Gauge transformations of constrained KP ows - Semantic Scholar

Journal of Mathematical Physics 36(6) (1995), pp. 2972{2984

Gauge transformations of constrained KP ows: new integrable hierarchies Anjan Kundu and Walter Strampp Fachbereich 17{Mathematik/Informatik GH{Universitat Kassel Hollandische Str. 36 34109 Kassel, Germany Walter Oevel Fachbereich 17{Mathematik/Informatik Universitat{GH Paderborn Warburger Str. 100 33095 Paderborn, Germany

Abstract

Integrable systems in 1+1 dimensions arise from the KP hierarchy as symmetry reductions involving square eigenfunctions. Exploiting the residual gauge freedom in these constraints new integrable systems are derived. They include generalizations of the hierarchy of the Kundu-Eckhaus equation and higher order extensions of the Yajima-Oikawa and Melnikov hierarchies. Constrained modi ed KP ows yield further integrable equations such as the hierarchies of the derivative NLS equation, the Gerdjikov-Ivanov equation and the Chen-Lee-Liu equation .

Running title: Gauge transformations of constrained KP ows PACS numbers: 03.40K, 11.10, 02.30



Permanent address: Saha Institute of Nuclear Physics, AF/1 Bidhan Nagar, Calcutta 700 064, India.

1

I Introduction It is well established that P the (scalar) KP hierarchy admits constraints [1-3] expressed through its symmetry ( i (i) (i) )x, where (i) and (i) are eigenfunctions and adjoint eigenfunctions of the corresponding linear system. In particular, the constraints

ux = (

m X i=1

(i) (i))x ; uy = (

m X i=1

(i) (i) )x and ut = (

m X i=1

(i) (i) )x ;

(1.1)

are known to produce vector versions of the AKNS , the Yajima-Oikawa and the Melnikov hierarchy, respectively [3]. A simple but crucial observation, which inspired the present investigation, is that the constraints (1.1) do not x uniquely the resulting equations obtained for (i) and (i) and allows a residual gauge freedom ~ (i) = e? i (i) ; ~ (i) = (i)e i (1.2) ( )

( )

with arbitrary functions (i) . These elds may be used to generate gauge transformed integrable systems. We will use this gauge freedom to obtain new integrable systems which include generalizations of the Kundu-Eckhaus [4-6] hierarchy, a higher-order Yajima-Oikawa hierarchy as well as a higher-order Melnikov hierarchy. We note that some of these extensions exhibit an interesting breaking of global symmetry, while integrability is still preserved. A similar gauge freedom is found for the constrained modi ed KP hierarchy, investigated in the recent past [7, 8]. The gauge freedom is even richer in this case, and even the simplest constraint incorporates dierent equations such as the derivative NLS system [9], the Gerdjikov-Ivanov equation [10] and the Chen-Lee-Liu [11] system. The organization of the paper is as follows. In Section 2 we introduce the main idea with a simple example leading to the hierarchy of the Kundu-Eckhaus equation. In Section 3 a consistent choice for the gauge function is proposed. This leads to a simple and general characterization of the integrable equations via an r-matrix formulation of their Lax representations. In Section 4 it is demonstrated that the simplest examples include extended Yajima-Oikawa and Melnikov systems. In Section 5 we brie y discuss alternative generalizations, while in Section 6 a similar construction is proposed for the constrained modi ed KP hierarchy.

II Constrained KP ows and the Kundu-Eckhaus hierarchy The integrable KP hierarchy can be formulated through Zakharov-Shabat type linear systems [12] (2.1) tl = Bl ; tl = ?Bl  ; l = 1; 2; : : :

with an in nite hierarchy of operators Bl = (Ll)+ ; where the subscript + denotes the projection of the powers of the micro-dierential Lax operator L = @ + u @ ?1 + u2 @ ?2 + u3 @ ?3 + : : : ; @ = @x (2.2) 2

