Miura-reciprocal transformations for non-isospectral Camassa-Holm hierarchies in 2+1 dimensions
aa r X i v : . [ m a t h - ph ] J un Miura-reciprocal transformationsfor non-isospectral Camassa-Holm hierarchiesin dimensions
P.G. Est´evez and C. Sard´onDepartment of Fundamental Physics, University of Salamanca,Plza. de la Merced s/n, 37.008, Salamanca, Spain.
Abstract
We present two hierarchies of partial differential equations in 2 + 1dimensions. Since there exist reciprocal transformations that connectthese hierarchies to the Calogero-Bogoyavlenski-Schiff equation andits modified version, we can prove that one of the hierarchies can beconsidered as a modified version of the other. The connection betweenthem can be achieved by means of a combination of reciprocal andMiura transformations.
Is there any way to decide whether an integrable nonlinear equation is ac-tually new, or merely a disguised form of another member of the class ofthe integrable zoo? The question is far from being trivial, given the amountof papers published on the subject and the number of different names of-ten given to the same equation in the current literature. A clear procedurehas to be found in order to relate the allegedly new equation to the rest ofits integrable siblings. This letter is an attempt to shed some light on thesolution of this seemingly simple, but nonetheless crucial, question.According to the Painlev´e criteria [19], a nonlinear PDE is said to beintegrable when its solutions turn out to be single valued in a neighborhoodof the movable singularities manifold. As is well known, the existence ofthis property can be checked through an algorithmic procedure which isvalid both in the case of nonlinear ordinary or partial differential equations[28]. For this so called Painlev´e property, a method has been developedby Weiss [29] that allows us to determine the set of solutions that giverise to a truncated Painlev´e expansion. In this situation, one can showthat the manifold of movable singularities should satisfy a set of equationsnamed singular manifold equations (SME). Long experience based on1any already worked examples by the present and other authors [5], [6], [7]indicates that SME could be the canonical form that identifies an integrablepartial differential equation (PDE). If this were the case, two seeminglydifferent integrable PDEs with the same set of SME must be related throughsome transformation. The remaining point is the nature and identificationof such a transformation.There is a caveat in this project that we shall deal with first. Thestandard form of the Painlev´e test cannot be applied to some interestingPDEs. Therefore one cannot be sure of whether in those cases the equationis or is not single-valued in the initial conditions. This is the case [13],for instance, of the celebrated Camassa-Holm equation [4]. This equationhas been known to be integrable for some time and it has an associatedlinear problem. Therefore, discarding any kind of pathology, one should beable to write down a transformation that brings it into a form in which thePainlev´e test can be applied. This is the case of the relation between theHarry-Dym and Korteweg de Vries equation that appears in [16]. Reciprocaltransformations for peakons equations like Camassa-Holm and Degasperis-Procesi were derived in [15] to transform these equations to equations withthe Painlev´e property. Actually, reciprocal transformations have provedto be a powerful tool to relate different equations and conservation laws[11], [12], [23], [24]. In chapter 3 of reference [22], reciprocal relations forgasdynamics equations as transformations of B¨acklund type were derived.For equations in 2 + 1 dimensions, it was proved in [26] that there existsa reciprocal link between the Harry-Dym equation in 2 + 1 dimensions andthe singular manifold equations [29] of the Kadomtsev-Petviashvili equation.Reciprocal transformations for 2 + 1 shallow water equations were identifiedin [14]. In reference [8] we introduced an integrable generalization to 2 + 1dimensions of the Camassa-Holm hierarchy (CH2+1). By using reciprocaltransformations, we were able to show explicitly that the n-component of thehierarchy was in fact equivalent to n copies of the Calogero-Bogoyavlenski-Schiff (CBS) equation [1], [18]. This CBS equation possesses the Painlev´eproperty and the singular manifold method can be applied to obtain its Laxpair and other relevant properties [8].It is tempting to assume that these and similar results appearing inother recent publications [9] are all but coincidence. The reciprocal trans-formations should be a useful method to transform equations in which thePainlev´e test cannot be applied to other fully tractable equations throughthe singular manifold method (SMM).To convince the reader that this conjecture has some grounds, we haverecently presented another non-trivial example of the procedure. Qiao has2eported an integrable equation [21] for which the Painlev´e test is neitherapplicable nor constructive. This equation is the second member of a hi-erarchy in 1 + 1 dimensions [20] that one of us has recently generalized to2 + 1 dimensions [10]. One can prove that a reciprocal transformation existswhich, as in the CH2+1 case, allows us to transform the n-component ofthe hierarchy into n copies of the modified Calogero-Bogoyavlenskii-Schiff(mCBS), which is known to have the Painlev´e property [7]. The last stepcan be shown by transforming mCBS into CBS by means of a Miura trans-formation. The singular manifold equations are the same for both sets ofequations. We shall denote in the following the hierarchy introduced in [10]as mCH(2+1) because it can be considered as a modified version of theCH(2+1) hierarchy introduced in [8].To summarize: CH(2+1) and mCH(2+1) are directly related to CBS andmCBS respectively through two different sets of reciprocal transformations.Aside from this property, there exists a Miura transformation relating CBSand mCBS. The relationship between these two hierarchies must necessarilyinclude both Miura and reciprocal transformations. Therefore, for obviousreasons, the name Miura-reciprocal transformations will be used extensivelythroughout the paper and they will be the subject of it.
