Non-isospectral 1+1 hierarchies arising from a Camassa--Holm hierarchy in 2+1 dimensions
aa r X i v : . [ m a t h - ph ] A ug Nonisospectral hierarchies arising froma Camassa–Holm hierarchy in 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
The non-isospectral problem (Lax pair) associated with a hierar-chy in 2+1 dimensions that generalizes the well known Camassa-Holmhierarchy is presented. Here, we have investigated the non-classicalLie symmetries of this Lax pair when the spectral parameter is con-sidered as a field. These symmetries can be written in terms of fivearbitrary constants and three arbitrary functions. Different similarityreductions associated with these symmetries have been derived. Ofparticular interest are the reduced hierarchies whose 1 + 1 Lax pair isalso non-isospectral.
The identification of the Lie symmetries of a given partial differential equa-tion (PDE) is an instrument of primary importance in order to solve suchan equation [21]. A standard method for finding solutions of PDEs is thatof reduction using Lie symmetries: each Lie symmetry allows a reductionof the PDE to a new equation with the number of independent variablesreduced by one [6], [18]. To a certain extent this procedure gives rise to theARS conjecture [2], which establishes that a PDE is integrable in the senseof Painlev´e [19] if all its reductions pass the Painlev´e test [22]. This meansthat the solutions of a PDE can be achieved by solving its reductions toordinary differential equations (ODE). Classical [21] and non-classical [6],[18] Lie symmetries are the usual way for identifying the reductions.
Lax pair
A generalization to 2 + 1 dimensions of the celebrated Camassa-Holm hi-erarchy (henceforth CHn2+1) was presented in [10]. By using reciprocal1ransformations, this hierarchy was proved to be equivalent to n copies ofthe AKNS equation in 2+1 variables [9], [16]. It is well known that the 2+1AKNS equation has the Painlev´e property, and its non-isospectral Lax paircan be obtained by means of the singular manifold method [9]. Thereforewe can use the inverse reciprocal transformation to obtain the Lax pair ofCHn2+1 [10]. This Lax pair is also a non-isospectral one that can be writtenin terms of n + 1 fields as follows: ψ xx − (cid:18) − λ M (cid:19) ψ = 0 ψ y − λ n ψ t + ˆ A ψ x − ˆ A x ψ = 0 , (1)where ˆ A = n X j =1 λ ( n − j +1) U [ j ] (2)and M = M ( x, y, t ) , U [ j ] = U [ j ] ( x, y, t ) , j = 1 ...n. Non-isospectrality and equations
The compatibility condition between equations (1) yields the non-isospectralcondition λ y − λ n λ t = 0 , λ x = 0 , (3)as well as the equations M y = U [ n ] x − U [ n ] xxx M t = U [1] M x + 2 M U [1] x (4) U [ j ] M x + 2 M U [ j ] x = U [ j − x − U [ j − xxx , j = 2 ..n. Recursion operator and hierarchy
The above equations can be written in more compact form by defining theoperators: J = ∂∂x − ∂∂x , K = M ∂∂x + ∂∂x M. (5)Equations (2.4) are therefore: M y = J U [ n ] M t = KU [1] (6)2 U [ j ] = J U [ j − , j = 2 ..n, which yields the hierarchy: M y = R n M t (7)where the recursion operator is: R = J K − . (8)Solutions of these equations were studied in [11]. The positive and negative,[1], [5], 1 + 1 Camassa-Holm hierarchies can be obtained by setting ∂∂y = ∂∂x or ∂∂y = ∂∂t respectively [10].The n = − Lie point symmetries
