Squared Eigenfunction Symmetries for Soliton Equations: Part I

Ž .JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 217, 161]178 1998ARTICLE NO. AY975707

Squared Eigenfunction Symmetries for SolitonEquations: Part I

Walter Oevel

FB17-Mathematik, Uni ersitat Paderborn, D 33095 Paderborn, Germany¨

and

Sandra Carillo

Dipartimento di Metodi e Modelli Matematici, per le Scienze Applicate,Uni ersita ‘‘La Sapienza,’’ I 00161 Rome, Italy`

Submitted by Colin Rogers

Received May 2, 1997

Three classes of soliton systems associated with scalar Lax operators are consid-ered. They represent, in turn, the 2 q 1-dimensional hierarchies of the KP equa-tion, the modified KP equation and the Dym equation. They are related via gaugetransformations and reciprocal links. For each class an algebraic construction of

Ž .the symmetry transformation generated by products of adjoint eigenfunctions isgiven. The links between the soliton hierarchies are extended to these symmetries.Q 1998 Academic Press

Key Words: integrable systems; squared eigenfunctions; Backlund transforma-¨tions; KP hierarchy; modified KP hierarchy; Dym hierarchy

1. INTRODUCTION

Backlund and reciprocal transformations represent powerful tools for¨investigating nonlinear evolution equations. They play a key role in estab-lishing structural properties such as Hamiltonian andror bi-Hamiltonianstructure, symmetry properties, etc. Also, they are a means of findingsolutions of nonlinear equations. An extensive bibliography concerningapplications of Backlund transformations in solving boundary value prob-¨lems, as well as initial value problems, arising in connection with models in

w xgas dynamics, fluid mechanics, and other areas, is given in 23 , and,w xsubsequently, in 22 .

161

0022-247Xr98 $25.00Copyright Q 1998 by Academic Press

All rights of reproduction in any form reserved.

OEVEL AND CARILLO162

For integrable nonlinear equations, arising as the compatibility of twolinear equations, Backlund and reciprocal transformations originate from¨transformations of the Lax pair. Certain Lax equations are associated witha Lie-algebraic decomposition of the space of pseudo-differential symbolsw x1, 9, 20 . This framework provides a convenient terminology to discuss thecorresponding nonlinear equations and their hierarchies. On the basis of

w xthe results established in 12, 18 , the links among hierarchies of nonlinearevolution equations in 2 q 1 dimensions are reconsidered. Section 2 isdevoted to notation and a short overview on background facts. In Section 3links among different hierarchies of 2 q 1-dimensional equations areconsidered. Section 4 comprises the introduction of a convenient terminol-ogy for squared eigenfunction symmetries associated with these hierar-chies. The links are extended to the squared eigenfunction symmetries.

However, as pointed out in Section 5, corresponding results exist for thereductions of the 2 q 1-dimensional hierarchies to soliton equations in

w x1 q 1 dimensions. The 1 q 1-dimensional case is studied in detail in 17 .Proofs of all lemmata and theorems are given in the appendix.

2. NOTATION AND BACKGROUND

Consider the algebra g of pseudo-differential symbols and its decompo-sitions,

g s u i s u i [ u i \ g [ g , 2.1Ž .Ý Ý Ýi x i x i x G k - k½ 5 ½ 5 ½ 5y`-ig` iGk i-k

into differential orders G k and - k, where,

` i i y 1 ??? i y j q 1Ž . Ž .i i Ž j. iyj u s u q u ,Ýx x xj!js1

juŽ j.u ' u ' .u jx ??? x x^ _

j

In particular, y1 u s uy1 y u y2 q u y3 y ??? . The coefficients ux x x x x x x iare scalar functions of x ' t and some further ‘‘time’’ parameters1t , t . . . . For L s Ý u i g g the projections to g and g are de-2 3 i i x G k - knoted by

L s u i , L s u i ,Ý ÝG k i x - k i xiGk i-k

SQUARED EIGENFUNCTION SYMMETRIES: PART I 163

further we introduce the zero-order term and the residue,

L s u , res L s u .Ž .0 0 y1

Ž .In 2.1 the cases k s 0, 1, 2 are distinguished by the fact that g andG kw x w xg define sub-Lie algebras of g : g , g ; g and g , g ;- k G k G k G k - k - k

Ž .g . Hence, in these cases, the decompositions 2.1 define classical- kw xr-matrices 25 ,

r : L ª L y L ,k G k - k

won g and give rise to three hierarchies of integrable Lax equations 2, 10,x18 which will be reviewed in Section 3.

w xRemark 2.1. For a differential operator M g g the symbol MfG 0denotes the action of M on the function f. This is not to be confused withMf, denoting the composition of M with the multiplication operator given

w x Ž .by f. We note Mf s Mf .0

The variational adjoint of an operator in g is given by

Uii iu s y1 u ,Ž .Ý Ýi x x iž /

i i

whence for any differential operator M g g and all functions f, c theG 0w x w U xexpression c Mf y M c f s v is a perfect x-derivative of some fieldx

v. In the context of pseudo-differential symbols v is conveniently ex-pressed by

v s res y1c Mfy1 .Ž .x x

One has the following lemma, which defines the ‘‘squared eigenfunctionpotential’’:

LEMMA 2.2. For any differential operator M g g and any pair ofG 0functions f, c sol ing the two adjoint equations,

w x w U xf s Mf , c s y M c , 2.2Ž .t t

Ž .there exists a potential V s V c , f defined by the compatible equations,

V s cf , V s res y1c Mfy1 .Ž .x t x x

This potential, to be regarded as a bilinear functional of f and c , iswell defined up to an additive constant. We remark that the evolution


parameter t may actually represent a collection of ‘‘times’’ t introducednw x w U xby f s M f and c s y M c , where M g g is a family oft n t n n G 0n n

differential operators satisfying the ‘‘zero curvature’’ conditions M yn t1 n2w xM s M , M granting the compatibility of the linear flows. Then alln t n n2 n 2 11

equations,

V s cf , V s res y1c M fy1Ž .x t x n xn

are compatible.

