy, (2.26) ms = −2mρuy + β2ρ2ρy, (2.27) for m given by m = u− β2ρ(ρuy)y + 1 2 κ. (2.28) An inverse reciprocal transformation (y, s) 7→ (x, t) is defined by relations: Fx = ρFy, Ft = Fs − ρuFy (2.29) for an arbitrary function F . Equations (2.26), (2.27) and (2.28) take a form ρt = − (uρ)x , (2.30) mt = −2mux −mxu+ β2ρρx, (2.31) m = u− β2uxx + 1 2 κ (2.32) in terms of the (x, t) variables. Equation (2.30) is called the compatibility condition, while equation (2.31) is the two-component Camassa–Holm equation [11], which agrees with standard Camassa–Holm equation (1.1) for ρ = 0. 3 General reduction scheme from AKNS system to the two-component Camassa–Holm equation Next, we perform the transformation Ψ → U(θ, f)Ψ = [ ϕ η ] (3.1) on AKNS two-component Ψ function from (2.2). U(θ, f) stands for an orthogonal matrix: U(θ, f) = Ω(θ) exp ( −1 2 fσ3 ) , 0 ≤ θ ≤ π 2 , (3.2) 6 H. Aratyn, J.F. Gomes and A.H. Zimerman where Ω(θ) is given by Ω(θ) = σ3e i θσ2 = [ cos θ sin θ sin θ − cos θ ] (3.3) and f is a function of y and s, which is going to be determined below for each value of θ. Note that Ω−1(θ) = Ω(θ) and Ω(0) = σ3, Ω(π/2) = σ1. Taking a derivative with respect to y and s on both sides of (3.1) one gets[ ϕ η ] y = ( UyU−1 + U [ λ q r −λ ] U−1 )[ ϕ η ] , (3.4)[ ϕ η ] s = ( UsU−1 + UD(−1)U−1 )[ ϕ η ] . (3.5) Thus, the flows of the new two-component function defined in (3.2) are governed by the gauge transformations of the AKNS matrices λσ3 + A0 and D(−1), respectively. This ensures that the original AKNS compatibility condition (2.4) still holds for the rotated system defined by equations (3.4) and (3.5). From equation (3.4) we derive that: λ (ϕ cos(2θ) + η sin(2θ)) = ϕy + 1 2 ϕ ( fy cos(2θ)− sin(2θ) ( qe−f + ref )) + η ( 1 2 fy sin(2θ)− ref sin2(θ) + qe−f cos2(θ) ) . (3.6) Repeating derivation with respect to y one more time yields[ ϕ η ] yy = [( UyU−1 + U [ λ q r −λ ] U−1 ) y + ( UyU−1 + U [ λ q r −λ ] U−1 )2] [ ϕ η ] = U [ λ2 − λfy + f2 y /4− fyy/2 + qr qy − fyq ry + fyr λ2 − λfy + f2 y /4 + fyy/2 + qr ] U−1 [ ϕ η ] . (3.7) For [ ϕ̄ η̄ ] = Ω(θ) [ ϕ η ] the result is[ ϕ̄ η̄ ] yy = [ λ2 − λfy + f2 y /4− fyy/2 + qr (qy − fyq)e−f (ry + fyr)ef λ2 − λfy + f2 y /4 + fyy/2 + qr ] [ ϕ̄ η̄ ] and shows in a transparent way that the condition for eliminating η̄ from the equation for ϕ̄yy requires (qy − fyq) exp(−f) = 0 or q = exp(f). Similarly, the condition for eliminating ϕ̄ from the equation for η̄yy requires (ry + fyr) exp(f) = 0 or r = exp(−f). Clearly these reductions reproduce results of the previous section. To obtain a more general result we return to equation (3.7). Projecting on the ϕ-component in equation (3.7) gives ϕyy = λ2ϕ− λfyϕ+ ( 1 4 f2 y + qr ) ϕ+ ( −1 2 fy cos(2θ) + 1 2 (qe−f + ref ) sin(2θ) ) y ϕ + ( −1 2 fy sin(2θ)− qe−f cos2 θ + ref sin2 θ ) y η. (3.8) On a Negative Flow of the AKNS Hierarchy 7 Next, we will eliminate η in order to obtain an equation for the one-component variable ϕ. This is analogous to the calculation made below equation (2.11), where the first order two-component AKNS spectral problem was reduced to second order equation for the one-component function ψ. To accomplish the task we must choose f so that the identity 1 2 fy sin(2θ) = ref sin2 