onto its dierential part. The leading coecient u plays a distinguished role, since it satis es the KP equation and its higher ows. The rst of the operators Bl are computed as B1 = @ ; B2 = @ 2 + 2u ; B3 = @ 3 + 3u@ + 3u2 + 3ux : (2.3) The Lax equations Ltl = [Bl; L] ; l = 1; 2; : : : (2.4) imply the compatibility conditions Bl;tk ? Bk;tl + [Bl ; Bk ] = 0 (2.5) of (2.1). They yield the eld equations along with dierential relations among the elds ui such as u2 = ? 21 ux + 21 @ ?1 uy ; (2.6) u3 = 41 uxx ? 12 uy ? 12 u2 + 41 @ ?2 uyy : Consequently all equations of this hierarchy can be expressed through the single eld u. We will refer to solutions of (2.1) as (adjoint) eigenfunctions, although in the following no spectral problem L =  , L  =   will be assumed. This corresponds to the Inverse Scattering Transform of the KP hierarchy, in which the evolution equation y = B2 associated with y = t2 actually is the \spatial" part of the scattering problem. Any (formal) spectral equation would only introduce some (asymptotic) dependence on a spectral parameter, which bears no relevance for the following. It has been observed that the product  of an eigenfunction and an adjoint eigenfunction represents a conserved covariant of the KP hierarchy. Hence, one may impose constraints on the KP ows by expressing the dynamical eld u through  [1-3]. In terms of the Lax operator (2.2) these constraints are characterized by the requirement that the negative dierential orders of a power Lk have the speci c form (Lk )? =

m X i=1

(i) @ ?1 (i) ;

(2.7)

where m pairs of (adjoint) eigenfunctions are considered. For given k this constraint leaves the coecients u; u2; ::; uk?1 in (2.2) and (i) , (i) as independent elds, whereas uk ; uk+1 ; :: become dierential expressions of these functions. One may replace L by the new Lax operator

L0 = Lk = @ k + k u @ k?2 +

kX ?3 j =0

Uj @ j +

m X i=1

(i) @ ?1 (i) ;

(2.8)

where U0 ; ::; Uk?3 are dierential expressions of u; u2; ::; uk?1. Thus the elds u; U0; ::; Uk?3 may be regarded as new independent elds related to u; u2; ::; uk?1 by a coordinate transformation. It is readily veri ed that the Lax equations (2.4) subject to the constraint 3

(2.7) automatically imply that (i) and (i) are (adjoint) eigenfunctions satisfying (2.1). We note that these constraints may indeed be regarded as symmetry reductions of the KP-hierarchy, since utk = res(L)tk = res([(Lk )+ ; L]) = res([L+ ; Lk ]) = res(Lk )x ; (2.9) so that (2.7) implies the relation

utk = (

m X i=1

(i) (i))x

(2.10)

between the kth ow of the KP hierarchy and the symmetry generated by square eigenfunctions. The simplest of these constraints k = 1, m = 1 is given by L = @ + @ ?1  = @ +  @ ?1 ? x @ ?2 + xx @ ?3  : : : ; (2.11) i.e. u =  ; u2 = ? x ; u3 = xx ; : : : : (2.12) In this case (2.1)/(2.4) yields the AKNS hierarchy for ;  , while u solves the KP equation and its higher ows. We now want to motivate the considerations of the next sections using the AKNS constraint (2.12). We observe that in the factorization u =  of the KP hierarchy there still remains a gauge freedom. One may introduce gauge transformed (adjoint) eigenfunctions ~ = h?1 ; ~  =  h (2.13) with an arbitrary function h. It is noted that under such a change the KP solution u remains the same, while the corresponding AKNS hierarchy is transformed into (2.14) ~ tl = B~l ~ ; ~ tl = ?B~l ~  ; l = 1; 2; : : : with the new dierential operators B~l = h?1  Bl  h ? h?1 htl : (2.15) We introduce the gauge eld in the form h = e , which simpli es the structure of the transformed operators B~l . For example, with t2 = y one obtains B~2 = B2 + 2x @ ? y + x2 + xx (2.16) and the corresponding linear equations ~ y = ~ xx + 2x ~ x + (2u ? y + x2 + xx ) ~ ; (2.17) ~ y = ? ~ xx + 2x ~ x ? (2u ? y + x2 ? xx ) ~  : 4