3. SOLITON HIERARCHIES AND THEIR LINKS

In this section we review established results on the algebraic construc-tion of the 2 q 1-dimensional hierarchies of the KP, the mKP, and theDym equations as well as the links which relate them.

Application of Backlund and reciprocal transformations to interconnect¨nonlinear evolution equations has its origin in the work by Rogers and

w xWong 24 . Subsequently, links among nonlinear evolution equations wereshown to represent a powerful tool to reveal interesting properties of

w xwhole hierarchies of nonlinear evolution equations 4, 8 . Specifically,equations connected to each other via Backlund charts admit symmetries,¨

w xbi-Hamiltonian structures, recursion operators etc., which are related 7 .In particular, the transformations relating the Dym equation and the KdV

w xequation 24 allow to recognize the interesting symmetry structures of thew x w xDym equation 8 . The corresponding hierarchy 3 was proved to exhibit

an invariance under a reciprocal transformation, which induces a genericauto-Backlund transformation for the KdV hierarchy. A reciprocal con-¨nection between the KP equation and the 2 q 1-dimensional Dym equa-

w xtion, as introduced by Konopelchenko and Dubrovsky 11 was noted inw x21 . A review on recent results in 2 q 1 dimensions is given by

w xKonopelchenko and Rogers 13 .Using the pseudo-differential approach, gauge and reciprocal links

among hierarchies of nonlinear evolution equations in 2 q 1 dimensionsw xwere established in 18 . In particular, the provide novel auto-Backlund¨

transformations for the Dym hierarchy.The algebraic construction of the KP hierarchy and its relatives is based

w xon the following simple observation 2, 10, 18 :

THEOREM 3.1. Let k s 0, 1, or 2. The equations of the Lax hierarchy,

nL s L , L ,Ž .Gktn


with

k s 0 : L s q uy1 q u y2 q u y3 q . . . ,KP x x 2 x 3 x

k s 1 : L s q ¨ q ¨ y1 q ¨ y2 q . . . ,mKP x 1 x 2 x 3.1Ž .k s 2 : L s w q w q w y1 q w y2 q . . . ,Dym x 0 1 x 2 x

commute.

w xFollowing the r-matrix scheme 25 this result is a simple consequence ofŽ .the Lie algebra decomposition 2.1 with k s 0, 1, or 2.

Each Lax equation represents an evolution equation for the infiniteŽ . Ž . Ž .vector u, u , . . . , ¨ , ¨ , . . . , and w, w , . . . , respectively. Equations in2 1 0

closed form can be extracted for the ‘‘prime’’ fields u, ¨ , w. The equationswŽ 2 . x Ž .resulting from L s L , L with y s t are used to express all fieldsy G k 2

in terms of u, ¨ , or w, respectively, and their y-derivatives. In particular,the equations defining the ‘‘auxiliary’’ fields u , ¨ , and w , respectively,2 1 0are computed as

k s 0 : u s u q 2u ,y x x 2 x

k s 1 : ¨ s ¨ q 2¨¨ q 2¨ ,y x x x 1 x 3.2Ž .k s 2 : w s w2 w q 2w2 w .y x x 0 x

The Lax equations for n G 3 represent equations in t , x, and y for thenfields u, ¨ , and w. In particular, the equations associated with n s 3 are

k s 0 : 4u s u q 12uu q 3u , KPŽ . Ž .t x x x x x y yx3

k s 1 : 4¨ s ¨ y 6¨ 2 ¨ q 3¨ q 6¨ ¨ q 6¨ Dy1 ¨ , mKPŽ . Ž .Ž .t x x x x x y y x y x x yx3

DymŽ .1 wy3 2 y1k s 2 : 4w s w w q w D ,t x x x 2ž /3 ž /2 w y

i.e., the KP equation, the modified KP equation and the 2 q 1-dimensionalDym equation. The integration Dy1 s H x arises from the elimination of

Ž .the auxiliary fields via 3.2 . For the derivation of these equations onecomputes

k s 0 : L2 s 2 q 2u ,Ž .KP xG0

L3 s 3 q 3u q 3 u q u ,Ž .Ž .KP x x x 2G0

k s 1 : L2 s 2 q 2¨ ,Ž .mKP x xG1

L3 s 3 q 3¨ 2 q 3 ¨ q ¨ 2 q ¨ ,Ž . Ž .mKP x x x 1G1

k s 2 : L2 s w2 2 ,Ž .Dym xG2

L3 s w3 3 q 3w2 w q w 2 .Ž .Ž .Dym x x 0 xG2


Eigenfunctions are defined as solutions of the evolution equations f stnwŽ n. xL f , which for n s 2, 3 yieldG k

k s 0 : f s f q 2uf , f s f q 3uf q 3 u q u f ,Ž .y x x t x x x x x 23

k s 1 : f s f q 2¨f , f s f q 3¨f q 3 ¨ q ¨ 2 q ¨ f ,Ž .y x x x t x x x x x x 1 x3

k s 2 : f s w2f , f s w3f q 3w2 w q w f . 3.3Ž . Ž .y x x t x x x x 0 x x3

The three hierarchies are linked through Darboux type transformationsgenerated by functions satisfying the eigenfunction dynamics:

w xTHEOREM 3.2. 12, 18 .

