Bilinear forms and Bäcklund transformations of the perturbation systems

Physics Letters A 341 (2005) 441–449 www.elsevier.com/locate/pla

Bilinear forms and Bäcklund transformations of the perturbation systems Wen-Xiu Ma a,∗ , Walter Strampp b a Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA b Department of Mathematics, University of Kassel, D-34109 Kassel, Germany

Received 11 March 2005; accepted 4 May 2005 Available online 13 May 2005 Communicated by A.R. Bishop

Abstract A class of bilinear forms and a class of Bäcklund transformations are presented for the perturbation systems generated from perturbations around solutions of a given system of integrable equations. The stability notion of bilinear structures is introduced to guarantee their hereditariness from the original system to their perturbation systems. Two special choices of the resulting bilinear forms and Bäcklund transformations are discussed. The Korteweg–de Vries equation is chosen as a model to illustrate the general idea.  2005 Elsevier B.V. All rights reserved. PACS: 05.45.-a; 02.30.Ik

1. Introduction The perturbation systems, which are generated from perturbations around solutions of a given system of integrable equations, are integrable couplings [1,2], and generalize the symmetry equations [3]. Study on the perturbation systems will help us in classifying integrable equations, based on triangular forms of the given equations. On the other hand, bilinear forms and Bäcklund transformations are extremely helpful in constructing solutions to nonlinear differential equations [4,5]. The existence of such bilinear structures has already been viewed as one of important integrable characteristics. In practice, bilinear forms bring various kinds of exact solutions such as solitons [4], positons [6] and complexitons [7]. * Corresponding author.

E-mail addresses: [email protected] (W.X. Ma), [email protected] (W. Strampp). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.05.013

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In this Letter, we would like to study the problem of bilinear forms and Bäcklund transformations for the perturbation systems. We shall use the standard notation [2]: x = (x1 , . . . , xp )T ,

u = (u1 , . . . , uq )T ,

ui = ui (x, t), 1
i
q,

(1.1)

where xi , t ∈ R, 1
i
p, and ui = ui (x, t), 1
i
q, are scalar functions. The perturbation series of order N is denoted by uˆ N =

N


ε i ηi ,

ηi = ηi (x, t), 0
i
N,

(1.2)

i=0

where ε is a small parameter, and ηi , 1
i
N , are column vectors of the same dimension as u. The corresponding perturbation vector of a given column vector-valued function K(u) is given by [8]   T  1 ∂K T (uˆ N )  1 ∂ N K T (uˆ N )  Kˆ N = Kˆ N (ηˆ N ) = K T (u0 ), (1.3) , . . . , , N  0,   1! ∂ε N! ∂ε N ε=0 ε=0 where   T T ηˆ N = η0T , . . . , ηN . Therefore, a system of evolution equations   ut = K(u) = K(x, t, u) = K x, t, u, Du, D 2 u, . . . ,

(1.4)

(1.5)

where D i u denotes the vector of the ith derivatives of u with respect to x, has its perturbation system of order N :  1 ∂ i  K(uˆ N ), i = 0, 1, . . . , N. ηˆ N t = Kˆ N (ηˆ N ), i.e., ηit = (1.6) i! ∂ε i ε=0 Now our question is what kind of bilinear forms and Bäcklund transformations can exist for the perturbation system (1.6), when the original system (1.5) possesses a bilinear form and Bäcklund transformation. In what follows, we would like to furnish a theory of bilinear forms and Bäcklund transformations for the perturbation system (1.6). A stability notion of bilinear structures will be introduced to guarantee their hereditariness, and two special choices for generating solutions to the perturbation system (1.6) will be presented. Moreover, the resulting theory will be applied to the perturbation systems of the Korteweg–de Vries (KdV) equation. All the results are also a supplement to the existing integrable theory of the perturbation systems [2,3].

2. Bilinear forms and Bäcklund transformations 2.1. Bilinear forms We first consider bilinear forms of the perturbation systems. Suppose that under a transformation u = P (f, g), where P is a vector-valued function depending on f and g, the system of evolution equations (1.5) possesses a bilinear form B1 (f · f ) + B2 (f · g) + B3 (g · g) = 0,

(2.1)

where Bi , 1
i
3, are three Hirota’s bilinear operators. That is, if f and g solve (2.1), then u = P (f, g) solves the original system of evolution equations (1.5). Note that Bi , 1
i
3, are often of multicomponent form, where (2.1) is a system of bilinear equations. We also assumed that only one pair of functions of f and g appeared in the

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bilinear form. For the KdV equation ut = 6uux − uxxx ,

(2.2) ∂2

we have its bilinear operators B1 = Dx (Dt + Dx3 ), B2 = B3 = 0, under the transformation u = −2 ∂x 2 log f . Starting with the system of bilinear equations (2.1), we introduce fˆN =

N


ε i fi ,

i=0

gˆ N =

N


ε i gi .

