Backlund transformations for integrable lattice equations

B¨ ac klund transformations fo r in te grable lattice equations James A tkinson Department of Applied Mathematics, Universit y o f Leeds, Leeds LS2 9 JT, UK E-mail: james @maths .leeds.ac.uk Abstract. W e give new B¨ acklund transformations (BTs) for some known int eg rable (in the sense of being multidimensionally consis ten t) q ua drilatera l lattice equatio ns . As opp osed to the natural auto-BT inherent in every such equation, these BTs are of t wo other kinds. Spec ific a lly , it is found that some equa tions admit a dditional auto - BTs (with B¨ acklund parameter), whilst so me pairs o f appa r ently distinct equations admit a BT which c onnects them. 1. In tro duction Multidimensional consis tency [1, 2] is the ess ence of inte g rabilit y found in examples of lattice equations whic h arise as the superp o sition principle for B¨ ac klund transformations (BTs). This prop ert y is deep enough to captur e fully the in tegrabilit y of a sy stem, but manageable enough to b e succes sfully emplo y ed in attempts to construct and clas sify in tegrable lattice equations [3, 4 , 5, 6]. In the presen t a rticle, relationships b et w een kno wn examples of m ultidimensionally consisten t equations are establishe d. These relationships are similar in spirit to the notion of m ultidimensional consistency . Ho w eve r, rather than an e quatio n b eing consisten t with copies of itself, distinct equations are consisten t with eac h other. This consistency is equiv a lent to the existence of a par t icular kind of BT, and it is this latt er p oin t of view w e adopt b ecause it lends more in the w ay of in tuition to the systems discusse d. The sense in whic h w e use the term BT throughout this ar t icle, is for an o ve rdetermined system in t w o v ariables whic h constitutes a t r ansformation b etw een solutions o f the tw o equations that emerge as the compatibilit y constrain ts. The term auto-BT will b e used to describe the case where the emerging equations coincide. W e refer to a fr ee parameter o f an auto- BT as a B¨ ac klund pa rameter if tra nsformations with differen t v alues of the parameter comm ute (in the sense that a superp osition principle exists - examples will b e giv en). B¨ acklund tr ansformations for inte gr able lattic e e quations 2 2. The degenerate cases of A dler’s equation A scalar m ultidimensionally consisten t lattice equation o f particular significance w as found b y Adler [7] as the superp o sition pr inciple for BTs of t he Krichev er-No vik ov equation [8, 9 ]. W e write Adler’s equation in the f o llo wing w ay , p ( u e u + b u b e u ) − q ( u b u + e u b e u ) = Qp − P q 1 − p 2 q 2  u b e u + e u b u − pq (1 + u e u b u e b u )  . (1) Here u = u ( n, m ), e u = u ( n + 1 , m ), b u = u ( n, m + 1) and b e u = u ( n + 1 , m + 1) denote v alues of the dependen t v ariable u as a function of the independen t v ariables n, m ∈ Z . The lattic e p ar am eters ( p, P ) a nd ( q , Q ) are p oints on an elliptic curv e, ( p , P ) , ( q , Q ) ∈ Γ, Γ =  ( x, X ) : X 2 = x 4 + 1 − ( k + 1 /k ) x 2  where k is an arbitrary constan t (the Jacobi elliptic mo dulus). The lattice para meters can b e view ed as having their origin in B¨ ac klund parameters asso ciat ed with comm uting BTs of the Kric hev er-No vik ov equation, they play a cen tral r o le in integrabilit y of (1). Equation (1) , the Jac obi form of Adler’s equation, was first giv en by Hietarin ta [6], it is equiv alen t (b y a c hange of v ariables) to the Weierstr ass form giv en originally by Adler [7], cf. [10 ]. Adler’s equation w as included in the list