Here, we are interested in the Lie symmetries of the Lax pair (1). Natu-rally, the symmetries of equations (2.4) are interesting in themselves, but wealso wish to know how the eigenfunction and the spectral parametertransform under the action of a Lie symmetry . More precisely, wewish to know what these fields look like under the reduction associated witheach symmetry. This is why we shall proceed to write the infinitesimal Liepoint transformation of the variables and fields that appear in the spectralproblem (1). We have proved the benefits of such a procedure [17] in aprevious paper [12].In the present case, it is important to note that the spectral parameter λ ( y, t ) is not a constant, and therefore that it should be considered as anadditional field satisfying (3). This means that we are actually looking forthe Lie point symmetries of equations (1) together with (3).The infinitesimal form of the Lie point symmetry that we are consideringis: x ′ = x + ε ξ ( x, y, t, λ, ψ, M, U [ j ] ) + O ( ε ) y ′ = y + ε ξ ( x, y, t, λ, ψ, M, U [ j ] ) + O ( ε ) t ′ = t + ε ξ ( x, y, t, λ, ψ, M, U [ j ] ) + O ( ε ) (9)3 ′ = ψ + ε φ ( x, y, t, λ, ψ, M, U [ j ] ) + O ( ε ) λ ′ = λ + ε φ ( x, y, t, λ, ψ, M, U [ j ] ) + O ( ε ) M ′ = M + ε Θ ( x, y, t, λ, ψ, M, U [ j ] ) + O ( ε )( U [ i ] ) ′ = U [ i ] + ε Θ j ( x, y, t, λ, ψ, M, U [ j ] ) + O ( ε ) , i, j = 1 ..n. where ǫ is the group parameter. The associated Lie algebra of infinitesimalsymmetries is the set of vector fields of the form: X = ξ ∂∂x + ξ ∂∂y + ξ ∂∂t + φ ∂∂ψ + φ ∂∂λ + Θ ∂∂M + n X j =1 Θ j ∂∂U [ j ] . (10)We also need to know how the derivatives of the fields transform underthe Lie symmetry. This means that we have to introduce the “prolongations”of the action of the group to the different derivatives that appear in (1)and (3). Exactly how to calculate the prolongations is a very well knownprocedure whose technical details can be found in [21].It is therefore necessary that the Lie transformation should leave (1) and(3) invariant. This yields an overdetermined system of equations for the in-finitesimals ξ ( x, y, t, λ, ψ, M, U [ j ] ), ξ ( x, y, t, λ, ψ, M, U [ j ] ), ξ ( x, y, t, λ, ψ, M, U [ j ] ), φ ( x, y, t, λ, ψ, M, U [ j ] ), φ ( x, y, t, λ, ψ, M, U [ j ] ) , and Θ i ( x, y, t, λ, ψ, M, U [ j ] ).This is the classical method [21] of finding Lie symmetries, and it canbe summarized as follows • Calculation of the prolongations of the derivatives of the fields thatappear in (1) and (3). • Substitution of the transformed fields (9) and their derivatives in (1)and (3). • Set all the coefficients in ǫ at 0. • Substitution of the prolongations. • ψ xx , ψ y and λ y can be substituted by using (1) and (3). • The system of equations for the infinitesimals can be obtained by set-ting each coefficient in the different remaining derivatives of the fieldsat zero. 4 on-classical symmetries
There is a generalization of the classical method that determines the non-classical or conditional symmetries [6], [18]. In this case we are lookingfor symmetries that leave invariant not only the equations but also the socalled “invariant surfaces”, which in our case are: φ = ξ ψ x + ξ ψ y + ξ ψ t φ = ξ λ y + ξ λ t Θ = ξ M x + ξ M y + ξ M t (11)Θ j = ξ U [ j ] x + ξ U [ j ] y + ξ U [ j ] t , j = 1 ..n. These non-classical symmetries are the symmetries that we address be-low. The method for calculating these symmetries is the same as the onewe have described for the classical ones complemented with equations (11),that must also be combined with step 4 to eliminate as many derivativesof the fields