θ − qe−f cos2 θ + c0 (3.9) holds, where c0 is an integration constant. The identity (3.9) ensures that terms with η drop out of equation (3.8). Note, that for θ = π/4 and c0 = 0 we recover identity fy = r exp(f)−q exp(−f) from [11, 19]. For θ = 0, c0 = 1 and θ = π/2, c0 = −1 we get, respectively, q = exp(f) and r = exp(−f) as in [20]. From now on we take c0 = 0 as long as 0 < θ < π/2. Let us shift a function f by a constant term, ln (tan θ): f −→ fθ = f + ln (tan θ) . (3.10) Then relation (3.9) can be rewritten for 0 < θ < π/2 as fθy = refθ − qe−fθ (3.11) which is of the same form as the relation found in reference [11]. It therefore appears that for all values of θ in the the 0 < θ < π/2 relation between function f and AKNS variables q and r remains invariant up to shift of f by a constant. Now, we turn our attention back to equation (3.5) rewritten as[ ϕ η ] s = U ( −1 2 fsσ3 + 1 λ [ A B C −A ]) U−1 [ ϕ η ] . For the ϕ component we find: ϕs = −1 2 fs (ϕ cos(2θ) + η sin(2θ)) + 1 2λ ϕ ( 2A cos(2θ) + Cef sin(2θ) +Be−f sin(2θ) ) + 1 λ η ( A sin(2θ) + Cef sin2 θ −Be−f cos2 θ ) . (3.12) For 0 < θ < π/2 we choose B = ( A− 1 4β ) efθ , C = − ( A+ 1 4β ) e−fθ , (3.13) which agrees with the determinant formula A2 +BC = 1/16β2 and implies identities: 2A−Be−fθ + Cefθ = 0, (3.14) Be−fθ + Cefθ = − 1 2β . (3.15) The first of these identities, (3.14), ensures that the last three terms containing η on the right hand side of equation (3.12) cancel. Recall at this point relation (3.6). Simplifying this relation by invoking identity (3.9) and plugging it into equation (3.12) gives ϕs = − fs 2λ ϕy + 1 λ ϕ ( −1 4 fsfy cos(2θ) + fs 4 sin(2θ) ( qe−f + ref ) 8 H. Aratyn, J.F. Gomes and A.H. Zimerman +A cos(2θ) + 1 2 Be−f sin(2θ) + 1 2 Cef sin(2θ) ) . (3.16) From (3.13) we find refθ = Cy 2A efθ = 1 2A ( fy ( A+ 1 4β ) −Ay ) , (3.17) qe−fθ = −By 2A e−fθ = −1 2A ( fy ( A− 1 4β ) +Ay ) (3.18) and therefore qe−fθ + refθ = fy 4Aβ − Ay A . (3.19) Due to the above relation and identity (3.15) equation (3.16) becomes ϕs = − fs 2λ ϕy + 1 λ ϕ ( − 1 4β ( 1− fsfy 4A ) − Ay A fs 4 ) . (3.20) Taking derivative of (3.9) with respect to s we find 1 2 fsy = Cefθ +Be−fθ + 1 2 fs ( qe−fθ + refθ ) = − 1 2β + fsfy 8Aβ − fsAy 2A . (3.21) Thus equation (3.20) becomes ϕs = − fs 2λ ϕy + fsy 4λ ϕ. (3.22) We now turn our attention to equation (3.8). The last term containing η vanishes due to the identity (3.9). In addition it holds that fsy 2fs + 1 2βfs = −1 2 fy cos(2θ) + 1 2 ( qe−f + ref ) sin(2θ) = 1 2 ( qe−fθ + refθ ) (3.23) as follows from relations (3.19) and (3.21). Also, it holds from relations (3.17)–(3.18) that for 0 < θ < π/2: rq = ( fsy 2fs + 1 2βfs )2 − 1 4 f2 y = g2 − f2 y /4, (3.24) where g = fsy 2fs + 1 2βfs . (3.25) Thus, the remaining constant (the ones which do not contain λ) terms on the right hand side of equation (3.8) are equal to 1 4 f2 y + qr + ( −1 2 fy cos(2θ) + 1 2 (qe−f + ref ) sin(2θ) ) y = 1 4 f2 y + qr + 1 2 ( qe−fθ + refθ ) y = g2 + gy. (3.26) Therefore, we can write equation (3.8) as: ϕyy = ( λ2 − λfy −Q ) ϕ, Q = −g2 − gy (3.27) with g given by (3.25). The above spectral problem together with equation (3.22) ensures via compatibility condition ϕyys − ϕsyy = 0, that Qs + 1 2 fyyfs + fyfsy = 0 (3.28) holds. The latter is equivalent to the two-component Camassa–Holm equation (2.19). On a Negative Flow of the AKNS Hierarchy 9 4 The θ = 0 case and Bäcklund transformation between different solutions We now consider θ at the boundary of the 0 < θ < π/2 interval. For illustration we take θ = 0, the remaining case θ = π/2 can be analyzed in a similar way. Plugging θ = 0 into relation (3.26) we obtain rq\|θ=0 = −1 4 f2 y + 1 2 fyy + g2 + gy = g2 − 1 4 f2 y + ( 1 2 fy + g ) y . Comparing with relation (3.24) we get rq\|θ=0 = rq\|θ + ( 1 2 fy + g ) y (4.1) which describes a relation between the product rq for zero and non-zero values of the angle θ, with rq\|θ being associated with θ within an interval 0 < θ < π/2. Recall that q = exp(f) for θ = 0. It follows that A = qsy/4q = (fsy + fsfy)/4 and equa- tion (2.7) is equivalent to (rq\|θ=0)s = 1 2 (fsy + fsfy)y. (4.2) On the other hand, it follows from (2.17) and C = 1/(16β2B)−A2/B that rq\|θ=0 = 1 2 ( fyy − 1 2 f2 y − f2 sy 2f2 s + 1 2β2f2 s + fsyy fs ) and accordingly equation (4.2) is equivalent to the two-component Camassa–Holm equation (2.19). From (3.18) one finds for 0 < θ < π/2 that: q = P−(fθ)efθ , (4.3) where P±(f) = ±1 2 fy + g = ±fy 2 + fsy 2fs + 1 2βfs . Obviously P±(fθ) = P±(f). We are now ready to show that f̄ = fθ + ln (P−(fθ)) = fθ + ln ( − fθ y 2 + fθ sy 2fθ s + 1 2βfθ s ) satisfies the two-component Camassa–Holm equation (2.19) for any f or fθ, which satisfies equation (2.19). For 0 < θ < π/2, it holds that q = exp(f̄) and therefore A = qsy/4q = (f̄sy + f̄sf̄y)/4 = (fsy + fsfy)/4 + fsP− y + P− ys + P− sfy 4P− . (4.4) We will now show that (rq\|θ)s = (rq\|θ=0)s − ( 1 2 fy + g ) ys = 1 2 (fsy + fsfy)y + ( fsP−y + P−ys + P−sfy 2P− ) y . (4.5) 10 H. Aratyn, J.F. Gomes and A.H. Zimerman Using equation (4.2) one can easily show that equation (4.5) holds if the following relation − (fy + P−)s = fsP− y + P− ys + P− sfy 2P− is true. We note that the above relation can be rewritten as (P2 −)s + 2fysP− + fsP− y + P− sy + P− sfy = 0 . The last equation is fully equivalent to the two-component Camassa–Holm equation (3.28) as can be seen by rewriting Q from relation (3.27) as Q = −(P− + fy/2)2 − (P− + fy/2)y. This completes the proof for relation (4.5). It follows from (2.17) and C = 1/(16β2B)−A2/B that rq\|θ = 1 2 ( f̄yy − 1 2 f̄2 y − f̄2 sy 2f̄2 s + 1 2β2f̄2 s + f̄syy f̄s ) . Thus, due to (4.4) and (4.5) we have proved explicitly that f̄ = f + ln ( tan θ ( −fy 2 + fsy 2fs + 1 2βfs )) = fθ + lnP−(fθ) (4.6) is a solution of a 2-component version of the Camassa–Holm equation. Thus the transformation f → f̄ maps a solution f of a 2-component version of the Camassa–Holm equation to a different solution f̄ . For example, let us consider, as in [21], the Camassa–Holm function: f(y, s) = ln a (1) 1 a (1) 2 z1e s 2z1 +2yz1 + a (2) 1 a (2) 2 z2e s 2z2 +2yz2 (z2 − z1)a (2) 1 a (1) 2 , (4.7) where a(j) i , i, j = 1, 2 and z1 and z2 are constants. The function f solves equation (2.19) for β2 = 1. Then, as an explicit calculation verifies, the map f → f̄ with f̄ given by expression (4.6) yields another solution of equation (2.19) for β2 = 1 and θ 6= 0. For θ = π/2 we have r = exp(−f) and comparing with the