We observe that one may choose the gauge eld  as the potential de ned by x = ~ ~  = u ; (2.18) y = ~ x ~  ? ~ ~ x + 2( ~ ~ )2 : The consistency xy = yx may be checked directly from the equations (2.17) and (2.18). Elimination of  and u nally yields the following gauge transformed AKNS system   ~ y = ~ xx + 2 ~ ~  + 2 ( ~ ~  )x ? ( ~ ~  )2 ~ ;   (2.19) ~ y = ? ~ xx ? 2 ~ ~  ? 2 ( ~ ~  )x ? ( ~ ~  )2 ~  ; which is recognized as the Kundu-Eckhaus equation [4, 5]. At this stage the de nition of  via (2.18) seems ad hoc. We will show in the next section that there is a systematic construction of the gauge eld in terms of the conserved densities of the KP hierarchy, i.e.  may be chosen to satisfy the potential KP hierarchy. Adjoining a suitable compatible time evolution with respect to the next time t = t3 to the de nition (2.18), one obtains the next higher ow of the Kundu-Eckhaus hierarchy. Any simultaneous solution of this system and (2.19), each of them being a system in 1+1 dimensions, gives rise to a KP solution u =  = ~ ~  .

III Consistent choice of the gauge function The transformation 7! ~ = e? ;  7! ~  =  e (3.1) of the (adjoint) eigenfunctions for the Lax operator (2.2) produces (adjoint) eigenfunctions of the gauge transformed Lax operator L~ = e?  L  e = @ + x + u(@ + x)?1 + u2(@ + x )?2 + : : : ; (3.2) where we note e?  @ i  e = (e?  @  e )i = (@ + x )i (3.3) for arbitrary powers i. The Lax equations (2.4) are mapped to corresponding equations for L~ with B~l given by (2.15), i.e. B~l = e?  (Ll )+  e ? tl = (L~ l)+ ? tl : (3.4) The choice of the gauge function  corresponding to the example of the last section is the following. One de nes  as the potential characterized by tl =  res(Ll ) =  res(L~ l ) ; l = 1; 2; : : : ; (3.5) where res is the usual residue of a micro-dierential operator, i.e. the coecient of @ ?1 . The arbitrary constant  is introduced as a deformation parameter. 5

We remark that any choice of  given by an arbitrary function of the KP elds, i.e. the coecients ui of L in (2.2), gives rise to a closed system of equations for the gauge transformed elds. However, in this case the transformation represents a mere \change of variables" ui 7! Fi (u1; u2; : : :) with some dierential expressions Fi . With the choice (3.5)  is a non-trivial potential, still preserving the local character of the reductions to be considered in the following. The compatibility tm tn = tn tm is easily veri ed, since the Lax equations (2.4) imply tm tn =  res([(Ln)+; Lm]) =  res(Lm+n ? (Lm)? Ln ? (Ln)?Lm ) ; (3.6) which is clearly symmetric in m and n. The de ning equations (3.5) determine  up to an integration constant. This does not contribute to the gauge transformation (3.2), and consequently L~ is characterized in terms of derivatives of . We note that the residues represent conserved densities of the KP hierarchy, since res(Ll )tn = res([(Ln)+ ; Ll]) (3.7) is a perfect x-derivative. For l = 1 equation (3.5) yields x =  u, so that  is a solution of the potential KP hierarchy. Consequently,  is the highest non-trivial coecient in the dressing operator W = 1+  @ ?1 +(:::)@ ?2 +