as possible, depending on whether all of the ξ i are differentfrom zero or not. This is why we have to distinguish three different types ofnon-classical symmetries. • ξ = 1 . • ξ = 0 , ξ = 1 . • ξ = 0 , ξ = 0 , ξ = 1 . Note that owing to (11), there is not restriction in selecting ξ j = 1 when ξ j = 0 [18]. In the following sections we shall determine these three types ofsymmetries of the Lax pair and its reduction to 1 + 1 dimensions by solvingthe characteristic equation dxξ = dyξ = dtξ = dψφ = dλφ = dM Θ = dU [ j ] Θ j . (12)The advantage of our approach of working with the Lax pair instead of theequations of the hierarchy lies in the fact that we can obtain the reducedeigenfuntion and the reduced spectral parameter at the same time. whichas we shall see, in many cases is not a trivial matter. The equations ofthe reduced hierarchies can be explicitly obtained from the reduced spectralproblem and we shall write them in all the cases.Of course the calculation of the symmetries is tedious , and we have usedthe MAPLE symbolic package to handle these calculations. For the benefitof the reader, we shall omit the technical details.5 Non-classical symmetries for ξ = 1 Calculation of symmetries
In this case (11) allows us to eliminate the derivatives with respect to tψ t = φ − ξ ψ x − ξ ψ y λ t = φ − ξ λ y M t = Θ − ξ M x − ξ M y (13) U [ j ] t = Θ j − ξ U [ j ] x − ξ U [ j ] y , j = 1 ..n. If we add (13) to the five steps listed above for the calculation of non-classical symmetries, we obtain (after long but straightforward calculations)the following symmetries: ξ = S S ξ = S S ξ = 1 φ = 1 S (cid:18) ∂S ∂x + a (cid:19) ψ (14) φ = 1 S (cid:18) a − a n (cid:19) λ Θ = 1 S (cid:18) − ∂S ∂x + a − a n (cid:19) M Θ = 1 S (cid:18) U [1] (cid:18) ∂S ∂x − a (cid:19) − ∂S ∂t (cid:19) Θ j = 1 S (cid:18) ∂S ∂x − a j − n − a n − j + 1 n (cid:19) U [ j ] , j = 2 ..n, where S = S ( x, t ) = A ( t ) + B ( t ) e x + C ( t ) e − x ,S = S ( y ) = a y + b , (15) S = S ( t ) = a t + b .A ( t ) , B ( t ) , C ( t ) are arbitrary functions of t . Furthermore, a , a , b , a , b are arbitrary constants, such that a and b cannot at the same time be 0.6 lassification of the reductions We have, therefore, several different reductions depending on which arbi-trary functions and/or constants are or are not zero. We shall use thefollowing classification • Type I: Corresponding to selecting A ( t ) = 0, B ( t ) = C ( t ) = 0. • Type II: Corresponding to selecting B ( t ) = 0, A ( t ) = C ( t ) = 0. Aswe shall show in Appendix I, this case yields the same reduced spectralproblems as those obtained for Type I, although the reductions aredifferent. • Type III: Corresponding to selecting C ( t ) = 0, A ( t ) = B ( t ) = 0. Itis easy to see that this case is equivalent to II owing to the invarianceof the Lax pair under the transformation x → − x , y → − y , t → − t .Below we only consider cases I and II.In each of the cases listed before we have different subcases, depending onthe values of the constants a j and b j . We have the following 5 independentpossibilities: • Case 1: a = 0 , a = 0; b = 0. • Case 2: a = 0 , a = 0; b = 0. • Case 3: a = 0 , a = 0; b = 0. • Case 4: a = 0 , a = 0; b = 0. • Case 5: a = 0;We can obtain 5 different non-trivial reductions: I.i i = 1 ..