Ž . Ž . wŽ n. x wŽ n. xa KP ª mKP : If L s L , L and f s L f , thent G 0 t G 0n n

˜ y1 ˜ ñ ˜wŽ . xL s f Lf satisfies L s L , L .t G1n

˜ ñ ˜ ˜ ñ ˜Ž . Ž . wŽ . x wŽ . xb mKP ª Dym : If L s L , L and f s L f , then,t G1 t G1n n

˜Ž .after the transformation x s f x, t and t s t of the independent äriables,n n nn˜ wŽ . x Ž .the operator L s L satisfies L s L , L , where the projection ?t G2 G2n

refers to powers of s r x.x

˜Ž . Ž .In b the change of variable x ª x s f x, t induces a transformationn˜y1 y1 y1 y1˜ ª s f , ª s f of the pseudo-differential symbolsx x x x x x x x

Ž . Ž . Ž .as well as a transformation w x, t ª w x, t s w x, t of functions. In˜ ñ n ni i i˜ ˜Ž .particular, this link maps L s Ý ¨ to L s Ý w s Ý ¨ f . Ini i x i i x i i x x

Ž .short, the transformation b is a pure change of the independent vari-ãbles: the new Lax operator L coincides with the original L, which has to

be rewritten in powers of the new symbol s r x associated with thexnew ‘‘space’’ variable x.

Ž .The transformation a of this theorem applied to the Lax operator ofthe KP hierarchy leads to

L s fy1L f s fy1 q uy1 q ??? f s q fy1f q uy1 q ??? ,Ž .mK KP x x x x xP

Ž .whence comparison with 3.1, k s 1 leads to the transformation u ª ¨ sy1 Žf f , where f is an eigenfunction of the KP hierarchy satisfying 3.3,x

. Ž .k s 0 . The inverse map ¨ ª u s ¨ with ¨ defined by 3.2, k s 1 is the1 1standard Miura transformation,

2u q ¨ q ¨ 2 s ¨ ,Ž .x yx

between the KP hierarchy and the modified KP hierarchy.


Ž .Similarly, the transformation b between the mKP and the Dym hierar-chies is obtained by

˜L s L s q ¨ q ??? s f q ¨ q ??? ,Dyn mKP x x x

Ž .whence comparison with 3.1, k s 2 leads to the transformation ¨ ª w s˜ ˜ ˜f , x ª x s f, y ª y s y, where f is any eigenfunction of the modifiedx

Ž .KP hierarchy satisfying 3.3, k s 1 . The inverse map is given by w ª ¨ sŽ .w with w defined by 3.2, k s 2 in the variables x, y, i.e.,0 0

w xy2¨ s y ww , s w , y s y.x x xw x

A 1 q 1-dimensional reciprocal link of such kind was first derived in thew xcontext of Painleve expansions 26, 27 .´

4. SQUARES EIGENFUNCTION SYMMETRIES ANDTHEIR LINKS

In this section we formulate the squared eigenfunction symmetriesassociated with the 2 q 1-dimensional hierarchies considered in the previ-ous section. The links of Section 3 are extended to these symmetries.

In 2 q 1 dimensions eigenfunctions are defined as solutions of thewŽ n. xlinear systems f s L f , where the equation associated with t s yt G k 2n

is regarded as the scattering problem used to solve the correspondingw xnonlinear equations via inverse scattering techniques 6 . The fact that

products of eigenfunctions and adjoint eigenfunctions can be regarded asinfinitesimal generators of symmetry transformations is crucial for suchconsiderations. In this section we formulate these symmetries algebraicallyfor the three hierarchies k s 0, 1, 2 considered in the previous section.

For the cases k s 1 and 2 it turns out that integrated versions of theusual adjoint linear problems are more appropriate. One observes that the

yk Ž n.U k Ž kŽ n. yk .Uoperators L s L are again differential opera-x G k x x G k xtors. The ‘‘adjoint’’ linear evolution equations to be considered are, in fact,

Ž n.Uassociated with these operators rather than with the usual adjoints L .G kWith L s L , L s L , or L s L , respectively, we shall defineKP mKP Dym

eigenfunctions and ‘‘adjoint eigenfunctions’’ as solutions of the linearproblems,

Un yk n kf s L f , c s y L c . 4.1Ž . Ž . Ž .Gk Gkt t x xn n

We remark that for k s 1 and 2, respectively, the fields c and c ,x x xŽ n.Urespectively, satisfy the usual adjoint problems given by L , whence cG k


Ž .defined by 4.1 should be regarded as an integrated version of a properadjoint eigenfunction. However, for simplicity, we will refer to solutions ofŽ . Ž .4.1 as adjoint eigenfunctions in the following.