(2.3)

i=0

Substituting fˆN and gˆ N into (2.1) and computing the coefficients of ε k , 0
k
N , leads to
B1 (fi · fj ) + B2 (fi · gj ) + B3 (gi · gj ) = 0, 0
k
N.

(2.4)

i+j =k i,j
0

To prove that (2.4) is a bilinear form for the perturbation system (1.6), we need the following stability of the bilinear form. Definition 2.1. A bilinear form (2.1) of a system of evolution equations (1.5) with a transformation u = P (f, g) is called stable under small perturbation, if       B1 f (ε) · f (ε) + B2 f (ε) · g(ε) + B3 g(ε) · g(ε) = o(ε), (2.5) implies that u = P (f (ε), g(ε)) satisfies ut = K(u) + o(ε).

(2.6)

Theorem 2.1. Suppose that a system of evolution equations (1.5) possesses a stable bilinear form (2.1) under a transformation u = P (f, g), where P is a vector-valued function depending on f and g. Then, the perturbation system (1.6) has the bilinear form (2.4) under the transformation   1 ∂P (fˆN , gˆ N )  1 ∂ N P (fˆN , gˆ N )  η1 = , . . . , η = η0 = P (f0 , g0 ), (2.7) N   , 1! ∂ε N! ∂ε N ε=0 ε=0 where fˆN and gˆ N are defined by (2.3). Proof. What we need to prove is that if fi and gi , 0
i
N , satisfy (2.4), then (2.7) gives a solution to the perturbation system (1.6). Let us first define vˆN =

  N
ε i ∂ i P (fˆN , gˆ N )   i! ∂ε i i=0

= P (f0 , g0 ) + ε=0

    ε ∂P (fˆN , gˆ N )  ε N ∂ N P (fˆN , gˆ N )  + · · · +  ,  1! ∂ε N! ∂ε N ε=0 ε=0

and thus   P (fˆN , gˆ N ) = vˆN + o ε N ,

(2.8)

by the Taylor series expansion. Note that fˆN and gˆ N solve (2.1) up to a precision o(ε N ), due to (2.4). Thus, it follows from the stability of the bilinear form that u = P (fˆN , gˆ N ) solves the original system (1.5) up to a precision o(ε N ). Now, because of (2.8), vˆN solves (1.5) up to a precision o(ε N ) as well, i.e.,   vˆN t = K(vˆN ) + o ε N .

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Finally, expanding K(vˆN ) as a series of ε and equating the coefficients of ε i , 0
i
N , on both sides shows that the vector-valued function ηˆ N defined through (2.7) solves the perturbation system (1.6). The proof is finished. 2 Let us now show two special solutions to the system of bilinear equations (2.4). First, we choose fˆN =

N
i=0

ε i fi =

N


ε i f,

gˆ N =

i=0

N


ε i gi =

i=0

N


ε i g,

(2.9)

i=0

where f and g satisfy the bilinear form (2.1). This yields a special solution to the system of bilinear equations (2.4). Therefore, a set of ηi , 0
i
N , determined by (2.7) solves the perturbation system (1.6). However, this choice is usually useless in practice, since we may have P (f, g) = P (cf, cg) for a constant c, i.e., P is a vector of homogeneous functions. If so, we just obtain a trivial solution of (1.6): η0 = P (f, g), ηi = 0, 1
i
N . Second, we note that if B is a system of Hirota’s bilinear equations, then
k    B ∂yl f · ∂yk−l g , ∂yk B(f, g) = l i+j =k i,j
0

where y can be any space variable xi , 1
i
p, or the time variable t. Therefore, when the system of bilinear equations (2.1) holds, we can have      
1 l 1 l 1 l 1 1 1 ∂y f · ∂yk−l g + B2 ∂y f · ∂yk−l g + B3 ∂y g · ∂yk−l g = 0. (2.10) B1 i! j! i! j! i! j! i+j =k i,j
0

It follows from Theorem 2.1 that (2.7) defines a solution to the perturbation system (1.6), where the functions fˆN and gˆ N are defined by fˆN =

N
εi ∂ i f , i! ∂y i

gˆ N =

i=0

N
εi ∂ i g , i! ∂y i

y = xj , 1
j
p, or t.