5. We shall seeeach reduction separately by obtaining the reduced variables, the reducedfields, the transformation of the spectral parameter and the eigenfuctionand, finally, the reduced spectral problem and the corresponding reducedhierarchy. Furthermore, there are several interesting reductions, especiallythose that also have a non-isospectral parameter in 1 + 1 dimensions . Letus summarize the results:
I.1) B ( t ) = C ( t ) = 0 , A ( t ) = 0 , a = 0 , a = 0 , b = 0 . By solving the characteristic equation (12), we have the following results7
Reduced variables: z = x − b R A ( t ) dt , z = y • Spectral parameter: λ ( y, t ) = λ • Reduced Fields: ψ ( x, y, t ) = e (cid:18) a tb (cid:19) e (cid:18) λ n a z b (cid:19) Φ( z , z ) M ( x, y, t ) = H ( z , z ) U [1] ( x, y, t ) = V [1] ( z , z ) − A b U [ j ] ( x, y, t ) = V [ j ] ( z , z ) • Reduced Spectral problem:Φ z z − (cid:18) − λ H (cid:19) Φ = 0Φ z + ˆ B Φ z − ˆ B z B = n X j =1 λ ( n − j +1)0 V [ j ] ( z , z ) (17) • Reduced Hierarchy: The compatibility condition of (16) yields: ∂ V [ n ] ∂z − ∂V [ n ] ∂z + ∂H∂z = 02 H ∂V [1] ∂z + V [1] ∂H∂z = 0 (18)2 H ∂V [ j +1] ∂z + V [ j +1] ∂H∂z + ∂ V [ j ] ∂z − ∂V [ j ] ∂z = 0 . which is the positive Camassa-Holm hierarchy , whose first com-ponent ( n = 1) is a modified Dym equation [1], [10]. I.2) B ( t ) = C ( t ) = 0 , A ( t ) = 0 , a = 0 , a = 0 , b = 0 . By solving the characteristic equation (12), we have the following results • Reduced variables: z = x − b R A ( t ) dt , z = yb − tb Spectral parameter: λ ( y, t ) = (cid:16) b b (cid:17) ( n ) λ • Reduced Fields: ψ ( x, y, t ) = e a tb ! e (cid:18) λ n a z λ n (cid:19) Φ( z , z ) M ( x, y, t ) = (cid:18) b b (cid:19) ( n ) H ( z , z ) U [1] ( x, y, t ) = (cid:18) b (cid:19) V [1] ( z , z ) − A b U [ j ] ( x, y, t ) = (cid:18) b (cid:19) (cid:18) b b (cid:19) ( − jn ) V [ j ] ( z , z ) • Reduced Spectral problem:Φ z z − (cid:18) − λ H (cid:19) Φ = 0Φ z (1 + λ n ) + ˆ B Φ z − ˆ B z B = n X j =1 λ ( n − j +1)0 V [ j ] ( z , z ) (20) • Reduced Hierarchy: The compatibility condition of (19) yields: ∂ V [ n ] ∂z − ∂V [ n ] ∂z + ∂H∂z = 02 H ∂V [1] ∂z + V [1] ∂H∂z + ∂H∂z = 0 (21)2 H ∂V [ j +1] ∂z + V [ j +1] ∂H∂z + ∂ V [ j ] ∂z − ∂V [ j ] ∂z = 0 . I.3) B ( t ) = C ( t ) = 0 , A ( t ) = 0 , a = 0 , a = 0 , b = 0 . By solving the characteristic equation (12), we have the following results • Reduced variables: z = x − R A ( t ) S ( t ) dt , z = a y Spectral parameter: In this case the reduction of the spectral param-eter is a non-trivial one that yields λ ( y, t ) = S ( n ) Λ( z )where Λ( z ) is the reduced spectral parameter. • Reduced Fields: ψ ( x, y, t ) = Λ( z ) (cid:18) a na (cid:19) S (cid:18) a a (cid:19) Φ( z , z ) M ( x, y, t ) = S ( − n ) H ( z , z ) U [1] ( x, y, t ) = a S V [1] ( z , z ) − A S U [ j ] ( x, y, t ) = a S ( j − n − ) V [ j ] ( z , z ) • Reduced Spectral problem:Φ z z − (cid:18) − Λ( z )2 H (cid:19) Φ = 0Φ z + ˆ B Φ z − ˆ B z B = n X j =1 Λ( z ) ( n − j +1) V [ j ] ( z , z ) (23)and Λ( z ) satisfies the non-isospectral condition n d Λ( z ) d z − Λ( z ) ( n +1) = 0 (24) • Reduced Hierarchy: The compatibility condition of (22) yields theautonomous hierarchy : ∂ V [ n ] ∂z − ∂V [ n ] ∂z + ∂H∂z = 02 H ∂V [1] ∂z + V [1] ∂H∂z + Hn = 0 (25)2 H ∂V [ j +1] ∂z + V [ j +1] ∂H∂z + ∂ V [ j ] ∂z − ∂V [ j ] ∂z = 0 . .4) B ( t ) = C ( t ) = 0 , A ( t ) = 0 , a = 0 , a = 0 , b = 0 . By solving the characteristic equation (12), we have the following results • Reduced variables: z = x − R A ( t ) S ( t ) dt , z = a yb − ln( S ) • Spectral parameter: In this case the reduction of the spectral param-eter