Squared eigenfunctions, i.e., products of eigenfunctions and their adjointgenerate a one parameter symmetry group of the Lax equations. This factis of fundamental importance in various aspects of KP theory. We willdenote the corresponding group parameter by z and we will characterizethis symmetry via its infinitesimal generator, which will be given as an

w xevolution equation in Lax form 19 :

THEOREM 4.1. Let k s 0, 1, or 2. Let L satisfy the hierarchy of LaxwŽ n. x Ž .equations L s L , L , let f , . . . , f and c , . . . , c be adjointt G k 1 m 1 mn

Ž .eigenfunctions satisfying 4.1 . Then the flow generated bym

y1 kL s f c , L 4.2Ž .Ýz i x i xis1

commuted with the Lax hierarchy.

For the three soliton hierarchies of Section 3 these symmetries imply thefollowing z-evolutions of the highest nontrivial coefficients in the Lax

Ž . Ž .operators 3.1 . From the highest orders of 4.2 with L s L , L , L ,KP mKP Dymrespectively, one obtains

k s 0 : u s y f c ,Ž .Ýz i i xi

f s f q 2uf ,i y i x x i

c s yc y 2uc ,i y i x x i

k s 1 : ¨ s y f c ,Ž .Ýz i i xi

f s f q 2¨f ,i y i x x i x

c s yc q 2¨c ,i y i x x i x

k s 2 : w s f c w y f c w ,Ž .Ž .Ýz i i x i i xi

f s w2f ,i y i x x

c s yw2c ,i y i x x

Ž .where the y s t evolutions of the adjoint eigenfunctions are generated2Ž 2 . Ž .by L . For the multicomponent version of the modified KP hierarchyG k

these are interesting 2 q 1-dimensional integrable systems of their ownw x19 , they include the Darboux system and a sine-Gordon equation with the

w xtwo space variables occurring on the same footing 14 .

Remark 4.2. The existence of the squared eigenfunction symmetry doesnot depend on the fact that the nonlinear soliton equations are con-


w x Ž n.structed from Lax equations L s M , L with M s L . Generally,t n n G kn

one may consider soliton equations in 2 q 1 dimensions given by a zerocurvature representation

D [ M y M q M , M s 0, 4.3Ž .n t n t n n2 n 1 n 2 11 2

with differential operators M , representing the compatibility condition ofnw x w U xthe linear problems f s M f and c s y M c , respectively, fort n t nn n

Ž .different choices of n. With adjoint eigenfunctions f , c the correspond-i iŽ Ž ..ing squared eigenfunction symmetry is characterized by see A.10, k s 0 ,

w X x X y1M s M , M , M s f c , 4.4Ž .Ýn z n i x iG0i

Ž .which is compatible with the zero curvature representation 4.3 . ThisŽ Ž ..follows from the pseudo-differential zero curvature equation see A.12

X w X xM y M s M , Mn z t nn

w X x Ž .leading to D s M , D s 0, whence 4.3 is invariant under the z-flow.zŽ . ŽIf the operators in 4.3 do not have terms of order 0 or 0 and 1,

. Ž .respectively , then 4.4 cannot be imposed because the right-hand sideincludes these orders. The squared eigenfunction symmetry then has to bemodified as indicated by the cases k s 1, 2 of the previous considerations.

The Lax equations and their adjoints

UU Un yk k yk n k yk kL s L , L , L s y L , L Ž . Ž .Gk Gkt x t x x x x xn n

wŽ n. xare the compatibility conditions of the linear equations f s L ft n G kw yk Ž n.U k xand their adjoints c s y L c for different choices of n. Alsot x G k xn

the squared eigenfunction symmetry may be regarded as the compatibilitycondition of linear equations, where, however, the z-evolution is not adifferential equation, but involves the squared eigenfunction potential of

<w y1 k x <Lemma 2.2. Formally, these linear equations are f s Ý f c fz i i x i xand

Ukyk y1 k k y1 kc s y f c c s y1 c f c ,Ž .Ý Ýz x i x i x x k x i xž /

i i

<w x <where ? means ‘‘application of an operator to a function.’’ However, thepseudo-differential symbol y1 is not supposed to act on ordinary func-x

<w y1 x <tions. Heuristically, ? has to be replaced by a squared eigenfunctionxpotential, which gives y1 the effect of an ordinary integration and, at thexsame time, fixes the time dependence of the integration constant:


LEMMA 4.3. The squared eigenfunction symmetry of Theorem 4.1 is thecompatibility condition of the linear problems:

n Žk .ˆf s L f , f s f V c , f ,Ž . Ž .ÝGkt z i ini

and

U kyk n k Žk .c s y L c , c s y1 V c , f c , 4.5Ž . Ž . Ž .Ž .ÝGkt x x z i ini

Ž .respecti ely. Here V ., . is the potential of Lemma 2.2, i.e.,

V c Žk . , f s c Žk .f , V c Žk . , f s res y1c Žk . Ln fy1 ,Ž .Ž . Ž . Ž .t Gkx xx n

4.6Ž .

where, in turn, c Žk . denotes c , c , or c for k s 0, 1, or 2. The potentialx x xˆ Ž .V ., . is defined by

¡V c , f k s 0 ,Ž . Ž .Žk . ~ˆ yV c , f q cf k s 1 ,Ž . Ž .V c , f s 4.7Ž .Ž . x¢V c , f y c f q cf k s 2 .Ž . Ž .x x x x

It satisfies the compatible equations:

ˆ Žk . Žk . ˆ Žk . y1 k n yk Žk . y1V c , f s cf , V c , f s res c L f ,Ž .Ž . Ž . Ž .t Gkx x x xx n

4.8Ž .

where, in turn, f Žk . denotes f, f , or f for k s 0, 1, or 2.x x x

We note the operator identity,

k k y 1Ž .ky1 k y1 Žk . y1 y1 Žk .ˆ c f s V c , f y y1 V c , f q cf .Ž .Ž . Ž .x x x x x 24.9Ž .