(2.11)

i=0

This gives the following corollary. Corollary 2.1. Let f and g satisfy the stable bilinear form (2.1). Suppose that fˆN = N i i=0 ε gi are defined by (2.11). Then (2.7) solves the perturbation system (1.6).

N

i=0 ε

if

i

and gˆ N =

2.2. Bäcklund transformations Second, let us consider the problem of bilinear Bäcklund transformations. Suppose that the original system (1.5) has a bilinear Bäcklund transformation 4


Bij (λ)(ri · rj ) = 0

(2.12)

i,j =1

with a solution link: u = P (r1 , r2 ) and v = Q(r3 , r4 ), where λ is a free parameter and each Bij (λ) is a Hirota’s bilinear operator. This means that when the non-zero functions ri , 1
i
4, satisfy (2.12), the vector-valued function v = Q(r3 , r4 ) defines a new solution to (1.5) if u = P (r1 , r2 ) solves (1.5). To speak about bilinear Bäcklund transformations for the perturbation systems generally, we need a stability notion of Bäcklund transformations.

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Definition 2.2. A bilinear form (2.12) of a system of evolution equations (1.5) with a solution link: u = P (r1 , r2 ) and v = Q(r3 , r4 ), is called stable under small perturbation, if 4


  Bij (λ) ri (ε) · rj (ε) = o(ε)

(2.13)

i,j =1

implies that if u = P (r1 (ε), r2 (ε)) solves (1.5) up to a precision o(ε), i.e., ut = K(u) + o(ε), so does v = Q(r3 (ε), r4 (ε)), i.e., vt = K(v) + o(ε). Theorem 2.2. Suppose that a system of evolution equations (1.5) possesses a stable bilinear Bäcklund transformation (2.12) with a solution link: u = P (r1 , r2 ) and v = Q(r3 , r4 ). Then the perturbation system (1.6) has the following bilinear Bäcklund transformation 4



Bij (λ)(rik · rj l ) = 0,

0
n
N,

(2.14)

k+l=n i,j =1 k,l
0

with the solution link:   T T ηˆ N = PˆN (r10 , r20 ; . . . ; r1N , r2N ) = η0T , . . . , ηN ,   T T ξˆN = Qˆ N (r30 , r40 ; . . . ; r3N , r4N ) = ξ0 , . . . , ξNT ,

(2.15) (2.16)

where ηi and ξi , 1
i
N , are defined by η0 = P (r10 , r20 ), ξ0 = Q(r30 , r40 ),

 1 ∂P (ˆr1N , rˆ2N )  η1 =  , 1! ∂ε ε=0  1 ∂Q(ˆr3N , rˆ4N )  ξ1 =  , 1! ∂ε ε=0

..., ...,

 1 ∂ N P (ˆr1N , rˆ2N )  ηN =  , N! ∂ε N ε=0  1 ∂ N Q(ˆr3N , rˆ4N )  ξN =  , N! ∂ε N

(2.17) (2.18)

ε=0

with rˆiN =

N


ε j rij ,

1
i
4.

(2.19)

j =0

Proof. Suppose that (2.14) holds and rˆiN , 1
i
4, are defined by (2.19). What we need to prove is that if ˆ N are defined through (2.17) ηˆ N = PˆN solves the perturbation system (1.6), so does ξˆN = Qˆ N , where PˆN and Q and (2.18), respectively. Let us define   N
ε i ∂ i P (ˆr1N , rˆ2N )  vˆN : =  i! ∂ε i ε=0 i=0     ε ∂P (ˆr1N , rˆ2N )  ε N ∂ N P (ˆr1N , rˆ2N )  = P (r10 , r20 ) + + ··· +  ,  1! ∂ε N! ∂ε N ε=0 ε=0   i N i
ε ∂ Q(ˆr3N , rˆ4N )  wˆ N : =  i! ∂ε i ε=0 i=0     ε ∂Q(ˆr3N , rˆ4N )  ε N ∂ N Q(ˆr3N , rˆ4N )  = Q(r30 , r40 ) + + · · · +  ,  1! ∂ε N! ∂ε N ε=0 ε=0

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and thus   P (ˆr1N , rˆ2N ) = vˆN + o ε N ,

  Q(ˆr3N , rˆ4N ) = wˆ N + o ε N .