In this section we shall briefly summarize and improve the results of [8]and [10] in order to establish the reciprocal transformation that connectsthe CH(2+1) and mCH(2+1) hierarchies with CBS and mCBS respectively.Many details (especially those referring to the detailed calculation) are omit-ted and can be obtained in the above cited references.It could be useful from the beginning to say that we shall use capital let-ters for the dependent and independent variables connected with CH(2+1).Henceforth, lower case letters will be used for mCH(2+1).
The CH(2+1) hierarchy can be written in a compact form as: U T = R − n U Y , (1)where R is the recursion operator defined as: R = J K − , K = ∂ XXX − ∂ X , J = −
12 ( ∂ X U + U ∂ X ) , where ∂ X = ∂∂X . (2)3ote that the factor − in the definition of J is not essential and it hasbeen introduced to make the later identification between the time variableseasier.This hierarchy was introduced in [8] as a generalization of the Camassa-Holm hierarchy. The recursion operator is the same as in the 1 + 1 di-mensional Camassa-Holm hierarchy. From this point of view, the spectralproblem is the same [3] and the Y -variable is just another “time” variable[14], [17].The n component of this hierarchy can also be written as a set of PDEsby introducing n dependent fields Ω [ i ] , ( i = 1 . . . n ) in the following way U Y = J Ω [1] J Ω [ i +1] = K Ω [ i ] , i = 1 . . . n − ,U T = K Ω [ n ] , (3)and by introducing two new fields, P and ∆, related to U as: U = P , P T = ∆ X , (4)we can write the hierarchy in the form of the following set of equations P Y = − (cid:16) P Ω [1] (cid:17) X , Ω [ i ] XXX − Ω [ i ] X = − P (cid:16) P Ω [ i +1] (cid:17) X , i = 1 . . . n − ,P T = Ω [ n ] XXX − Ω [ n ] X P = ∆ X . (5)It was shown in [8] that (2.1) can be reduced to the the negative Camassa-Holm hierarchy under the reduction ∂∂t = 0. The positive flow can be ob-tained under the reduction ∂∂x = ∂∂y . The conservative form of the first twoequations allows us to define the following exact derivative dz = P dX − P Ω [1] dY + ∆ dT. (6)A reciprocal transformation [14], [25], [26] can be introduced by consideringthe former independent variable X as a field depending on z , z = Y and z n +1 = T . From (6) we have dX = 1 P dz + Ω [1] dz − ∆ P dz n +1 ,Y = z , T = z n +1 , (7)4nd therefore X = 1 P ,X = Ω [1] ,X n +1 = − ∆ P , (8)where X i = ∂X∂z i . We can now extend the transformation by introducing anew independent variable z i for each field Ω [ i ] by generalizing (8) as X i = Ω [ i ] , i = 1 . . . n. (9)Therefore, the new field X = X ( z , z , . . . z n , z n +1 ) depends on n + 2 inde-pendent variables, where each of the former dependent fields Ω i , ( i = 1 . . . n )allows us to define a new dependent variable z i through definition (9). Itrequires some calculation (see [8] for details) but it can be proved that thereciprocal transformation (7)-(9) transforms (5) to the following set of n PDEs: − (cid:18) X i +1 X (cid:19) = (cid:20) X , X + X (cid:21) − (cid:20) X , X + X (cid:21) ! i , i = 1 . . . n. (10)Note that each equation depends on only three variables z , z i , z i +1 . This re-sult generalizes the one found in [14] for the first component of the hierarchy.The conservative form of (10) allows us to define a field M ( z , z , . . . z n +1 )such that M i = − (cid:18) X i +1 X (cid:19) , i = 1 . . . n,M = 14 (cid:20) X , X + X (cid:21) − (cid:20) X , X + X (cid:21) ! . (11)It is easy to prove that each M i should satisfy the following CBS equation[1], [2]: M ,i +1 + M , , ,i + 4 M i M , + 8 M M ,i = 0 , i = 1 . . . n. (12) The CBS equation has the Painlev´e property [18] and the SMM canbe successfully used to derive its Lax pair [7]. In [8] it was proved that5he Lax pair of CBS yields the following spectral problem for the CH(2+1)hierarchy (3) Φ XX + 14 ( λU −