We now consider the commutativity of two squared eigenfunction sym-Ž .metries generated by different pairs of adjoint eigenfunctions c , f ,i i


Ž . Ž .i s 1, . . . , m and r , q j s 1, . . . , m , respectively:1 j j 2

Ž . Ž .THEOREM 4.4. Let f , c i s 1, . . . , m and q , r j s 1, . . . , m sat-i i 1 j j 2isfy

Un yk n kf s L f , c s y L c ,Ž . Ž .Gk Gki t i i t x x in n

Un yk n kq s L q , r s y L r ,Ž . Ž .Gk Gkjt j jt x x jn n

and

kˆf s q V r , f , c s y1 V c , q r ,Ž .Ž . Ž .Ý Ýi z j j i i z i j j2 2j j

kˆq s f V c , q , r s y1 V r , f c ,Ž .Ž . Ž .Ý Ýj z i i j j z j i i1 1i i

ˆŽ . Ž .with V ?,? and V ?,? as in Lemma 4.3. Then

M X s f y1c k and MY s q y1 r kÝ Ýi x i x j x j xi j

X Y w Y X xsatisfy the zero cur ature equation M y M s M , M , whence the Laxz z2 1

w X x w Y xequations L s M , L and L s M , L commute.z z1 2

X Y w Y X xRemark 4.5. The zero curvature equation M y M s M , M indi-z z2 1

cates the compatibility of the linear equations,

ˆ ˆf s f V c , f and f s r V q , f .Ž . Ž .Ý Ýz i i z j j1 2i j

Indeed, these equations commute, provided the z-dependence of thepotentials is chosen according to

ˆ ˆ ˆV r , f s V r , f V c , f ,Ž .Ž . Ž .Ýj j i iz1i

ˆ ˆ ˆV c , f s V c , q V r , f ,Ž . Ž . Ž .Ýzi i j j2j

equivalent to

kV r , f s y1 V r , f V c , f ,Ž . Ž .Ž . Ž .Ýj j i iz1

i

kV c , f s y1 V c , q V r , f .Ž . Ž . Ž . Ž .Ýzi i j j2

j

It may be verified that these evolutions are compatible with the definingŽ . Ž . Ž .equations 4.6 , 4.7 , and 4.8 .


The following theorem extends the links between the three hierarchiesas given by Theorem 3.2 to the squared eigenfunction symmetries:

ˆŽ . Ž .THEOREM 4.6. Let V ., . , V ., . be the potentials of Lemma 4.3.

Ž . Ž .a KP ª mKP : Let L be a solution of the KP hierarchy L stn

wŽ n. x Ž .L , L with adjoint eigenfunctions f , . . . , f , c , . . . , c satisfyingG 0 1 m 1 mŽ .4.1, k s 0 and generating the squared eigenfunction symmetry L sz

y1 ˆw x Ž .Ý f c , L . Let f be a further eigenfunction satisfying f s Ý f V c , f .i i x i z i i i˜ y1Ž .Then, according to Theorem 3.2 a , L s f Lf satisfies the modified KP

˜ ˜n ˜wŽ . xhierarchy L s L , L andt G1n

˜ y1 ˜ ˆf s f f , c s yV c , f s yV c , fŽ . Ž .i i i i i

˜are eigenfunctions of L , i.e.,i

Un y1 n˜ ˜ ˜ ˜ ˜ ˜f s L f , c s y L c .Ž . Ž .G1 G1i t i i t x x in n

˜ ˜ ˜ y1˜w xFurther, L satisfies L s Ý f c , L .z i i x i x

˜Ž . Ž .b mKP ª Dym : Let L be a solution of the modified KP hierarchy˜ ˜ ˜ ˜ ˜ ˜ ˜wŽ . x Ž .L s L , L with adjoint eigenfunctions f , . . . , f , c , . . . , c satis-t n G1 1 m 1 mn

˜Ž .fying 4.1, k s 1, L replaced by L and generating the squared eigenfunction˜ ˜ y1˜ ˜ ˜w xsymmetry L s Ý f c , L . Let f be a further eigenfunction satisfyingz i i x i x

˜ ˜ ˆ ˜ ˜ ˜ ˜ ˜ ˜ ˜Ž . Ž Ž ..f s Ý f V c , f s Ý f c f y V c , f . Then, according to Theoremz i i i x i i i i x˜Ž .3.2 b , after the transformation x s f, t s t , z s z, the operator L s Ln n

n ˜wŽ . xsatisfies the Dym hierarchy L s L , L andt G2n

˜ ˆ ˜ ˜ ˜ ˜ ˜ ˜f s f c s yV c , f s V c , f y c fž / ž /i i i i x i x i

are eigenfunctions of L, i.e.,

Un y2 n 2f s L f , c s y L c ,Ž . Ž .G2 G2i t i i t x x in n

Ui i iŽ . Ž .where the transposition ) refers to w s y1 w. Further, L satisfiesx xy1 2w xL s Ý f c , L .z i i x i x