(2.20)

Note that (2.14) implies that (2.12) is true up to a precision o(ε N ) for rˆiN , 1
i
4. Therefore, by the stability of the Bäcklund transformation, if u = P (ˆr1N , rˆ2N ) is a solution to (1.5) up to a precision o(ε N ), so is v = Q(ˆr3N , rˆ4N ). On the other hand, because of (2.20), u = P (ˆr1N , rˆ2N ) and v = Q(ˆr3N , rˆ4N ) solve (1.5) up to a precision o(ε N ) if and only if     vˆN t = K(vˆN ) + o ε N , wˆ N t = K(wˆ N ) + o ε N , respectively. Thus, when (2.17) defines a solution to the perturbation system (1.6), so does (2.18). The proof is finished. 2 Let us similarly mention two special choices for the system of bilinear Bäcklund transformations (2.14): rik = ri , 1
i
4, 0
k
N, 1 ∂ k ri rik = , 1
i
4, 0
k
N, y = xj , 1
j
p, or t, k! ∂y k

(2.21) (2.22)

where ri , 1
i
4, are determined by (2.12). The first choice often leads to a trivial solution ξ0 = Q(r3 , r4 ), ξi = 0, 1
i
N . But the second choice gives a non-trivial solution, which is summarized in the following. Corollary 2.2. Let ri , 1
i
4, be determined by a stable bilinear Bäcklund transformation (2.12). Suppose that rˆiN =

N
ε k ∂ k ri , k! ∂y k

1
i
4,

k=0

where y = xj , 1
j
p, or t. Then if the vector-valued functions ηi , 1
i
N , defined by (2.17) solve the perturbation system (1.6), so does the vector-valued functions ξi , 1
i
N , defined by (2.18).

3. Application to the KdV equation The KdV equation (2.2) has its bilinear form [4,9]   Dx Dt + Dx3 (f · f ) = 0

(3.1)

under the transformation ∂2 log f, ∂x 2 where Dx and Dt are the Hirota bilinear operators:      ∂ m ∂ ∂ n ∂  m n −  −  f (x, t)g(x  , t  ) x  =x . Dt Dx (f · g) = ∂t ∂t ∂x ∂x t  =t u = P (f ) = −2

Moreover, it has the bilinear Bäcklund transformation  (Dt − 3λDx + Dx3 )(f · g) = 0, (Dx2 + λ)(f · g) = 0,

(3.2)

(3.3)

W.X. Ma, W. Strampp / Physics Letters A 341 (2005) 441–449

where λ is a free parameter, with the solution link u = P (f ) and v = P (g), where P is defined by (3.2). Note that under (3.2), we have

2(Dx Dt + Dx4 )f · f , ut − 6uux + uxxx = − f2 x

447

(3.4)

and that the Bäcklund transformation (3.3) is equivalent to (see, say, [9])     f 2 Dx Dt + Dx3 (g · g) − g 2 Dx Dt + Dx3 (f · f ) = 0. Therefore, the bilinear form (3.1) and the bilinear Bäcklund transformation (3.3) are all stable under small perturbation. Based on Theorem 2.1, the N th order perturbation system of the KdV equation [10] ηˆ N t = Kˆ N = Φˆ N ηˆ N x ,

(3.5)

with a hereditary symmetry operator [10]   0 Φ0   Φ1 Φ0 , Φˆ N =  .. ..   ... . . ΦN

···

Φ1

(3.6)

Φ0

∂ −1

where Φi = −δ0i + 2ηix + 4ηi , 0
i
N , has the bilinear form
  Dx Dt + Dx3 (fi · fj ) = 0, 0
k
N, ∂2

(3.7)

i+j =k i,j
0

under the transformation η0 = P (f0 ),

 1 ∂P (fˆN )  η1 = , 1! ∂ε ε=0

...,

 i where fˆN = N i=0 ε fi . In particular, the first-order perturbation system  η0t = 6η0 η0x − η0xxx , η1t = 6(η0 η1 )x − η1xxx ,

 1 ∂ N P (fˆN )  ηN = , N ! ∂ε N ε=0

possesses the bilinear form  Dx (Dt + Dx3 )(f0 · f0 ) = 0, Dx (Dt + Dx3 )(f0 · f1 ) = 0, under the transformation  2  2 η0 = − 2 f0 f0xx − f0x , f0

(3.9)

(3.10)

η1 = −

 2  2 2 f0 f1xx − f0 f0xx f1 − 2f0 f0x f1x + 2f0x f1 . 3 f0

Based on Theorem 2.2, the perturbation system (3.5) possesses the bilinear Bäcklund transformation:  3   k+l=n (Dt − 3λDx + Dx )(fk · gl ) = 0, 0
n
N, k,l
0   0
n
N,  k+l=n (Dx2 + λ)(fk · gl ) = 0, k,l
0

(3.8)

(3.11)

(3.12)