, which generates L = W ?1 @W by dressing the bare operator @ . Also, we note the link  = x = to the  function of the KP hierarchy. One nally obtains the deformed Lax operator L~ = @ +  u + u (@ +  u)?1 + u2 (@ +  u)?2 + : : : (3.8) = @ +  u + u @ ?1 + (u2 ?  u2 ) @ ?2 + : : : : With (3.4) and (3.5) the new Lax equations are given by (3.9) L~ tl = [B~l; L~ ] = [r(L~ l); L~ ] with r(A) = A+ ?  res(A) ? 21 A = 21 (A+ ? A? ) ?  res(A) : (3.10) We note that this map satis es the modi ed Yang-Baxter equation [13] (3.11) [r(A); r(B )] + 41 [A; B ] = r([r(A); B ] + [A; r(B )]) for all micro-dierential operators A; B and any  2 IR, so that (3.10) yields the r-matrix of the ows (3.9). Up to this point one has not gained any new results from the gauge transformation of the KP hierarchy. However, we now look for reductions to 1+1 dimensions, where 6

the new formulation yields results which are dierent from the standard reductions. The

k-constraint (2.7) yields

m

m

i=1

i=1

X X (L~ k )? = e?  (Lk )?  e = e? (i) @ ?1 (i)e = ~ (i) @ ?1 ~ (i) :

(3.12)

One may replace the Lax operator (3.8) by its kth power L~ 0 = L~ k and obtains the Lax representation L~ 0tl = [B~l; L~ 0] ; B~l = (L~ 0l=k )+ ?  res(L~ 0l=k ) ; l = 1; 2; : : : (3.13) for Lax operators of the form

L~ 0 = (@ +  u)k + k u (@ +  u)k?2 +

kX ?3 j =0

Uj (@ +  u)j +

m X i=1

~ (i) @ ?1 ~ (i) ;

(3.14)

which are deformations of the operators (2.8). By construction the negative dierential orders of (3.13) are compatible with the assumption that ~ (i) and ~ (i) are (adjoint) eigenfunctions satisfying (3.15) ~ (tli) = B~l ~ (i) ; ~ (tli) = ?B~l ~ (i) :

We nally note that the deformation process induced by the parameter  may also be regarded as a Miura type transformation between the equations given by (2.1) and (3.15), respectively. Elimination of  from (3.1) yields the Backlund relations     (i)  ~ (i)  = tl =  res(L~ l ) ; (3.16) ln ~ (i) t = ln (i) tl l with the spatial part     (i)  ~ (i)  = x =  u : (3.17) ln ~ (i) x = ln (i) x They provide the transformation from (i) ; (i) associated with  = 0 to the gauge transformed (adjoint) eigenfunctions ~ (i) ; ~ (i) associated with arbitrary .

IV Examples: extended AKNS, Yajima-Oikawa and Melnikov hierarchies Depending on the choice of the integer k we nd the following examples, where we use the notation t1 = x, t2 = y and t3 = t:

7

Example 1: For k = 1 one nds x =  res(L~ ) =  u =  y =  res(L~ 2) =  for the Lax operator

m X i=1

( ~ xi

( )

m X

~ (i) ~ (i) ;

i=1 ~ (i) ? ~ (i) ~ (xi)) + 2 2 u2

m

m

i=1

i=1

(4.1)

;

X X L~ = @ +  u + ~ (i) @ ?1 ~ (i) = @ + ~ (i) (@ ?1 + ) ~ (i) :

(4.2)

The basic equation of the corresponding vector Kundu-Eckhaus hierarchy for the elds ~ (1) ; ::; ~ (m) is ~ (yi) = ~ yi  ( )

~ (xxi) + 2  u ~ (xi) +



2 u ?  2 u2 + 2 

m X



~ (j ) ~ x(j ) ~ (i) ;

j =1 m   X i  ( i )  = ? ~ xx + 2  u ~ x + ? 2 u + 2 u + 2  ~ (xj ) ~ (j ) ~ (i) j =1 P has to insert the constraint u = j ~ (j ) ~ (j ) of the KP hierarchy. ( )

(4.3)

;

where one This equation reduces to (2.19) for a single pair of (adjoint) eigenfunctions and  = 1. For  = 0 one obtains the usual AKNS system.