of multidime nsionally consisten t equations giv en later b y Adler, Bob enk o and Suris (ABS) in [3] (where it w a s denoted Q 4). Here w e repro duce the remaining equations in that list: Q 3 δ : ( p − 1 p )( u e u + b u b e u ) − ( q − 1 q )( u b u + e u b e u ) = ( p q − q p )( e u b u + u b e u + δ 2 4 ( p − 1 p )( q − 1 q )) , Q 2 : p ( u − b u )( e u − b e u ) − q ( u − e u )( b u − b e u ) = p q ( q − p )( u + e u + b u + b e u − p 2 + pq − q 2 ) , Q 1 δ : p ( u − b u )( e u − b e u ) − q ( u − e u )( b u − b e u ) = δ 2 pq ( q − p ) , A 2 : ( p − 1 p )( u b u + e u b e u ) − ( q − 1 q )( u e u + b u b e u ) = ( p q − q p )(1 + u e u b u b e u ) , A 1 δ : p ( u + b u )( e u + b e u ) − q ( u + e u )( b u + b e u ) = δ 2 pq ( p − q ) , H 3 δ : p ( u e u + b u b e u ) − q ( u b u + e u b e u ) = δ ( q 2 − p 2 ) , H 2 : ( u − b e u ) ( e u − b u ) = ( p − q )( u + e u + b u + b e u + p + q ) , H 1 : ( u − b e u ) ( e u − b u ) = ( p − q ) , (2) where it app ears, δ is a constant para meter o f the equation. The equations in the list (2) are all degenerate sub cases of the equation (1). T able 1 contains the details of these degenerations. T o clarify the meaning of the en tries in this table w e include an example here. Let us make the substitutions u → ǫu, p → ǫp, q → ǫq in (1 ) and consider the leading term in the small- ǫ expansion of the resulting expression. F or this calculation it is necessary to write the parameters P and Q as a series in ǫ , P = ± (1 − ǫ 2 1 2 ( k + 1 /k ) p 2 + . . . ) , Q = ± (1 − ǫ 2 1 2 ( k + 1 /k ) q 2 + . . . ) , B¨ acklund tr ansformations for inte gr able lattic e e quations 3 Eq u k p P Q 3 δ 2 iǫ δ u − 4 ǫ 2 ǫ ( p − 1 p ) 1 2 ( p + 1 p ) + O ( ǫ 4 ) Q 2 1 ǫ + ǫ 2 u ǫ 2 ǫ 2 p 1 − ǫ 2 2 p 2 − ǫ 4 8 p 4 + O ( ǫ 6 ) Q 1 δ ǫ δ u k ǫp 1 + O ( ǫ 2 ) A 2 u − 4 ǫ 2 1 ǫ ( p − 1 p ) − 1 − 1 2 ǫ 2 ( p + 1 p )( p − 1 p ) − 2 + O ( ǫ 2 ) A 1 δ ǫ δ u k ǫp − 1 + O ( ǫ 2 ) H 3 δ 1 + ǫ √ − δ u 1 1 − ǫ 2 2 p − ǫ 2 p + O ( ǫ 4 ) H 2 1 ǫ + ǫ − ǫ 2 u − 4 ǫ 4 1 − ǫ 2 2 p − 1 2 ǫ 2 + 1 4 p − 2 ǫ 2 + ǫ 4 p − ǫ 6 p + O ( ǫ 10 ) H 1 1 + ǫu k 1 − ǫ 2 2 p k − 1 √ − k − ǫ 2 k − 1 2 √ − k p + O ( ǫ 4 ) T able 1. Substitutions which le ad to the indic ate d de gener ate su b c ase (Eq) of A d ler’s e quation (1) in the limit ǫ → 0 . Cho ose δ = ǫ r ather than 0 to arrive at Eq with δ = 0 . so there is some choice of sign. The rest of the calculation is stra ig h tforward and the leading order expression that results is exactly the equation Q 1 1 or A 1 1 dep ending on this c hoice of sign. It was p ointed out in [3] that one can desc end throug h the lists ‘Q’, ‘A’ and ‘H’ in (2) by degeneration from Q 4, A 2 and H 3 δ resp ectiv ely . The degenerations from Adler’s equation in W eierstrass f orm to t he equations in the ‘Q’ list are giv en explicitly in [11]. P art of what gives (1) its particular significance is that, as far as w e are a ware, all kno wn scalar mu ltidimensionally consisten t lat t ice equations are either linearisable or transformable to (1) or one of its degenerate sub cases ( 2) (p ossibly by a non- autonomous, or gauge, transformation). Note, t his apparen t ubiquity of Adler’s equation is partially explained b y the main result in [4]. 