yields λ ( y, t ) = (cid:18) S b (cid:19) ( n ) Λ( z )where Λ( z ) is the reduced spectral parameter: • Reduced Fields: ψ ( x, y, t ) = Λ( z ) (cid:18) na a (cid:19) S (cid:18) a a (cid:19) Φ( z , z ) M ( x, y, t ) = (cid:18) b S (cid:19) ( n ) H ( z , z ) U [1] ( x, y, t ) = a S V [1] ( z , z ) − A S U [ j ] ( x, y, t ) = a S (cid:18) S b (cid:19) ( j − n ) V [ j ] ( z , z ) • Reduced Spectral problem:Φ z z − (cid:18) − Λ( z )2 H (cid:19) Φ = 0Φ z (1 + Λ( z ) n ) + ˆ B Φ z − ˆ B z B = n X j =1 (Λ( z )) ( n − j +1) V [ j ] ( z , z ) (27)and Λ( z ) satisfies the non-isospectral condition n (1 + Λ( z ) n ) d Λ d z − Λ( z ) n +1 = 0 (28)11 Reduced Hierarchy: The compatibility condition of (26) yields: ∂ V [ n ] ∂z − ∂V [ n ] ∂z + ∂H∂z = 02 H ∂V [1] ∂z + V [1] ∂H∂z + Hn + ∂H∂z = 0 (29)2 H ∂V [ j +1] ∂z + V [ j +1] ∂H∂z + ∂ V [ j ] ∂z − ∂V [ j ] ∂z = 0 . Therefore, although the Lax pair is non-isospectral, the re-duced hierarchy is autonomous
I.5) B ( t ) = C ( t ) = 0 , A ( t ) = 0 , a = 0 By solving the characteristic equation (12), we have the following results : • Reduced variables: z = x − R A ( t ) S ( t ) dt , z = S S (cid:18) − a a (cid:19) • Spectral parameter: In this case the reduction of the spectral param-eter yields λ ( y, t ) = S (cid:18) a − a a n (cid:19) Λ( z )where Λ( z ) is the reduced spectral parameter: • Reduced Fields: ψ ( x, y, t ) = Λ( z ) (cid:18) na a − a (cid:19) S (cid:18) a a (cid:19) Φ( z , z ) M ( x, y, t ) = S (cid:18) a − a a n (cid:19) H ( z , z ) U [1] ( x, y, t ) = (cid:18) a S (cid:19) V [1] ( z , z ) − A S U [ j ] ( x, y, t ) = (cid:18) a S (cid:19) S (cid:18) ( a − a )( j − a n (cid:19) V [ j ] ( z , z )12 Reduced Spectral problem:Φ z z − (cid:18) − Λ( z )2 H (cid:19) Φ = 0Φ z (1 + z Λ( z ) n ) + ˆ B Φ z − ˆ B z B = n X j =1 Λ( z ) ( n − j +1) V [ j ] ( z , z ) (31)and Λ( z ) satisfies the non-isospectral condition n (1 + z Λ( z ) n ) d Λ d z − a − a a Λ( z ) ( n +1) = 0 (32) • Reduced Hierarchy: The compatibility condition of (30) yields: ∂ V [ n ] ∂z − ∂V [ n ] ∂z + ∂H∂z = 02 H ∂V [1] ∂z + V [1] ∂H∂z + a − a a Hn + z ∂H∂z = 0 (33)2 H ∂V [ j +1] ∂z + V [ j +1] ∂H∂z + ∂ V [ j ] ∂z − ∂V [ j ] ∂z = 0 . the Lax pair is non-isospectral and the reduced hierarchy isnon-autonomous • Note that the singularity, which apparently appears in the reductionswhen a = 0, can be easily removed by considering thatlim a → (cid:18) a t + b b (cid:19) (1 /a ) = e t/b We refer readers to Appendix I so that they can check that the spectralproblems obtained in case II are not different from those of case I ξ = 0 , ξ = 1 We can now write ψ y = φ − ξ ψ x y = φ M y = Θ − ξ M x (34) U [ j ] y = Θ j − ξ U [ j ] x , j = 1 ..n We can combine (34) with (3) and (2.4). This allows us to remove ψ xx , ψ y , ψ t , λ y ,λ t , M y , U [ j ] y from the equation of the symmetries. In this case, we obtain thefollowing symmetries ξ = S S ξ = 1 ξ = 0 φ = 1 S (cid:18) ∂S ∂x + a (cid:19) ψ (35) φ = 1 S (cid:18) − a n (cid:19) λ Θ = 1 S (cid:18) − ∂S ∂x + a n (cid:19) M Θ = 1 S (cid:18) U [1] (cid:18) ∂S ∂x (cid:19) − ∂S ∂t (cid:19) Θ j = 1 S (cid:18) ∂S ∂x − a j − n (cid:19) U [ j ] , j = 2 ..n. where S and S are those given in (15).Evidently we should consider that a and b cannot be 0 at the sametime. Classification of the reductions