5. CONCLUSIONS

Here we are concentrating our investigations on 2 q 1-dimensionalsoliton systems. On adopting Sato’s approach it turns out that Backlund as¨well as reciprocal links can be easily represented for whole hierarchies.Squared eigenfunction symmetries, according to their definition in the


previous sections, represent further integrable equations augmenting thehierarchies. They also arise as the compatibility conditions of several linearproblems. The links between the soliton hierarchies extend naturally to thecorresponding squared eigenfunction symmetries.

Various 1 q 1-dimensional soliton equations can be obtained from thew x2 q 1-dimensional hierarchies via suitable constraints 5, 15, 16, 28 . The

approach proposed here also provides a unifying picture of the Backlund¨and the reciprocal links for these reductions. Details of the 1 q 1-

w xdimensional case are discussed in 17 .

APPENDIX

In the computations the following rules are used which hold for arbi-trary operators A, B g g and arbitrary functions f ,

res A s res A y A , A.1Ž . Ž . Ž .x x x

res Ay1 s A , A.2Ž .Ž .x 0

res A s yres AU , A.3Ž . Ž . Ž .A s Ay1 , A.4Ž .Ž .G1 x G0

U UA s A , A.5Ž . Ž . Ž .G0G 0

w x w xA B f s A B f , A.6Ž .G 0 G 0 G 0 G 0

w xA f s Af , A.7Ž . Ž .0G 0

A y1 s A y1 , A.8Ž .Ž .G 0 x 0 x-0

y1A s y1 AU . A.9Ž . Ž .Ž . 0x G 0 x-0

They are all checked by simple and straightforward computations using therepresentation

A s ??? qA q A q A y1 q ??? .1 x 0 y1 x

Proof of Lemma 2.2. Using the previous formulae one finds

Ž .A.1y1 y1 y1 y1res c Mf s res c Mf y res c MfŽ . Ž . Ž .x x x xx

Ž .A.2, A.3 U y1w xs c Mf y res fM cŽ .x

Ž .A.2 Uw x w xs c Mf y M c f s cf ,Ž . t

i.e., V s V , whence the two equations defining V are compatible.t x x t


Ž n. X y1 kProof of Theorem 4.1. We put M s L and M s Ý f c n G k i i x i xw X x n w X n xfor convenience. The Lax equation L s M , L leads to L s M , L ,z z

w X n xwhence projection to differential orders G k yields M s M , L .n z G kŽ n. Ž X w xSince L does not contribute note that M g g and g , g ;- k - k - k - k

. ng for k s 0, 1, 2 , we can replaced L by M and obtain- k n

w X xM s M , M . A.10Ž .n z n Gk

On the other hand, from

M f y1c k y f y1c kMŽ . Ž .Ý Ýn i x i x i x i x n-k -ki i

s M f y1 c k y f y1c kM yk kŽ . Ž .Ý Ýn i x i x i x i x n x x-0 -0j j

Ž .A.8, A.9 Uy1 k y1 yk k kw xs M f c y f M c Ý Ýn i x i x i x x n x i xi i

one concludes

w X x XM , M s M . A.11Ž .n t-k n

This leads to the pseudo-differential zero curvature equation:X w X xM y M s M , M , A.12Ž .n z t nn

Ž . Ž .where A.10 and A.11 , in turn, represent the differential orders G k andŽ .- k of A.12 . This establishes the commutativity of the squared eigen-

function flow and the Lax hierarchy:X X Xw x w x w xL y L s M , L y M , L s M y M y M , M , L s 0.tz t t z n t n z nznn n n

Ž .Proof of Lemma 4.3. The compatibility of 4.6 was verified in Lemma1 ˆŽ .2.2. A straightforward computation shows that 4.8 holds for V defined

Ž . Ž . Ž n. Xby 4.7 , ensuring the compatibility of 4.8 . With M s L , M sn G ky1 k ˆ ˆ Žk .Ž .Ý f c , and V s V c , f one obtainsi i x i x j i

ˆ y1 k yk Žk . y1w xf s M f V q res f c M f Ž .Ý Ýz t n i i i x i x n x xni i

ˆ X yk Žk . y1w xs M f V q res M M f .Ž .Ý n i i n x xi

1 It is useful to note that

Ž .A.3U Uyk n k yk n k y1 y1 n ykc s y L c s yres L c s y res c L .Ž . Ž . Ž .Ž . Ž . Ž .Gk Gk Gkt n x x x x x x x0


X X X Ž Ž ..Insertion of M M s M y M q M M from A.12 into the last equa-n n z t nn

tion produces

ˆ yk Žk . y1w xf s M f V q res M f Ž .Ýz t n i i n z x xni

y res M X ykf Žk .y1 q res M M Xykf Žk .y1 .Ž .Ž .t x x n x xn

Ž . w xWith A.2 the second term yields M f . The third term vanishesn zbecause M X yk g g . The fourth term yieldst x - 0n

y1 Žk . y1 y1 ˆ ˆ y1res M f c f s res M f V y V Ž .Ý Ý ž /ž /n i x i x n i x x i i x xi i

Ž .A.2 ˆ ˆw xs M f V y M f V ,Ý Ýn i i n i ii i

ˆw x w x w xwhence f s M f q Ý M f V s M f q M f s f . Thez t n z i n i i n z n z t zn nŽ .compatibility of 4.5 is checked by a similar computation.

