Integrable equations exhibit interesting conformal properties and can be
written in terms of the so-called conformal invariants. The most basic and
important example is the KdV equation and the corresponding Schwarz-KdV
equation. Other examples, including the Camassa-Holm equation and the
associated Camassa-Holm equation are investigated in this paper. It is shown
that the Baecklund transform is related to the conformal properties of these
equations. Some particular solutions of the Associated Camassa-Holm Equation
are discussed also.Comment: 13 pages, 1 figure. Physics Letters A (accepted

Beals

Camassa

Constantin

Depireux

Dickey

Dorfman

Fisher

Fokas

Galas

Gieres

Hille

Hone

Johnson

Lenells

Matveev

Misiolek

Nucci

Rasinariu

Rossen Ivanov

Schiff

Shabat

Weiss

Wilson

English

arXiv

Integrable equations exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants. The most basic and important example is the KdV equation and the corresponding Schwarz-KdV equation. Other examples, including the Camassa-Holm equation and the associated Camassa-Holm equation are investigated in this paper. It is shown that the B¨acklund transform is related to the conformal properties of these equations. Some particular solutions of the Associated Camassa-Holm Equation are discussed also

Ivanov, Rossen

Irish Universities

Conformal Properties and Baecklund Transform for the Associated Camassa-Holm Equation