In this case, one of the reduced variables is t . This means that the integralsthat involve S can be performed without any restrictions for the functions A ( t ) , B ( t ) , C ( t ). We have four different cases IV.1: a = 0 , E = p A − B C = 0 . • Reduced variables: z = R dxS ( x,t ) − yb , z = tb • Spectral parameter: λ ( y, t ) = λ Reduced Fields: ψ ( x, y, t ) = p S e (cid:18) a yb (cid:19) e (cid:18) a tb λ n (cid:19) Φ( z , z ) M ( x, y, t ) = H ( z , z ) S U [1] ( x, y, t ) = S b V [1] ( z , z ) + S ddt (cid:18)Z dxS ( x, t ) (cid:19) U [ j ] ( x, y, t ) = S b V [ j ] ( z , z ) • Reduced Spectral problem:Φ z z + λ H Φ = 0 λ n Φ z = ( ˆ B − z − ˆ B z B = n X j =1 λ ( n − j +1)0 V [ j ] ( z , z ) (37) • Reduced Hierarchy: The compatibility condition of (36) yields theautonomous hierarchy : ∂ V [ n ] ∂z − ∂H∂z = 02 H ∂V [1] ∂z + V [1] ∂H∂z − ∂H∂z = 0 (38)2 H ∂V [ j +1] ∂z + V [ j +1] ∂H∂z + ∂ V [ j ] ∂z = 0 . IV.2: a = 0 , E = p A − B C = 0 . • Reduced variables: z = E (cid:16)R dxS − yb (cid:17) , z = b R E ( t ) dt • Spectral parameter: λ ( y, t ) = λ • Reduced Fields: ψ ( x, y, t ) = r S E e (cid:18) a yb (cid:19) e (cid:18) a tb λ n (cid:19) Φ( z , z )15 ( x, y, t ) = E S H ( z , z ) U [1] ( x, y, t ) = S b V [1] ( z , z ) + S ddt (cid:18)Z dxS ( x, t ) (cid:19) + S E t E z U [ j ] ( x, y, t ) = S b V [ j ] ( z , z ) • Reduced Spectral problem:Φ z z + (cid:18) λ H − (cid:19) Φ = 0 λ n Φ z = ( ˆ B − z − ˆ B z B = n X j =1 λ ( n − j +1)0 V [ j ] ( z , z ) (40) • Reduced Hierarchy: The compatibility condition of (39) yields theautonomous hierarchy : ∂ V [ n ] ∂z − ∂V [ n ] ∂z − ∂H∂z = 02 H ∂V [1] ∂z + V [1] ∂H∂z − ∂H∂z = 0 (41)2 H ∂V [ j +1] ∂z + V [ j +1] ∂H∂z + ∂ V [ j ] ∂z − ∂V [ j ] ∂z = 0 , which is the celebrated negative Camassa-Holm hierarchy [1], [3],[4], [5] . IV.3: a = 0 , E = p A − B C = 0 . • Reduced variables: z = R dxS ( x,t ) − ln( S ) a , z = t • Spectral parameter: λ ( y, t ) = S ( − n ) Λ( z ) where Λ( z ) satisfies thenon-isospectral condition n d Λ( z ) dz + a Λ( z ) (1 − n ) = 016 Reduced Fields: ψ ( x, y, t ) = p S λ (cid:18) − a na (cid:19) Φ( z , z ) M ( x, y, t ) = S ( n ) S H ( z , z ) U [1] ( x, y, t ) = S V [1] ( z , z ) + S ddt (cid:18)Z dxS ( x, t ) (cid:19) U [ j ] ( x, y, t ) = S S ( − jn ) V [ j ] ( z , z ) • Reduced Spectral problem:Φ z z + Λ( z )2 H Φ = 0Λ( z ) n Φ z = ( ˆ B − z − ˆ B z B = n X j =1 Λ( z ) ( n − j +1) V [ j ] ( z , z ) (43) • Reduced Hierarchy: The compatibility condition of (42) yields theautonomous hierarchy : ∂ V [ n ] ∂z − ∂H∂z + a n H = 02 H ∂V [1] ∂z + V [1] ∂H∂z − ∂H∂z = 0 (44)2 H ∂V [ j +1] ∂z + V [ j +1] ∂H∂z + ∂ V [ j ] ∂z = 0 . IV.4: a = 0 , E = p A − B C = 0 . • Reduced variables: z = E (cid:16)R dxS ( x,t ) − ln( S ) a (cid:17) , z = R E ( t ) dt • Spectral parameter: λ ( y, t ) = S ( − n ) Λ( z ) where Λ( z ) satisfies thenon-isospectral condition n d Λ( z ) dz + a E ( z ) Λ( z ) (1 − n ) = 017 Reduced Fields: ψ ( x, y, t ) = r S E λ (cid:18) − a na (cid:19) Φ( z , z ) M ( x, y, t ) = E S S ( n ) H ( z , z ) U [1] ( x, y, t ) = S V [1] ( z , z ) + S ddt (cid:18)Z dxS ( x, t ) (cid:19) + S E t E z U [ j ] ( x, y, t ) = S S ( − jn ) V [ j ] ( z , z ) • Reduced Spectral problem:Φ z z + (cid:18) Λ( z )2 H − (cid:19) Φ = 0Λ( z ) n Φ z = ( ˆ B − z − ˆ B z B = n X j =1 Λ( z ) ( n − j +1) V [ j ] ( z , z ) (46) • Reduced Hierarchy: The compatibility condition of (45) yields thenon-autonomous hierarchy : ∂ V [ n ] ∂z − ∂V [ n ] ∂z − ∂H∂z + a nE ( z ) H + = 02 H ∂V [1] ∂z + V [1] ∂H∂z − ∂H∂z = 0 (47)2 H ∂V [ j +1] ∂z + V [ j +1] ∂H∂z + ∂ V [ j ] ∂z − ∂V [ j ] ∂z = 0 . ξ = ξ = 0 , ξ = 1 We can now