result for 0 < θ < π/2: r = P+(fθ)e−fθ , (4.8) we get a Bäcklund transformation f → fθ − ln (P+(fθ)) = fθ − ln ( fθy 2 + fθsy 2fθs + 1 2βfθs ) . Additional Bäcklund transformations can be obtained by comparing expressions for q and r variables in terms of f for the boundary values of θ. We first turn our attention to the case of θ = 0 for which we have q = exp(f) and r = 1 2 ( fyy − 1 2 f2 y − f2 sy 2f2 s + 1 2β2f2 s + fsyy fs ) e−f = ( P2 + − P+fy + P+y ) e−f . (4.9) From the AKNS equation (2.18) we see immediately that f = ln q must satisfy the 2-component Camassa–Holm equation (2.19). Note, in addition, that the AKNS equation (2.18) is still valid if we replace q by r and therefore f − ln ( P2 + − P+fy + P+y ) must satisfy the 2-component Camassa–Holm equation (2.19) as well. On a Negative Flow of the AKNS Hierarchy 11 Next, for θ = π/2 we have r = exp(−f) and q = 1 2 ( −fyy − 1 2 f2 y − f2 sy 2f2 s + 1 2β2f2 s + fsyy fs ) ef = ( P2 − + P−fy + P−y ) ef . (4.10) Comparing expressions for q and r we find find that if f is a solution of the 2-component Camassa–Holm equation (2.19) then so is also f + ln ( P2 − + P−fy + P−y ) . To summarize we found the following Bäcklund maps f → { fθ ± ln (P∓(fθ)) , fθ = f + const, f ± ln ( P2 ∓ ± P∓fy + P∓y ) . The top row lists maps between θ = 0, π/2 cases and θ within the interval 0 < θ < π/2 [20]. The bottom row shows new maps derived for the θ = 0 and π/2 cases only. 5 Conclusions These notes describe an attempt to construct a general and universal formalism which would realize possible connections between the 2-component Camassa–Holm equation and AKNS hie- rarchy extended by a negative flow. Construction yields gauge copies of an extended AKNS model connected by a continuous parameter (angle) θ taking values in an interval 0 ≤ θ ≤ π/2. Eliminating one of two components of the sl(2) wave function gives a second order non-linear partial differential equation for a single function f of the two-component Camassa–Holm model. Functions f corresponding to different values of θ in an interior of interval 0 ≤ θ ≤ π/2 differ only by a trivial constant and fall into a class considered in [11]. Two remaining and separate cases correspond to θ equal to 0 and π/2 and agree with a structure described in [20]. 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[21] Aratyn H., van de Leur J., Clifford algebra derivations of tau functions for two-dimensional integrable models with positive and negative flows, nlin.SI/0605027. http://arxiv.org/abs/patt-sol/9305002 http://arxiv.org/abs/nlin.SI/0501028 http://arxiv.org/abs/math.DG/0405146 http://arxiv.org/abs/nlin.SI/0505059 http://arxiv.org/abs/nlin.SI/0604008 http://arxiv.org/abs/nlin.SI/0601066 http://arxiv.org/abs/nlin.SI/0312043 http://arxiv.org/abs/nlin.si/0507062 http://arxiv.org/abs/nlin.SI/0605027 1 Introduction 2 A simple derivation of a relation between AKNS and two-component Camassa--Holm models 3 General reduction scheme from AKNS system to the two-component Camassa--Holm equation 4 The =0 case and Bäcklund transformation between different solutions 5 Conclusions

On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation

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