1) Φ = 0 , Φ T − λ n Φ Y − λ C Φ X + λ C X Φ = 0 . (13)where C = n X i =1 λ n − i Ω [ i ] and λ ( Y, T ) is a non-isospectral parameter that satisfies λ X = 0 , λ T − λ n λ Y = 0 . (14)Consequently the problems that we meet when we try to apply the Painlev´etest to CH(2+1) [13] can be solved owing to the existence of a reciprocaltransformation that transforms the CH(2+1) hierarchy to n copies of theCBS equation, for which the Painlev´e methods are applicable. In [10], one of us introduced the following 2 + 1 hierarchy (mCH(2+1) inwhat follows) u t = r − n u y , (15)where r is the recursion operator, defined as: r = jk − , k = ∂ xxx − ∂ x , j = − ∂ x u ( ∂ x ) − u ∂ x . (16)where ∂ x = ∂∂x . This hierarchy generalizes the one introduced by Qiao in[20], whose second positive member was studied in [21]. We shall brieflysummarize the results of [10] when a procedure similar to that describedabove for CH(2+1) is applied to mCH(2+1).If we introduce 2 n auxiliary fields v [ i ] , ω [ i ] defined through u y = jv [1] ,jv [ i +1] = kv [ i ] , ω [ i ] x = uv [ i ] x , i = 1 . . . n − ,u t = kv [ n ] , (17)6he hierarchy can be written as the system: u y = − (cid:16) uω [1] (cid:17) x ,v [ i ] xxx − v [ i ] x = − (cid:16) uω [ i +1] (cid:17) x , i = 1 . . . n − ,u t = v [ n ] xxx − v [ n ] x = δ x , (18)which allows to define the exact derivative dz = u dx − uω [1] dy + δ dt (19)and z = y, z n +1 = t . We can define a reciprocal transformation such thatthe former independent variable x is a new field x = x ( z , z , . . . . . . z n +1 )depending on n + 2 variables in the form x = 1 u ,x i = ω [ i ] ,x n +1 = − δu . (20)The transformation of the equations (18) yields the system of equations (cid:18) x i +1 x + x i, , x (cid:19) = (cid:18) x (cid:19) i , i = 1 . . . n. (21)Note that each equation depends on only three variables: z , z i , z i +1 .The conservative form of (21) allows us to define a field m = m ( z , z , . . . z n +1 )such that m = x , m i = x i +1 x + x i, , x , i = 1 . . . n. (22)Equation (21) has been extensively studied from the point of view of Painlev´eanalysis [7] and it can be considered as the modified version of the CBSequation (12). Actually, in [7] it was proved that the Miura transformationthat relates (12) and (22) is: 4 M = x − m (23)A non-isospectral Lax pair was obtained for (22) in [7]. By inverting thisLax pair through the reciprocal transformation (20) the following spectralproblem was obtained for mCH(2+1). This Lax pair reads [7]:7 φ ˆ φ (cid:19) x = 12 (cid:18) − I √ λuI √ λu (cid:19) (cid:18) φ ˆ φ (cid:19) , (cid:18) φ ˆ φ (cid:19) t = λ n (cid:18) φ ˆ φ (cid:19) y + λa (cid:18) φ ˆ φ (cid:19) x ++ I √ λ (cid:18) b xx − b x b xx + b x (cid:19) (cid:18) φ ˆ φ (cid:19) . (24)where a = n X i =1 λ n − i ω [ i ] , b = n X i =1 λ n − i v [ i ] , I = √− λ ( y, t ) is a non-isospectral parameter that satisfies λ x = 0 , λ t − λ n λ y = 0 . (25)Although the Painlev´e test cannot be applied to mCH(2+1), reciprocaltransformations are a tool that can be used to write the hierarchy as a setof mCBS equation to which the Painlev´e analysis (the SMM in particular)can be successfully applied. As stated in the previous section, there are two reciprocal transformations(8) and (20) that relate CH(2+1) and mCH(2+1) hierarchies with CBS (12)and mCBS (22) respectively. Furthermore, it is known that a Miura trans-formation (23) relates CBS and mCBS. The natural question that arises iswhether the mCH(2+1) hierarchy can be considered as the modified versionof CH(2+1). Evidently the relationship between both hierarchies cannot bea simple Miura transformation because they are written in different vari-ables (