3. Alternativ e auto-B¨ ac klund t ransformations T able 2 lists auto-BTs for some par ticular equations from t he list (2). The BTs listed are distinct from the natural auto-BT asso ciated with ev ery m ultidimensionally consisten t equation (for example, this is desc rib ed for Adler’s equation in [1 0]), one significan t difference is tha t the sup erp osition principle asso ciated with these alternative auto-BTs coincides with some other equation presen t in t he list (2). T o explain the implemen tation of the BTs in table 2 w e giv e an example here (the last entry in the ta ble). Consider the following system of equations in the tw o v ariables u ( n, m ) and v ( n, m ), ( u − e u )( v − e v ) = − p ( u + e u + v + e v + p + 2 r ) , ( u − b u )( v − b v ) = − q ( u + b u + v + b v + q + 2 r ) (3) (the second equation here is implicit from the first and so is omitted f rom the table for brevit y). With u fixed throughout the lattice (i.e., for all n, m ), (3) constitutes B¨ acklund tr ansformations for inte gr able lattic e e quations 4 Eq B¨ ac klund transformation SP Q 3 0 ( pr − 1 pr )( uv + e u e v ) − ( r − 1 r )( u e v + e uv ) = ( p − 1 p )(1 + u e uv e v ) A 2 Q 1 δ p ( u + v )( e u + e v ) − r ( u − e u )( v − e v ) = δ 2 pr ( p + r ) A 1 δ Q 3 0 p ( u e v + e uv ) − uv − e u e v = δ r (1 − p 2 ) H 3 δ Q 1 1 ( u − e u )( v − e v ) = − p ( u + e u + v + e v + p + 2 r ) H 2 T able 2. Each e quation, Eq, admits the given auto-BT (with B¨ acklund p ar ameter r ). The e quation S P emer ges as the sup erp osition principle for solutions of Eq r elate d by this BT. It turns out that the c onverse asso ciations also hold ( se e main text ). an ov erdetermine d system f o r v . This is resolve d ( e b v = b e v ) if u is c hosen to satisfy the equation Q 1 1 throughout the lattice, moreo v er, v whic h then emerges in the solutio n of (3) also satisfies Q 1 1 . W e say that the solutions u and v of Q 1 1 are related by the BT (3) and for con ve nience write u r ∼ v . (4) Here r is the par a meter presen t in (3), this is a f r ee parameter of the transformatio n. The relation (4) is symm etric b ecause (3) is in v arian t under the in terc hange u ↔ v . T ransformations (3) with differen t c hoices of t he parameter r comm ute in the sense that a superp osition principle exists . That is, giv en a solution u ( n, m ) of Q 1 1 , suppose w e compute other solutio ns u ( n, m ), ˙ u ( n, m ), ˙ u ( n, m ) and ˙ u ( n, m ) for whic h u r ∼ u, u s ∼ ˙ u, u s ∼ ˙ u, ˙ u r ∼ ˙ u. (5) Then the solutions ˙ u and ˙ u coincide throughout the la t t ice provided t hey coincide at a single p oin t where the equation ( u − ˙ u )( u − ˙ u ) = ( r − s )( u + u + ˙ u + ˙ u + r + s ) (6) also holds ( a nd in the computation o f these new solutions we can alwa ys c ho ose the in tegration constan ts to make this so). F urthermore, the relation (6) then contin ues to hold throughout the lattice. In this sense w e regard (6) as the sup erp osition principle for solutions of Q 1 1 related b y t he BT (3 ), up to a change in notation (6) coincides with the lattice equation H 2 from the list (2). T o conclude our description of the BT (3) w e r ecognise that the preceding facts are also true in the con v erse sense. O bserv e first tha t the system (5) implies (amongst others) the following equations, ( u − e u )( u − e u ) = − p ( u + e u + u + e u + p + 2 r ) , ( u − e u )( ˙ u − ˙ e u ) = − p ( u + e u + ˙ u + ˙ e u + p + 2 s ) . (7) No w consider (6) as a lattice equation, so that u = u ( l , k ), u = u ( l + 1 , k ), ˙ u = u ( l , k + 1) and ˙ u = u ( l + 1 , k + 1) for new independen t v ariables l , k ∈ Z . Then the sys tem B¨ acklund tr ansformations for inte gr able lattic e e quations 5 (7) fo rms a BT, with B¨ ac klund parameter p , betw een solutions u ( l, k ) and e u ( l , k ) of the equation (6) (i.e., constitutes an auto-BT for H 2 ) . This BT comm utes with its coun terpart with B¨ ac klund parameter q (whic h relates solutions u ( l , k ) and b u ( l, k ) o f (6)), the sup erp o sition principle in this case is exactly the equation Q 1 1 . As describ ed for this example, all the BTs giv en in table 2 establish a kind of dualit y b et we en a particular pair of equations, sp ecifically , o ne equation eme rg es as the sup erp osition principle for BTs that relate solutions of the other. This can b e compared to the natural auto-BT of a m ultidimensionally consisten t equation, for whic h the superp o sition principle coincides with the equation itself. 