Arrow@TUDublin

Technological University Dublin ARROW@TU Dublin Articles School of Mathematics 2005-01-01 Conformal Properties and Baecklund Transform for the Associated Camassa-Holm Equation Rossen Ivanov Technological University Dublin, rossen.ivanov@tudublin.ie Follow this and additional works at: https://arrow.tudublin.ie/scschmatart  Part of the Dynamic Systems Commons, Mathematics Commons, Non-linear Dynamics Commons, and the Partial Differential Equations Commons Recommended Citation R. Ivanov, Conformal Properties and Baecklund Transform for the Associated Camassa-Holm Equation, Phys. Lett. A 345, 2005, pp 235-243 This Article is brought to you for free and open access by the School of Mathematics at ARROW@TU Dublin. It has been accepted for inclusion in Articles by an authorized administrator of ARROW@TU Dublin. For more information, please contact yvonne.desmond@tudublin.ie, arrow.admin@tudublin.ie, brian.widdis@tudublin.ie. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License arXiv:nlin/0507005v1  [nlin.SI]  3 Jul 2005Conformal Properties and Ba¨cklundTransform for the AssociatedCamassa-Holm EquationRossen Ivanov∗1School of Mathematics, Trinity College,Dublin 2, IrelandTel: + 353 - 1 - 608 2898Fax: + 353 - 1- 608 2282∗e-mail: ivanovr@maths.tcd.ieAbstractIntegrable equations exhibit interesting conformal properties and canbe written in terms of the so-called conformal invariants. The most ba-sic and important example is the KdV equation and the correspondingSchwarz-KdV equation. Other examples, including the Camassa-Holmequation and the associated Camassa-Holm equation are investigatedin this paper. It is shown that the Ba¨cklund transform is related tothe conformal properties of these equations. Some particular solutionsof the Associated Camassa-Holm Equation are discussed also.PACS: 05.45.Yv, 02.30.IkKey Words: Schwarz Derivative, Conformal Invariants, Lax Pair,Integrable Systems, Solitons, Positons, Negatons.1On leave from the Institute for Nuclear Research and Nuclear Energy, BulgarianAcademy of Sciences, Sofia, Bulgaria.11 IntroductionIntegrable equations exhibit many extraordinary properties like infinitelymany conservation laws, multi- Hamiltonian structures, soliton solutionsetc. Many integrable equations in 1+1 dimensions like KdV, MKdV, Harry-Dym, Boussinesq equations possess interesting conformal properties as well[1, 2, 3, 4, 5]. They can be written in terms of the so-called independentconformal invariants of the function φ = φ(x, t):p1 =φtφxp2 = {φ;x} ≡ φxxxφx− 32φ2xxφ2x(1)Here {φ;x} denotes the Schwarz derivative. A quantity is called conformallyinvariant if it is invariant under the Mo¨bius transformationφ → αφ+ βγφ+ δ, αδ 6= βγ. (2)For example, the KdV equationut + auxxx + 3uux = 0 (3)(a is a constant) can be written in a Schwarzian form, i.e. in terms of theconformal invariants (1) as p1 + ap2 = 0 orφtφx+ a{φ;x} = 0 (4)whereu = a{φ;x}. (5)The KdV and Camassa-Holm (CH) [6] equations arise also as equationsof the geodesic flow for the L2 and H1 metrics correspondingly on the Bott-Virasoro group [7, 8, 9]. The conformal properties of these equations andtheir link to the Bott-Virasoro group originate from the Hamiltonian oper-atorD3 ≡ a∂3 + ∂u(x) + u(x)∂ (6)2where ∂ ≡ ∂/∂x. D3 is also known as the third Bol operator [10] and is theconformally covariant version of the differential operator ∂3, see also [11].D3 defines a Poisson bracket [1]{f, g} = 12pii∫ ((a∂3 + ∂u(x) + u(x)∂)δfδu).