write ψ x = φ M x = Θ (48) U [ j ] x = Θ j j = 1 ..n ξ = 1 ξ = 0 ξ = 0 φ = 12 (cid:16) ± i √ λM (cid:17) ψ (49) φ = 0Θ = − M Θ j = U [ j ] , j = 1 ..n, which holds only if M t = M y = 0 and yields the following reductions: z = y, z = tλ ( y, t ) = Λ( z , z ) ψ = e (cid:18) x ± i √ H e − x (cid:19) Φ( z , z ) M ( x, y, t ) = H e − x (50) U [ j ] ( x, y, t ) = e x V [ j ] ( z , z ) , j = 1 ..n, (51)and it is easy to see that (50) satisfies (2.4) for V [ j ] ( z , z ) , ( j = 1 ..n ) arbi-trary and H constant. • We started with the spectral problem (although non-isospectral) asso-ciated with a Camassa-Holm hierarchy in 2 + 1 dimensions • Non-classical Lie symmetries of this CHn2+1 spectral problem havebeen obtained. • Each Lie symmetry yields a reduced spectral 1 + 1 problem whosecompatibility condition provides a 1 + 1 hierarchy. • The main achievement of this paper is that our procedure also providesthe reduction of the eigenfunction as well as the spectral parameter. Inmany cases, the reduced parameter also proves to be non-isospectral,even in the 1 + 1 reduction [13].19
There are several different reductions but they can be summarized in 9different non-trivial cases: I.i (i=1..5) and IV.j (j=1..4). Five of thesehierarchies (I.3,I.4,I.5,IV.3,IV,4) have a non-isospectral Lax pair andtwo of them (I.1 and IV.2) are the positive and negative Camassa-Holm hierarchies respectively. The equations for all these reducedhierarchies have been explicitly written in each case.
Acknowledgements
This research has been supported in part by the DGICYT under projectFIS2009-07880 and JCyL under contract GR224. [1] M. Antonowitz and A. Fordy,
J. Phys A: Math. Gen. , (1988), L269–L275.[2] M. J. Ablowitz, A. Ramani and H. Segur, Lett. Nuov. Cim. , (1978),333–338.[3] R. Camassa and D. D. Holm, Phys. Rev. Lett , (1993), 1661–1664.[4] R. Camassa, D. D. Holm and J. M. Hyman, Adv. in Appl. Mech. ,(1994), 1–33.[5] P. A. Clarkson, P. R. Gordoa and A. Pickering , Inverse Problems ,(1997), 1463–1476.[6] G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equa-tions , Springer Verlag, (1974).[7] P. G. Est´evez,
Phys. Lett.
A171 , (1992), 259–261.[8] P. G. Est´evez,
Studies in Appl. Math. , (1995), 73–113.[9] P. G. Est´evez and J. Prada, Journal of Nonlinear Math. Phys. ,(2005), 164–179, (2004).[10] P. G. Est´evez and J. Prada, Journal of Phys. A: Math and Gen ,1–11.[11] P. G. Est´evez, Theor. and Math. Phys. , (2005), 1132–1137.2012] P. G. Est´evez, M. L. Gandarias and J. Prada,
Physics Letters A ,(2005), 40–47.[13] PR. Gordoa and A. Pickering,
Phys. Lett A , (2003), 223-227.[14] RI Ivanov
Theor. and Math. Phys. , (2009), 952-959.[15] RA. Kraenkel and A. Zenchuk
Phys. Lett A , (1999), 218-224.[16] N. Kudryashov and A. Pickering,
J. Phys A: Math. Gen. , (1998),9505–9518.[17] M. Legare, Journal of Nonlinear Math. Phys. , (1996), 266–285.[18] P. J. Olver, Applications of Lie Groups to Differential Equations ,Springer Verlag, (1999).[19] P. Painlev´e,
Acta Mathematica , (1902) 1–85.[20] G. Ortenzi, SIGMA , (2007), 47-57.[21] H. Stephani, Differential equations. Their solutions using symmetries ,edited by M. Mac Callum, Cambridge University Press, (1989).[22] J. Weiss ,
J. Math. Phys. , (1983), 1405–1413. Appendix
Let us go on to prove that the reduced hierarchies obtained by means of thereductions related to the symmetries of type II are the same as type I, eventhough the reductions of variables and fields are different.