Example 2: For k = 2 , i.e for the constraint uy = (

m X i=1

~ (i) ~ (i))x

(4.4)

of the KP hierarchy one obtains

x =  res(L~ ) =  u ; y =  res(L~ 2) = 

m X i=1

~ (i) ~ (i) :

(4.5)

The rst ow of the corresponding hierarchy of equations (3.13) for u; ~ (i); ~ (i) associated with the Lax operator

L~ 0 = L~2 = (@ +  u)2 + 2 u + is given by ~ (yi) =

m X i=1

~ (i) @ ?1 ~ (i)

(4.6) m





X ~ (xxi) + 2  u ~ (xi) + 2 u +  ux + 2 u2 ?  ~ (j ) ~ (j ) ~ (i) ;

j =1 m X



~ (yi) = ? ~ (xxi) + 2  u ~ (xi) + ? 2 u +  ux ? 2 u2 +  8

j =1



~ (j ) ~ (j ) ~ (i) ;

(4.7)

complemented by (4.4). For  = 0 one obtains the Yajima-Oikawa system.

Example 3: The next higher constraint (2.7) with k = 3 implies ut = (

m X i=1

~ (i) ~ (i))x ;

(4.8)

so that the KP equation becomes m  X 1 4 ~ (i) ~ (i) x ; uy = vx ; vy = ? 3 uxxx ? 4uux + 3 i=1 where we have put u2 = (v ? ux )=2 in (2.2). One nds x =  res(L~ ) =  u ; y =  res(L~ 2) =  v ;

t =  res(L~ 3) = 

m X i=1

~ (i) ~ (i) ;

(4.9)

(4.10)

and the rst equations of the gauge transformed hierarchy for u; v; ~ (i); ~ (i) are given by   ~ (yi) = ~ (xxi) + 2  u ~ (xi) + 2 u +  ux ? v + 2 u2 ~ (i) ; (4.11)   ~ (yi) = ? ~ (xxi) + 2  u ~ (xi) + ? 2 u +  ux +  v ? 2 u2 ~ (i) ; together with the equations (4.9). The next higher ow is computed as i) + 3  u ~ (xxi) + 3 (u +  ux + 2 u2 ) ~ (xi) ~ (ti) = ~ (xxx m   X + 23 ux + 32 v +  uxx + 3  u2 + 3 2 uux + 3 u3 ?  ~ (j ) ~ (j ) ~ (i) j =1 (4.12) ( i )  i) ? 3  u ~ (xxi) + 3 (u ?  ux + 2 u2 ) ~ (xi) ~ t = ~ (xxx m   X + 32 ux ? 23 v ?  uxx ? 3  u2 + 3 2 uux ? 3 u3 +  ~ (j ) ~ (j ) ~ (i) j =1 along with the t-evolution equations given by the constraint (4.8) and

vt =

m  X j =1



~ (xj ) ~ (j ) ? ~ (j ) ~ x(j ) + 2  u ~ (j ) ~ (j ) x :

This last equation follows from (4.11) with

vt = res(L~ 2)t = res(L~ 3)t = 3

2

m X



~ (i) ~ (i) y : i=1

(4.13) (4.14)

The associated Lax operator is m X L~ 0 = L~3 = (@ +  u)3 + 23 u (@ +  u) + 32 (@ +  u) u + 32 v + ~ (i) @ ?1 ~ (i) : (4.15) i=1 For  = 0 one obtains the standard Melnikov hierarchy. 9

V Multicomponent generalizations It was shown that the vector AKNS equation (4.3) with  = 0 is gauge transformed into the vector Kundu-Eckhaus equation (4.3) with arbitrary . Apparently the invariance of the original AKNS system under the SU (m) transformations ( ~ (i) ; ::; ~ (m)) 7! ( ~ (i); ::; ~ (m)) U ; ( ~ (i); ::; ~ (m))y 7! U y ( ~ (i); ::; ~ (m))y (5.1) with UU y = I is preserved in this deformation. We note that this symmetry is easily broken, while still preserving integrability, when more general gauge transformations (1.2) are considered, where (i) is dierent for each pair of (adjoint) eigenfunctions. The gauge elds may be chosen consistently as the potentials de ned by

t(li) = (i) res(@ ?1 (i)Ll (i) @ ?1 ) ; l = 1; 2; : : :

(5.2)

implying

x(i) = (i) (i) (i)