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W.X. Ma, W. Strampp / Physics Letters A 341 (2005) 441–449

with the solution link η0 = P (f0 ),

 1 ∂P (fˆN )  η1 = , 1! ∂ε ε=0

...,

 1 ∂ N P (fˆN )  ηN = , N ! ∂ε N ε=0

(3.13)

and

  1 ∂P (gˆ N )  1 ∂ N P (gˆ N )  ξ0 = P (g0 ), (3.14) ξ1 = , ..., ξN = , 1! ∂ε ε=0 N ! ∂ε N ε=0  N i i where fˆN = N i=0 ε fi , gˆ N = i=0 ε gi and P is defined by (3.2). In particular, for the first-order perturbation system (3.9), we obtain the following bilinear Bäcklund transformation  (Dt − 3λDx + Dx3 )(f0 · g0 ) = 0,     (D − 3λD + D 3 )(f · g ) + (D − 3λD + D 3 )(f · g ) = 0, t x 0 1 t x 1 0 x x (3.15) 2  (Dx + λ)(f0 · g0 ) = 0,    2 (Dx + λ)(f0 · g1 ) + (Dx2 + λ)(f1 · g0 ) = 0, with the solution link  2 ),  η0 = − 22 (f0 f0xx − f0x f0  ξ0 = − 22 (g0 g0xx g0

2 ), − g0x

η1 = −

2 2 f ), (f 2 f − f0 f0xx f1 − 2f0 f0x f1x + 2f0x 1 f03 0 1xx 2 g ). ξ1 = − 23 (g02 g1xx − g0 g0xx g1 − 2g0 g0x g1x + 2g0x 1 g0

(3.16)

4. Concluding remarks A class of bilinear forms and a class of bilinear Bäcklund transformations were furnished for the perturbation systems, based on the existence of bilinear forms and Bäcklund transformations for the initial systems of evolution equations. The stability concept was introduced to guarantee the hereditariness of bilinear properties. Two special situations of the resulting bilinear forms and Bäcklund transformations were analyzed. The Korteweg–de Vries equation is discussed as an illustrative example, and the corresponding bilinear forms and Bäcklund transformations were obtained, which are not easy to prove directly. We remark that if we assume that f = fˆN =

N
i=0

ε i fi ,

g = gˆ N =

N


ε i gi ,

i=0

satisfy (2.1), then comparing the powers ε i , 0
i
2N , in the resulting equation (2.1) leads to
B1 (fi · fj ) + B2 (fi · gj ) + B3 (gi · gj ) = 0, 0
k
2N.

(4.17)

i+j =k 0i,j N

Conversely, if we have this system, obviously (2.1) exactly holds for f = fˆN and g = gˆ N . This implies that the perturbation system (1.6) can have a bilinear form (4.17) under the same transformation (2.7), and the stability of the bilinear form (2.1) is not required. The bilinear system (4.17) seems more natural, but it has N more conditions than the expanded bilinear form (2.4), which are not expected. However, of the bilinear Bäcklund transformation is different. Although from the assumption that the situation j ri = rˆiN = N j =1 ε rij , 1
i
4, satisfy the Bäcklund transformation (2.12), we can generate a bigger Bäcklund

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449

transformation


4


Bij (λ)(rik · rj l ) = 0,

0
n
2N,

(4.18)

k+l=n i,j =1 0k,lN

with the same solution link defined by (2.15)–(2.18), the stability of the bilinear Bäcklund transformation (2.12) cannot be waived to guarantee its hereditariness. This is because if there is no stability condition, Q(ˆr3N , rˆ4N ) may not be an approximate solution to the original system (1.5) up to a precision o(ε N ) even if P (ˆr1N , rˆ2N ) is, and thus ξˆN defined by (2.16) may not solve the perturbation system (1.6) even if ηˆ N defined by (2.15) solves, noting that ηˆ N and ξˆN solve (1.6) iff P (ˆr1N , rˆ2N ) and Q(ˆr3N , rˆ4N ) are approximate solutions of (1.5) up to a precision o(ε N ), respectively. Finally, we would like to mention that the results in the previous section are also applicable to other soliton equations, since all bilinear forms and Bäcklund transformations with analytical transformations and solution links are stable under small perturbation. Moreover, a similar theory on bilinear forms and Bäcklund transformations can be furnished for triangular systems by perturbations [11] or by semi-products of loop Lie algebras [12]. Also, noting that the transformation (3.11) and the solution link (3.16) are quite complicated, we wonder whether there exist any simple transformations and solution links in the bilinear theory for the perturbation systems of soliton equations.

Acknowledgements One of the authors (Ma) would like to thank Xing-Biao Hu for stimulating discussions. Financial support from Dean’s office of the College of Arts and Sciences of the University of South Florida is also gratefully acknowledged.

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