X, Y, T ) and ( x, y, t ). The answer is provided by the relationship ofboth sets of variables with the same set z , z , z n +1 . By combining (6) and(19) we have P dX − P Ω [1] dY + ∆ dT = u dx − uω [1] dy + δ dt,Y = y, T = t, (26)8hich yields the required relationship between the independent variables ofCH(2+1) and those of mCH(2+1). The Miura transformation (23), alsoaffords the following results4 M = x , − m = ⇒ X , X + X = x , M i = x ,i − m i = ⇒ − X i +1 X = x ,i − x , ,i x − x i +1 x , i = 1 . . . n, (27)where (11) and (22) have been used. With the aid of (8), (9) and (20), thefollowing results arise from (27) (see appendix)1 u = (cid:18) P (cid:19) X + 1 P ,P Ω i +1 = 2( v [ i ] − v [ i ] x ) , = ⇒ ω [ i +1] = Ω [ i +1] X + Ω [ i +1] , i = 1 . . . n −
1∆ = v [ n ] x − v [ n ] . (28)Furthermore, (26) can be written as: dx = (cid:20) − P X P (cid:21) dX + " ω [1] − Ω [1] (cid:18) − P X P (cid:19) dY + ∆ − δu dT. (29)The cross derivatives of (29) imply (see appendix) that: (cid:20) − P X P (cid:21) Y = " ω [1] − Ω [1] (cid:18) − P X P (cid:19) X = ⇒ ω [1] = Ω [1] X + Ω [1] , (cid:20) − P X P (cid:21) T = (cid:20) ∆ − δP (cid:18) − P X P (cid:19)(cid:21) X , = ⇒ δu = (cid:18) ∆ P (cid:19) X + ∆ P (30)and therefore with the aid of (30), (29) reads dx = (cid:20) − P X P (cid:21) dX − P Y P dY − P T P dT. (31)This exact derivative can be integrated as x = X − ln P. (32)By summarizing the above conclusions, we have proved that the mCH(2+1)hierarchy u t = r − n u y , u = u ( x, y, t ) , U T = R − n U Y , U = U ( X, Y, T ) . The transformation that connects the two hierarchies involves the reciprocaltransformation x = X −
12 ln U (33)as well as the following transformation between the fields1 u = 1 √ U (cid:18) − U X U (cid:19) , ⇓ ω [ i ] = Ω [ i ] X + Ω [ i ] , i = 1 . . . n,δu = (cid:18) ∆ √ U (cid:19) X + ∆ √ U . (34)
We are now restricted to the first component of the hierarchies n = 1 in thecase in which the field u is independent of y and U is independent of Y. • From (4) and (5), for the restriction of CH(2+1) we have U = P ,U T = Ω [1] XXX − Ω [1] X , ( P Ω [1] ) X = 0 . (35)which can be summarized asΩ [1] = k P = k √ U ,U T = k (cid:20)(cid:18) √ U (cid:19) XXX − (cid:18) √ U (cid:19) X (cid:21) (36)which is the Dym equation • The reduction of mCH(2+1) can be achieved from (17) and (18) inthe form ω [1] x = uv [1] x ,u t = v [1] xxx − v [1] x , (cid:16) uω [1] (cid:17) x = 0 , (37)10hich can be written as ω [1] = k u , = ⇒ v [1] = k u u t = k (cid:20)(cid:18) u (cid:19) xx − (cid:18) u (cid:19)(cid:21) x (38)which is the Qiao equation • From (29) and (33) it is easy to see that k = 2 k . By setting k = 1,we can conclude that the Qiao equation. u t = (cid:18) u (cid:19) xxx − (cid:18) u (cid:19) x is the modified version of the Dym equation U T = (cid:18) √ U (cid:19) XXX − (cid:18) √ U (cid:19) X • From (8) and (20), it is easy to see that the independence from y implies that X = X and x = x , which means that the CBS andmodified CBS equations (12) and (22) reduce to the following potentialversions of the Korteweg de Vries and modified Korteweg de Vriesequations (cid:0) M + M , , + 6 M (cid:1) = 0 ,x + x , , − x = 0 . If we are restricted to the n = 1 component when T = X and t = x , thefollowing results hold • From (4) and (5), for the restriction of CH(2+1) we have∆ = P = √ U ,U = Ω [1] XX − Ω [1] ,U Y + U Ω [1] X + 12 Ω [1] U X = 0 (39)which is the Camassa Holm equation equation.11 The reduction of mCH(2+1) can be obtained from From (17) and (18)in the form δ = u = v [1] xx − v [1] ,u y + ( uω [1] ) x = 0 ,ω [1] x − uv [1] x = 0 , (40)which can be considered as a modified Camassa-Holm equation. • From (8) and (20), it is easy to see that X = x = −