Ž .Proof of Theorem 4.4. Using 4.9 one computes

X Y w Y X x kM y M y M , M Ž .z z x2 1

ˆ y1s f y q V r , f cŽ .Ý Ýi z j j i x iž /2i i

ky1q f c y y1 V c , q rŽ . Ž .Ý Ýi x i z i j j2ž /i j

ˆ y1y q y f V c , q rŽ .Ý Ýj z i i j x jž /1j i

ky1q q r y y1 V r , f c .Ž . Ž .Ý Ýj x j z j i i1ž /j i

Ž .Proof of Theorem 4.6. a From

˜ ˜n y1 y1 y1w xM [ L s f M f s f M f y f M f ,Ž . Ž .G1n n n nG1

Ž n.with M [ L , one findsn G 0

y1 y1 y1˜ ˜ ˜ w x w xf y M f s f f y M f y f f y M f f f s 0.Ž . Ž .i t n i i t n i t n in n n


˜Ž . w xFrom the definition of V c , f and fM s M f y M f one computesi n n n

˜ y1 y1 y1 ˜ y1c s yres c M f s yres c fM Ž . Ž .i t x i n x x i n xn

y1 ˜ ˜ ˜ y1 y1˜ ˜ y1s res c y c M s yres c M ž / ž /ž /x x i i x n x x i x n x

Ž . Ž .A.3 A.2U Uy1 y1 y1˜ ˜ ˜ ˜s y res M c s y M c .ž /x n x i x x n x i

Further,

y1 y1 y1 y1 y1˜ ˜ ˜ ˜L s yf f , L q f L f s f f c q f f c f , LŽ .Ýz z z i i i x ii

y1 y1˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜s f c y f c , L s f c , L .Ý Ýž /i i i x i x i x i xi i

n n˜Ž . Ž . Ž .b Since the first order coefficient of L s L is given byG1 G1n n ˜ ˜ ˜wŽ . x Ž . w xL x , one has M [ L s M y M f , whenceG1 n G2 n n x

˜ ˜ ˜ ˜ ˜ ˜f s f y f f s M f y M f f s M f .i t i t i t i x n i n i x n in n n

ˆ ˆ ˜ ˜ ˜ ˜ ˜ ˜Ž . Ž . Ž .According to 4.8 the potential V s V c , f s c f y V c , f s yci i x i i x i

ˆ ˜ ˜ ˆ y1˜ ˜ y1˜ y1Ž .satisfies V s c f and V s res c M f , whencei x i x i t x i x n x x xn

y1 y1 y1 y1ˆ ˆ ˜ ˜ ˜ ˜ ˜ ˜ ˜c s yV q V f f s yres c M f q c fž /i t i t i x x t x i x n x x x i tn n n n

Ž . Ž .) ))y1 y1 y1 y1 y1 y2 y1˜ ˜ ˜ ˜ ˜ ˜ ˜s res c M f s res f c M fž /x i x n x x x x x i x n x xž /Ž .))) y1 y1 y2˜ ˜s res f c M .ž /x x i x n x

˜ ˜ ˜ ˜ ˜ ˜ ˜ y1˜ y1 y1˜ y1w x Ž . Ž .We note c f s c M f s res c M f in ) and f s i t i n i n x x x x x xn

Ž . Ž .in )) . The residue res related to the symbol is given by res A sxy1˜ ˜Ž . Ž .res Af for all symbols A s A. In ))) one may replace M by M ,x n n

˜ ˜ ˜w xsince the difference M y M s M f does not contribute to then n n xy1ˆ ˜ ˜ ˜ ˜Ž .residue. Observing c s yV c , f f s yc one obtainsi x i x x x i

y1 y1 y2 y1 y2 y2 2˜ ˜c s res f c M s yres c M s y M c .ž / ž /i t x x i x n x x i x x n x x n x in


y1˜ ˜ ˜ ˜ ˜ ˜w x w x w xFinally, L s , L s q f , L s L q f f , L yieldsz z z z x z z x x

y1 y1 y1˜ ˜ ˜ ˜ ˜ ˜ ˜ ˆ ˜ ˜L s L y f f , L s f c y f V f , LÝz z z x x i x i x i i x xž /i

y1 y1 2s yf c q f c , L s f c , L .Ý Ýi x i x x i i x i x i xži i

ACKNOWLEDGMENTS

This work was partially supported by G.N.F.M.]C.N.R. and by the C.N.R. ContractŽ .n.95.01082.01. One of the authors W.O. wishes to acknowledge the Department Metodi e

Modelli Matematici per le Scienze Applicate, University ‘‘La Sapienza,’’ Rome, Italy, for theirkind hospitality.

REFERENCES

1. M. Adler, On a trace functional for pseudo differential operators and the symplecticŽ .structure of the Korteweg]de Vries equation, In¨ent. Math. 50 1979 , 219]248.