(5.3) for l = 1. Here not only the potentials, but also the deformation parameters (i) are chosen independently for each pair. The proof of the integrability condition t(lit)m = t(mi)tl may be found in [14] (Lemma 1) or [8]. For the simplest constraint k = 1, i.e.

u=

m X i=1

one obtains

(i) (i) =

m X i=1

m

X ~ (i) ~ (i) ; L = @ + (i) @ ?1 (i) ;

i=1

x(i) = (i) (i) (i) = (i) ~ (i) ~ (i) ; y(i) = (i) ( (xi) (i) ? (i) (xi) ) = (i) ( ~ (xi) ~ (i) ? ~ (i) ~ (xi) ) + 2 (i)2 ( ~ (i) ~ (i) )2 ;

(5.4)

(5.5)

and the transformations (1.2) map (2.1) into the following multicomponent generalization of the Kundu-Eckhaus equation   ~ (yi) = ~ (xxi) + 2 u + 2 (i) ( ~ (i) ~ (i))x ? (i)2 ( ~ (i) ~ (i) )2 ~ (i) ; (5.6)   ~ (yi) = ? ~ (xxi) + ? 2 u + 2 (i) ( ~ (i) ~ (i) )x + (i)2 ( ~ (i) ~ (i))2 ~ (i) ; where u has to be inserted from (5.4). It is signi cant to observe that the original global SU (m) symmetry has been broken down to just U (1) due to the presence of anisotropic

10

P

extensions. We remark that the eld  =  i (i) =(i) is the potential of the last section satisfying tl =  res(Ll ). This is easily seen from integrating the relation m X

m X ? 1 (i) l (i) ?1 tl x =  res(@ L @ )x =  res( [ @ ; @ ?1 (i) Ll (i) @ ?1 ] ) i=1 i=1 m X (5.7) l (i) ?1 (i) =  res( [ L ; @ ] ) =  res( [ Ll ; L? ] ) i=1 =  res( [ L+ ; Ll ] ) =  res( [ @ ; Ll ] ) =  res(Ll )x :

Hence, for a single pair of (adjoint) eigenfunctions m = 1, it is no surprise that the multicomponent extensions (4.3) and (5.6) of the Kundu-Eckhaus equation coincide.

VI Gauge transformations of the constrained modi ed KP hierarchy The modi ed KP hierarchy arises from the Lax representation ([15, 8]) Ltl = [Bl; L] ; Bl = (Ll)1 ; l = 1; 2; ::: (6.1) where (:)1 denotes the projection to dierential orders strictly larger than zero. The Lax operator is given by L = @ + v + u @ ?1 + v2 @ ?2 + v3 @ ?3 +


; (6.2) where v solves the modi ed KP equation and u solves the KP equation. The formal adjoints of the linear problems tl = Bl can be integrated, so that we regard (6.3) tl = Bl ; tl = ?@ ?1 Bl @  as the associated linear (adjoint) problems. With t2 = y the rst non-trivial linear problem is given by (yi) =

(xxi) + 2 v (xi) ;

(6.4) (yi) = ? (xxi) + 2 v (xi) : Our aim is to demonstrate that the gauge transformation formulated for the constrained KP system is applicable also to this case. As before, we consider the gauge transformation ~ (i) = e? (i) ; ~ (i) = (i) e ; (6.5) which maps (6.4) into ~ (yi) = ~ (xxi) + 2 (v + x ) ~ (xi) + (xx + x2 + 2 v x ? y ) ~ (i) ; (6.6) ~ (yi) = ? ~ (xxi) + 2 (v + x ) ~ (xi) + (xx ? x2 ? 2 v x + y ) ~ (i) : 11

A consistent choice of the gauge eld  is given with the potential de ned by tl = 1 res( Ll ) + 2 res( Ll @ ?1 ) (6.7) with two arbitrary deformation parameters 1;2. The compatibility tn tm = tm tn is veri ed from the relation tm tn = res( [ (Ln)1 ; Lm ] (1 + 2 @ ?1 ) )   (6.8) = res( Lm+n ? (Lm )