II.1) A ( t ) = C ( t ) = 0 , B ( t ) = 0 , a = 0 , a = 0 , b = 0 . The reductions are now • Reduced variables: z = − ln (cid:16) e − x + b R B ( t ) dt (cid:17) , z = y • Spectral parameter: λ ( y, t ) = λ • Reduced Fields: ψ ( x, y, t ) = e (cid:18) a tb (cid:19) e (cid:18) λ n a z b (cid:19) (cid:18) e x e z (cid:19) Φ( z , z )21 ( x, y, t ) = (cid:18) e z e x (cid:19) H ( z , z ) U [1] ( x, y, t ) = (cid:18) e x e z (cid:19) V [1] ( z , z ) − B b e x U [ j ] ( x, y, t ) = (cid:18) e x e z (cid:19) V [ j ] ( z , z ) , which yield the same spectral problem as in case I.1. II.2) A ( t ) = C ( t ) = 0 , B ( t ) = 0 , a = 0 , a = 0 , b = 0 . This case affords the reductions • Reduced variables: z = − ln (cid:16) e − x + b R B ( t ) dt (cid:17) , z = yb − tb • Spectral parameter: λ ( y, t ) = (cid:16) b b (cid:17) ( n ) λ • Reduced Fields: ψ ( x, y, t ) = e a tb ! e (cid:18) λ n a z λ n (cid:19)(cid:18) e x e z (cid:19) Φ( z , z ) M ( x, y, t ) = (cid:18) b b (cid:19) ( n ) (cid:18) e z e x (cid:19) H ( z , z ) U [1] ( x, y, t ) = (cid:18) b (cid:19) (cid:18) e x e z (cid:19) V [1] ( z , z ) − B b e x U [ j ] ( x, y, t ) = (cid:18) b (cid:19) (cid:18) b b (cid:19) ( − jn ) (cid:18) e x e z (cid:19) V [ j ] ( z , z ) , and the spectral problem is the same as in I.2. II.3) A ( t ) = C ( t ) = 0 , B ( t ) = 0 , a = 0 , a = 0 , b = 0 . In this case, the reductions are • Reduced variables: z = − ln (cid:16) e − x + R B ( t ) S dt (cid:17) , z = a y • Spectral parameter: In this case, the reduction of the spectral param-eter is a non-trivial one that yields λ ( y, t ) = S ( n ) Λ( z )22 Reduced Fields: ψ ( x, y, t ) = Λ( z ) (cid:18) a na (cid:19) S (cid:18) a a (cid:19) (cid:18) e x e z (cid:19) Φ( z , z ) M ( x, y, t ) = S ( − n ) (cid:18) e z e x (cid:19) H ( z , z ) U [1] ( x, y, t ) = a S (cid:18) e x e z (cid:19) V [1] ( z , z ) − B S e x U [ j ] ( x, y, t ) = a S ( j − n − ) (cid:18) e x e z (cid:19) V [ j ] ( z , z ) , and the spectral problem is exactly the same as in I.3. II.4) A ( t ) = C ( t ) = 0 , B ( t ) = 0 , a = 0 , a = 0 , b = 0 . We have the following results • Reduced variables: z = − ln (cid:16) e − x + R B ( t ) S ( t ) dt (cid:17) , z = a yb − ln( S ) • Spectral parameter: In this case the reduction of the spectral param-eter yields λ ( y, t ) = (cid:18) S b (cid:19) ( n ) Λ( z )where Λ( z ) is the reduced spectral parameter: • Reduced Fields: ψ ( x, y, t ) = Λ( z ) (cid:18) na a (cid:19) S (cid:18) a a (cid:19) (cid:18) e x e z (cid:19) Φ( z , z ) M ( x, y, t ) = (cid:18) b S (cid:19) ( n ) (cid:18) e z e x (cid:19) H ( z , z ) U [1] ( x, y, t ) = a S (cid:18) e x e z (cid:19) V [1] ( z , z ) − B S e x U [ j ] ( x, y, t ) = a S (cid:18) S b (cid:19) ( j − n ) (cid:18) e x e z (cid:19) V [ j ] ( z , z ) , which yields the spectral problem I.4.23 I.5) A ( t ) = C ( t ) = 0 , B ( t ) = 0 , a = 0 The same spectral problem I.5 is obtained through the following reductions • Reduced variables: z = − ln (cid:16) e − x + R B ( t ) S ( t ) dt (cid:17) , z = S S (cid:18) − a a (cid:19) • Spectral parameter: In this case the reduction of the spectral param-eter yields λ ( y, t ) = S (cid:18) a − a a n (cid:19) Λ( z ) , where Λ( z ) is the reduced spectral parameter: • Reduced Fields: ψ ( x, y, t ) = Λ( z ) (cid:18) na a − a (cid:19) S (cid:18) a a (cid:19) (cid:18) e x e z (cid:19) Φ( z , z ) M ( x, y, t ) = S (cid:18) a − a a n (cid:19) (cid:18) e z e x (cid:19) H ( z , z ) U [1] ( x, y, t ) = (cid:18) a S (cid:19) (cid:18) e x e z (cid:19) V [1] ( z , z ) − B S e x U [ j ] ( x, y, t ) = (cid:18) a S (cid:19) S (cid:18) ( a − a )( j − a n (cid:19) (cid:18) e x e z (cid:19) V [ j ] ( z , z ) ..