2. H. Aratyn, E. Nissimov, S. Pacheva, and I. Vaysburd, R-Matrix formulation of the KPŽ .hierarchies and their gauge equivalence, Phys. Lett. B 294 1992 , 167]176.

3. F. Calogero and A. Degasperis, Spectral transform and solitons I, Stud. Math. Appl. 13Ž .1982 .

4. S. Carillo and B. Fuschssteiner, The abundant symmetry structure of hierarchies ofŽ .nonlinear equations obtained by reciprocal links, J. Math. Phys. 30 1989 , 1606]1613.

Ž .5. Y. Cheng, Constraints of the KP hierarchy, J. Math. Phys. 33 1992 , 3774]3782.6. A. S. Fokas and M. J. Ablowitz, On the inverse scattering transform of the time

Ž .dependent Schrodinger equation and the associated Kadomtsev]Petviashvili I equation,¨Ž .Stud. Appl. Math. 69 1983 , 211]228.

7. A. S. Fokas and B. Fuchssteiner, Backlund transformations for hereditary symmetries,¨Ž .Nonlinear Anal. TMA 5 1981 , 423]432.

8. B. Fuchssteiner and S. Carillo, Soliton structure versus singularity analysis: third orderŽ .completely integrable nonlinear equations in 1 q 1 dimensions, Phys. A. 152 1989 ,

467]510.9. I. M. Gelfand and L. A. Dikii, Fractional powers of operators and Hamiltonian systems,

Ž .Funct. Anal. Appl. 10 1976 , 259]273.10. K. Kiso, A remark on the commuting flows defined by Lax equations, Prog. Theor. Phys.

Ž .83 1990 , 1108]1114.11. B. G. Konopelchenko and V. G. Dubrovsky, Some new integrable nonlinear evolution

Ž .equations in 2 q 1 dimensions, Phys. Lett. A 102 1984 , 15]17.12. B. G. Konopelchenko and W. Oevel, An r-matrix approach to nonstandard classes of

Ž .integrable equations, Publ. Res. Inst. Math. Sci. 29, Kyoto Univ. 1993 , 581]666.13. B. G. Konopelchenko and C. Rogers, Backlund and reciprocal transformations: Gauge¨

Žconnections, in ‘‘Nonlinear Equations in the Applied Sciences,’’ W. F. Ames and C..Rogers, Eds. , p. 317]362, Academic Press, Boston, 1992.

Ž .14. B. G. Konopelchenko, W. Schief, and C. Rogers, A 2 q 1 -dimensional sine-GordonŽ .system: its auto-Backlund transformation, Phys. Lett. A 172 1992 , 39]49.¨


Ž .15. B. G. Konopelchenko, J. Sidorenko, and W. Strampp, 1 q 1 -dimensional integrableŽ . Ž .systems as symmetry constraint of 2 q 1 -dimensional systems, Phys. Lett. A 157 1991 ,

17]21.16. B. G. Konopelchenko and W. Strampp, The AKNS hierarchy as symmetry constraint of

Ž .the KP hierarchy, In¨erse Problems 7 1991 , L17]L24.17. W. Oevel and S. Carillo, Squared eigenfunction symmetries for soliton equations: Part II,

following article.18. W. Oevel and C. Rogers, Gauge transformation and reciprocal links in 2 q 1 dimensions,

Ž .Re¨ . Math. Phys. 5 1993 , 299]330.Ž .19. W. Oevel and W. Schief, Squared eigenfunctions of the modified KP hierarchy and

Ž .scattering problems of Loewner type, Re¨ . Math. Phys. 6 1994 , 1301]1338.20. Y. Ohta, J. Satsuma, D. Takahashi, and T. Tokihiro, An elementary introduction to Sato

Ž .theory, Prog Theor. Phys. Suppl. 94 1988 , 210]241.21. C. Rogers, The Harry Dym equation in 2 q 1 dimensions: A reciprocal link with the

Ž .Kadomtsev]Petviashvili equation, Phys. Lett. A 120 1987 , 15]18.22. C. Rogers and W. F. Ames, ‘‘Nonlinear Boundary Value Problems in Science and

Engineering,’’ Academic Press, Boston, 1989.23. C. Rogers and W. F. Shadwick, Backlund transformations and their applications, Math.¨

Ž .Sci. and Eng. 161, Academic Press 1982 .24. C. Rogers and P. Wong, On reciprocal Backlund transformations and the Korteweg]de¨

Ž .Vries hierarchy, Phys. Scr. 30 1984 , 10]14.Ž .25. M. A. Semenov-Tian-Shansky, What is a classical R-matrix? Funct. Anal. Appl. 17 1983 ,

259]272.26. C. Weiss, Painleve property for partial differential equations II: Backlund transforma-´ ¨

Ž .tions, Lax pairs and the Schwarzian derivative, J. Math. Phys. 24 1983 , 1405]1413.Ž .27. C. Weiss, Backlund transformation and Painleve-property, J. Math. Phys. 27 1986 ,¨ ´

1293]1305.Ž .28. B. Xu, 1 q 1 -dimensional integrable Hamiltonian systems reduced from symmetry

Ž .constraints of the KP hierarchy, In¨erse Problems 9 1993 , 355]363.

Squared Eigenfunction Symmetries for Soliton Equations: Part I

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