1. Therefore,the reductions of (12) and (21) are: M , , , + 4 M M , + 8 M M , = 0 , which is the AKNS equation and (cid:18) x , , − x (cid:19) = (cid:18) x (cid:19) , which is the modified AKNS equation. In the first part of this paper previous results concerning the CH(2+1) hi-erarchy for a field U ( X, Y, T ) and the mCH(2+1) hierarchy for u ( x, y, t ) isdiscussed. Reciprocal transformations that connect both hierarchies withthe CBS and mCBS are constructed. The main advantage of this method isthat these reciprocal transformations allow us to transform the hierarchiesinto a set of equations that can be studied through Painlev´e analysis andthe methods derived from it. In particular, the Lax pairs of both hierarchiescan be obtained in this way.The Miura transformation that connects CBS and modified CBS is thekey that allows us to establish a reciprocal transformation that relates thetwo fields U and u as well as the two sets of variables ( x, y, t ) and ( X, Y, T ).This reciprocal transformation is carefully constructed and allows us to provethat mCH(2+1) is a modified version of CH(2+1).As a particular case, the relationship between the two components of thehierarchies when they are independent of Y and y respectively is shown. Itshows that the Qiao equation is a modified Dym equation.In a similar way, we determine the modified Camassa Holm equationwhen the reduction T = X is applied to the first component ( n = 1) of theCH(2+1) hierarchy. 12 cknowledgements This research has been supported in part by the DGICYT under projectFIS2009-07880. We thank Professor J.M. Cerver´o for some interesting sug-gestions and a careful reading of the manuscript. We also thank the refereesfor their interesting suggestions and references.
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Rev. Math. Phys. (1993) 299-300.[28] J. Weiss, M. Tabor and G. Carnevale, The Painlev´e property for partialdifferential equations, J. Math. Phys (1983) 522-526.[29] J. Weiss, The Painlev´e property for partial differential equations II.B¨acklund Transformation, Lax pairs and the Schwartzian derivative, J.Math. Phys (1983) 1405-1413. Appendix • Method for obtaining equation (29)Equation (27) provides x = X + ∂ (ln X ) . If we use the fact that X = P and x = u we get obtain1 u = 1 P − ∂ (ln P ) , and by using (6) we have1 u = 1 P − P (ln P ) X which yields (29). • Method for obtaining equation (30)By taking i = 1 . . . n − − X i +1 X = x ,i − x , ,i x − x i +1 x , i = 1 . . . n − − P Ω [ i +1] ∂ ( ω [ i ] ) − u∂ ( ω [ i ] ) − uω [ i +1] , i = 1 . . . n − − P Ω [ i +1] ω [ i ] x u − ω [ i ] x u ! x − uω [ i +1] , i = 1 . . . n − ω [ i ] x = uv [ i ] x , uω [ i +1] = v [ i ] − v [ i ] xx , i = 1 . . . n − , the result is − P Ω [ i +1] v [ i ] x − v i , i = 1 . . . n − uω [ i +1] = v [ i ] − v [ i ] xx , i = 1 . . . n − uω [ i +1] = (cid:16) v [ i ] − v [ i ] x (cid:17) + (cid:16) v [ i ] x − v [ i ] xx (cid:17) = P Ω [ i +1] ! + P Ω [ i +1] ! x , i = 1 . . . n − ∂ x = uP ∂ X . Therefore, uω [ i +1] = P Ω [ i +1] ! + uP P Ω [ i +1] ! X , i = 1 . . . n − ω [ i +1] = P Ω [ i +1] u ! + 12 P (cid:16) P X Ω [ i +1] + P Ω [ i +1] X (cid:17) , i = 1 . . . n − u with the aid of (29). The result is ω [ i +1] = Ω [ i +1] X + Ω [ i +1] , i = 1 . . . n − . • Method for obtaining equation (31)By taking i = n in (28) we have − X n +1 X = x ,n − x , ,n x − x n +1 x . ∂ ( ω [ n ] ) − u∂ ( ω [ n ] ) + δ With the aid of (19), it reads∆ = ω [ n ] x u − ω [ n ] x u ! x + δ If ω [ n ] x = uv [ n ] x and δ = v [ n ] xx − v [ n ]]