δgδudx (7)and KdV, CH and many other equations [12] can be written in the formut = {u,H} (8)for some Hamiltonian H. On the other hand, suppose for simplicity that uis 2pi periodic in x, i.e.u(x) =∞∑−∞Lneinx − a2(9)then the Fourier coefficients Ln close a classical Virasoro algebra with respectto the Poisson bracket (7):{Ln, Lm} = (n −m)Ln+m + a(n3 − n)δn+m,0 (10)Therefore, it is natural to expect that equations which can be written in theform (8) exibit interesting conformal properties, and this is the case indeed[5, 11, 13, 14, 15]. In what follows we will concentrate on the Camassa-Holmand the associated Camassa-Holm equation.2 The Camassa-Holm equation in Schwarzian formThe Camassa-Holm equation [6, 16]ut − uxxt + auxxx + 3uux − 2uxuxx − uuxxx = 0, (11)describes the unidirectional propagation of shallow water waves over a flatbottom [6, 17]. CH is a completely integrable equation [18, 19, 20, 21],describing permanent and breaking waves [22, 23]. Its solitary waves arestable solitons [24, 25, 26, 27]. CH arises also as an equation of the geodesicflow for the H1 metrics on the Bott-Virasoro group [7, 8, 9]. The equation(11) admits a Lax pair [6]Vxx =(14− λ(m+ a2))V (12)Vt = −( 12λ+ u− a)Vx +ux2V (13)3wherem = u− uxx. (14)In order to find the Schwarz form for the CH equation we proceed as follows.Let V1 and V2 be two linearly independent solutions of the system (12), (13)and let us defineφ =V2V1(15)Then, from (13) it follows thatφtφx= −u+(a− 12λ)(16)According to the Theorem 10.1.1 from [28] due to (12) we also have{φ;x} = 2λm+ aλ− 12(17)From (14), (16) and (17) we obtain the Schwarz-Camassa-Holm (S-CH)equation:(1− ∂2) φtφx+12λ{φ;x} = 32a− 34λ(18)Since λ is an arbitrary constant, one can take λ = 1/2a (if a 6= 0) and thenthe S-CH equation (18) acquires the form (1− ∂2)p1 + ap2 = 0 or(1− ∂2) φtφx+ a{φ;x} = 0 (19)Applying the hodograph transformx→ φ, t→ t, φ→ x (20)to the S-CH (19) and using the transformation properties of the Schwarzianderivative [28]{φ;x} = −φ2x{x;φ} (21)we obtain the following integrable deformation of the Harry Dym equationfor the variable v = 1/xφ:4vt + v2[v(v∂−1φ (v−1)t)φ]φ = av3vφφφThe equations (11) and (19) with u = − φtφx are not equivalent – as amatter of fact (19) implies (11), cf. [3]. It is often convenient to think thatthe Lax operator belongs to some Lie algebra, and the corresponding Jostsolution- to the corresponding group. Thus the relation between u and φ (see(15)) resembles the relation between the Lie group and the correspondingLie algebra, as pointed out in [3]. More precisely, the following propositionholds:Proposition 1 Let φ be a solution of (18). Then V1 = φ−1/2x and V2 =φφ−1/2x are solutions of (12), (13) with u = − φtφx − ( 12λ − a)Proof : It follows easily by a direct computation.Note that the Proposition 1 is consistent with (15). From Proposition 1it followsProposition 2 The general solution of (12), (13) isV =Aφ+B√φx(22)where A and B are two arbitrary constants, not simultaneously zero.Note that the expression (22) is covariant with respect to the Mo¨biustransformation (2), i.e. under (2), the expression (22) transforms into anexpression of the same form but with constantsA→ A′ = αA+ γB√αδ − βγ , B → B′ =βA+ δB√αδ − βγ . (23)3 The associated Camassa-Holm equations and theBa¨cklund TransformAn inverse scattering method, which can be applied directly to the spectralproblem (12) is not developed completely yet. However, the solutions of theCH (in parametric form) can be obtained by an implicit change of variables,which maps (12) to the well known spectral problem of the KdV hierarchy.This change of variables leads to the so-called Associated Camassa-Holmequation (ACH) [29]pt′ = p2fx′ , f =p4(log p)x′t′ − p22. (24)5Indeed, the ACH is related to the CH equation (11) through the followingchange of variables:p =√m+a2, f = −u2− a4(25)dx′ =12pdx+ (−u2+a2)pdt, dt′ = dt. (26)In what follows we will omit the dash from the new variables x′ and t′ andwe will write simply x and t instead.The ACH equation (24) is related to the KdV hierarchy [30, 31] since itmay be written in the formUt = −2px, U = −12(ppxx − 12p2x + 2p2)(27)and the Lax pair for (27) isVxx +(U − 1λ)V = 0, (28)Vt = λ(pVx − 12pxV). (29)The second part of (27) is the Ermakov-Pinney equation for p (for a knownU) [31, 20, 27, 32]. In order to obtain the Schwarzian form of the ACHequation, we introduce again φ = V2/V1 as in (15), where now V1 and V2are two linearly independent solutions of the system (28), (29). Then, byanalogous arguments, from (28) it followsU =12{φ;x} + 1λ, (30)and from (29)p =φtλφx. (31)Using the relation between p and U given in the first part of (27), (30) and(31) we obtain the Schwarz-ACH (S-ACH) equation p2,t + (4/λ)p1,x = 0, or{φ;x}t + 4λ( φtφx)x= 0. (32)Now one can easily check by a direct computation the validity of thefollowing6Proposition 3 Let φ be a solution of S-ACH (32). Then V1 = φ−1/2x andV2 = φφ−1/2x are solutions of (28), (29) withp =φtλφx. (33)As a corollary we find that the general solution of (28), (29), expressedthrough the solution of (32) isV =Aφ+B√φx(34)where A and B are two arbitrary constants, not simultaneously zero. Aswe know the expression (34) is covariant under the Mo¨bius transformations(2).Definition 1 A transformation, which connects the solution u of one equa-tion to the solution u˜ of the same (or another) equation is called a Ba¨cklundTransform (BT).Such type of transformations were first discovered in relation to the sine-Gordon equation by A. Ba¨cklund around 1883. Usually BT depends on anarbitrary parameter which is called Ba¨cklund parameter.The Lax pair (28), (29) has the following important property [31, 33],namely the BT for the ACH :Proposition 4 If V is a solution of the Lax pair (28), (29) with potentialsU and p, then V −1 is a solution of (28), (29) with potentialsU˜ = U + 2(log V )xx, (35)p˜ = p− (log V )xt. (36)From (34) and (36) it now follows that the BT for the solution (33) of theACH equation can be expressed through the solution of the S-ACH equationas follows:p˜ =φtλφx−(logAφ+B√φx)xt. (37)Since the constants A, B are not simultaneously zero, one can alwaysfactor out one of the two constants, i.e. (37) obviously depends only on theratio of these two constants (which is the Ba¨cklund parameter in this case).7We conclude this section with the following remark for the KdV equation.It is not difficult to see through similar considerations, that the Ba¨cklundTransform for the solution (5) of the KdV equation (3) isu˜ = a{φ;x} + 4a(logAφ+B√φx)xx, (38)where φ is the solution of the S-KdV equation (4). Then the choice A = 0in (38) corresponds to the ’exotic’ Ba¨cklund Transform found by Galas in[34] and mentioned in [35].4 On the solutions of the ACH and S-ACH equa-tionsThe solutions of the ACH and S-ACH equations can be constructed, follow-ing the scheme for the construction of the solutions of the KdV hierarchy[31, 36, 37, 38, 39, 40].Let v1, v2, . . . vn be n different solutions of the Lax pair (28), (29) withpotentials U and p, taken at λ = λ1, λ2, . . . λn respectively and let V1 andV2 be two linearly independent solutions for some value λ. Consider theWronskian determinantsW = W (v1, v1;1, . . . , v1;m1 , v2, v2;1, . . . , v2;m2 , . . . , vn, vn;1, . . . , vn;mn)Wk = W (v1, v1;1, . . . , v1;m1 , . . . , vn, vn;1, . . . , vn;mn , Vk), k = 1, 2,(39)where vi;l ≡ ∂lλvi, mi are arbitrary nonnegative integers and the Wronskiandeterminant of N functions ϕ1, ϕ2, . . . ϕN is defined byW (ϕ1, ϕ2, . . . ϕN ) = detA, Aij =di−1ϕjdxi−1, i, j = 1, 2, . . . , N. (40)The following generalization of the Crum theorem is valid for the KdV hi-erarchy and in particular for the ACH equation [37]:Proposition 5 V˜k = Wk/W , k = 1, 2 are two linearly independent solu-tions of the Lax pair (28), (29) with potentialsU˜ = U + 2(logW )xx, (41)p˜ = p− (logW )xt. (42)From Proposition 5 it follows immediately thatφ˜ =V˜2V˜1=W2W1, φ =V2V1(43)8are the solutions of the S-ACH (32), which correspond to the potentialsp˜ =φ˜tλφ˜x, p =φtλφx. (44)In addition to the solutions mentioned in [31] one may construct alsothe so-called negaton and positon solutions (which are singular) and mixedsoliton-positon-negaton solutions. For a detailed discussion on the positonand negaton solutions of KdV we refer to [40].Let us return to (28) with p = h = const, i.e. U = −1/h2. Then it hasthe formVxx +E(h, λ)V = 0, E(h, λ) = −( 1h2+1λ). (45)The type of the solutions of (45) clearly depends on the sign of the constantE(h, λ). If E < 0 the independent solutions of (45) and (29) arev1(λ) = cosh(√ 1λ+1h2(x− hλt + x1(λ))), (46)v2(λ) = sinh(√ 1λ+1h2(x− hλt + x2(λ))), (47)where x1,2(λ) are arbitrary functions. The simplest choice for W in (41),(42) is W = v1 or W = v2 (Wronskian of order one), which corresponds tothe one-soliton solution. The case W = W (v1(λ1), v2(λ2)) corresponds tothe two-soliton solution, i.e. this solution describes the interaction of twosolitons. However, according to (39) it is possible to merge two or moreidentical solitons, takingW = W (v1, v1;1, . . . , v1;m1), (48)orW = W (v2, v2;1, . . . , v2;m2). (49)In physical terms E(h, λ) is the energy of the soliton. Since the energyof the solitons is negative, (48) is called negaton of type C and order m1,and (49) is called negaton of type S and order m2, cf. [40]. The analogousconsiderations resulting from the following solutions of (45),v1(λ) = cos(√ 1λ− 1h2(x− hλt + x1(λ))), (50)v2(λ) = sin(√ 1λ− 1h2(x− hλt + x2(λ)))(51)9(0 < λ ≤ h2), whose energy is positive, E > 0 leads naturally to thedefinition of positon of type C and order m1 via (48) and (42), and positonof type S and order m2, (49) and (42); cf. [40].Due to (39) it is possible to construct solutions, which describe the inter-action between solitons, positons and negatons. I.e. the interaction betweenn positons (of orders m1,m2, . . . ,mn) and N solitons is given by the n-positon – N soliton solution npN of the ACH (24) which can be constructedfrom (42) taking the following solutions of the Lax pair (28), (29):vi = cos(√ 1λi− 1h2(x− hλit+ yi(λi))), i = 1, . . . n (52)ηj = exp(√ 1λn+j+1h2(x− hλn+jt+ xj))+ (53)cj exp(−√1λn+j+1h2(x− hλn+jt+ xj)), j = 1, . . . N, (54)where yi(λ) are arbitrary functions of λ, cj , xj are arbitrary constants and0 < λi ≤ h2 for i = 1, . . . , n. Then (cf. [39])npN (x, t) =h −(logW (v1, v1;1, . . . , v1;m1 , . . . , vn, vn;1, . . . , vn;mn , η1, . . . , ηN ))xt.(55)As an example, we provide the one-positon solution of order one for theACH, see also Fig. 1:1p0(x, t) = h−8h(h2 − λ1)(λ1 + λ1 cos z(x, t) + r(x, t) sin z(x, t))(2r(x, t) + h2 sin z(x, t))2 (56)where ω =√1λ1− 1h2, y1 = aλ, a is an arbitrary constant;z(x, t) = 2ω(x− hλ1t+ aλ1),r(x, t) = ω(h2x+ λ1(h2 − 2λ1)(ht− a)).5 AcknowledgementsThe author is grateful to Prof. A. Constantin for helpful discussions and tothe referees for valuable suggestions. The author also acknowledges fundingfrom the Science Foundation Ireland, Grant 04/BR6/M0042.10References[1] L. Dickey, Soliton Equations and Hamiltonian Systems, World Scien-tific, New Jersey, 2003.[2] I. Ya. Dorfman, Phys. Lett. A 125 (1987) 240.[3] G. Wilson, Phys. Lett. A 132 (1988) 445.[4] D. A. Depireux and J. Schiff, Lett. Math. Phys. 33 (1995) 99.[5] S.-y. Lou, X. Tang, Q.-P. Liu and T. Fukuyama, Z. 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Shabat, J. Nonlin. Math. Phys. 12 (2005) 614.[33] A. N. W. Hone, V. B. Kuznetsov and O. Ragnisco, J. Phys. A: Math.Gen. 32 (1999) L299.[34] F. Galas, J. Phys. A: Math. Gen. 25 (1992) L981.[35] J. Schiff, preprint solv-int/9606004 (1996).[36] V. Matveev and M. Salle, Darboux Transformations and Solitons,Springer-Verlag, Berlin-Heidelberg, 1991.[37] V. Matveev, Phys. Lett. A 166 (1992) 205; erratum, ibid 168 (1993)463.[38] V. Matveev, Phys. Lett. A 166 (1992) 209.[39] V. Matveev, J. Math. Phys. 35 (1994) 2955.[40] C. Rasinariu, U. Sukhatme and A. Khare, J. Phys. A: Math. Gen. 29(1996) 1803.12−20 −15 −10 −5 0 5 10 15 20−10−5051015x1p0(x)Figure 1: One-positon solution of the ACH equation, a = 0, h = 2, λ1 = 1:t = 0 – solid line; t = −4 – dashed line; t = 4 – dotted line.13

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Conformal properties and Bäcklund transform for the associated Camassa–Holm equation

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Conformal Properties and Baecklund Transform for the Associated
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Conformal Properties and Baecklund Transform for the Associated Camassa-Holm Equation

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