4. B¨ ackl und t ransformations b etw een distinct equations T able 3 lists BTs that connect par ticular pairs of equations from the list (2). T o b e precise ab out the meaning of the en tries in this table w e again giv e an example. Consider the sys tem of equations 2 u e u = v + e v + p, 2 u b u = v + b v + q . (8) This system is compatible in v if the v a riable u satisfies the equation H 1. G iv en suc h u , it can be verified that v whic h emerges in the solution of (8) then satisfies the equation H 2. Con v ersely , if v satisfies H 2 then solving (8) yie lds u whic h satisfies H 1. In this w a y the system (8) constitutes a BT b et we en the equations H 1 and H 2 , whic h corresp onds to the fifth en try in table 3 (where w e giv e only one equation from the pair (8 ), the other b eing implicit). Eq in u B¨ ac klund transformation Eq in v Q 3 0 uv + e u e v − p ( u e v + e u v ) = ( p − 1 p )( u e u + δ 2 4 p ) Q 3 δ Q 1 1 ( u − e u )( v − e v ) = p ( v + e v − 2 u e u ) + p 2 ( u + e u + p ) Q 2 Q 1 0 ( u − e u )( v − e v ) = p ( u e u − δ 2 ) Q 1 δ H 3 0 pu e u − uv − e u e v = δ H 3 δ H 1 2 u e u = v + e v + p H 2 A 1 0 ( u + e u )( v + e v ) = p ( u e u + δ 2 ) A 1 δ A 1 0 ( u + e u )( v − e v ) = p ( u − e u ) Q 1 1 † A 1 δ u + e u = 2 pv e v + δ p 2 H 3 δ † A 1 0 ( u + e u ) v e v = p (1 − δ 2 u )(1 − δ 2 e u ) H 3 δ T able 3. BTs b etwe en distinct latt ic e e qu ations. The BT b etwe en Q 1 0 and Q 1 δ was given ori ginal ly by ABS in [4]. † indic ates appl ic ation of the p oint tr ansformation p → p 2 , q → q 2 to t he lattic e p ar ameters. B¨ acklund tr ansformations for inte gr able lattic e e quations 6 The BT (8) can b e explained as a non- symmetric degeneration of the natura l auto- BT for the equation H 2, which is defined b y the syste m ( u − e v )( e u − v ) = ( p − r )( u + e u + v + e v + p + r ) , ( u − b v )( b u − v ) = ( q − r )( u + b u + v + b v + q + r ) (9) ( r is t he B¨ acklund parameter). No w, the substitution u → 1 ǫ 2 + 2 ǫ u in the equation H 2 leads to the equation H 1 in the limit ǫ − → 0 . This substitution in the system (9) together with the pa r ticular choice r = − 1 ǫ 2 yields t he system ( 8) in the limit ǫ − → 0. Note that it is not a priori ob vious that the BT will b e preserv ed in this limit, by whic h w e mean that once the syste m (8) has b een found, it remains to v erify the result. It can b e confirmed that a ll but the last three entrie s in table 3 are explained as a non-symmetric degeneration of the natura l auto- BT for the equation in v , ho w ev er the last three en tries hav e b een found b y ad ho c methods. The transformations in table 3 are stated up to comp o sition with p oin t symmetries of the equations in u and v . T o conclude this section w e g iv e tw o more BTs. These connect multidimens ionally consisten t lattice equations whic h lie outside the list ( 2 ). Consider first the system (with 2-comp onent lattice parameters) ( u + p 1 ) v = ( e u + p 2 ) e v , ( u + q 1 ) v = ( b u + q 2 ) b v . (10) This constitutes a BT b etw een the pair o f lattice equations ( u + q 1 )( e u + p 2 )( b u + p 1 )( b e u + q 2 ) = ( u + p 1 )( b u + q 2 )( e u + q 1 )( b e u + p 2 ) , (11 ) ( p 1 − q 1 ) v + ( p 2 − q 2 ) b e v = ( p 2 − q 1 ) e v + ( p 1 − q 2 ) b v . (12) The equation (11) w as give n originally b y Hietarinta in [5] and subse quen tly sho wn to b e linearisable by Ramani et al. in [12]. The BT (10) provides an alternativ e linearisation b y connecting it with the equation (12). The o ther example is a BT b et wee n equations of rank-2 (i.e., 2 comp onen t systems) and is therefore o utside the list giv en b y ABS [3] where o nly scalar equations are considered. It is defined b y the system (with scalar lattice parameters) ( v 1 − e v 1 ) u 2 e u 2 = pu 1 , ( v 2 − e v 2 ) u 1 e u 1 = p e u 2 , ( v 1 − b v 1 ) u 2 b u 2 = q u 1 , ( v 2 − b v 2 ) u 1 b u 1 = q e u 2 , (13) and connects the equations p ( u 2 e u 2 b u 1 − u 1 b u 2 b e u 2 ) = q ( u 2 b u 2 e u 1 − u 0 e u 2 b e u 2 ) , p ( u 1 e u 1 b e u 2 − e u 2 b u 1 b e u 1 ) = q ( u 1 b u 1 b e u 2 − b u 2 e u 1 b e u 1 ) , (14) and p 3 ( v 1 − b v 1 )( e v 1 − b e v 1 )( e v 2 − b e v 2 ) = q 3 ( v 1 − e v 1 )( b v 1 − b e v 1 )( b v 2 − b e v 2 ) , p 3 ( v 2 − b v 2 )( e v 2 − b e v 2 )( v 1 − b v 1 ) = q 3 ( v 2 − e v 2 )( b v 2 − b e v 2 )( v 1 − e v 1 ) . (15) The equation (14) is the lattice modified Boussinesq eq ua t ion given orig inally as a second order scalar equation in [15], the rank-2 v ersion (14) is attributable to Nijhoff in [13]. B¨ acklund tr ansformations for inte gr able lattic e e quations 7 The equation (15) is a rank-2 vers ion of the lattice Sch w arzian Boussinesq equation whic h w as give n originally as a second order scalar equation in [14] (a second-order scalar equation can b e reco v ered fr om (14) or (1 5) b y elimination of one of the v ariables from t he tw o-comp onen t system, b y second order here we mean a lattice equation on a square nine p oin t stencil). The BT (13 ) nat ur a lly g eneralises a scalar BT given in [16] whic h connects the lattice mo dified and Sc h warzian Kortew eg-de V ries equations. (Not e, when transformed to a BT b etw een equations from the list (2) this b ecomes a non-autonomous BT, a type of BT not considered in t he presen t article.) 5. Discussion In the preceding sec t io ns w e ha ve giv en systems of equations whic h ma y b e written generically in t he form f p ( u, e u, v , e v ) = 0 , f q ( u, b u, v , b v ) = 0 , (16) and that constitute a BT b etw een a lattice equation in u = u ( n, m ) and a p ossibly differen t lattice equation in v = v ( n, m ), sa y Q pq ( u, e u, b u, b e u ) = 0 , (17) Q ∗ pq ( v , e v , b v , b e v ) = 0 . (18) (Here w e supp o se that u and v are scalar fie lds and f , Q and Q ∗ are p olynomials of degree 1 in whic h the co efficien ts are functions of the lattice parameters.) In this generic (scalar) case it can b e deduc ed (b y considering an initial v alue problem on the cub e) that t he multidim ensional consistency of ( 1 7) implies the m ultidimensional consistency of ( 1 8). F urthermore, when (17) and (18) are multidim ensionally consisten t, the BT (16) comm utes with the natural auto- BTs for t hese equations, the sup erp osition principle b eing the equation f r ( u, u, v , v ) = 0 . (19) Here u and u are solutions of (17) related b y its natural auto-BT (with B¨ ack lund parameter r ), similarly v and v are solutions of (18) related b y its natural auto -BT ( a lso with B¨ ack lund parameter r ), a nd finally , u and u are related to v and v resp ectiv ely b y the BT (16). W e remark that not a ll lattice equations (17), (18) whic h arise in this w a y are m ultidimensionally consisten t. Consider the f ollo wing example (whic h inv o lv es 2- comp onen t lat t ice pa r a meters), p 1 u e u = v + e v + p 2 , q 1 u e u = v + e v + q 2 . (20) This syste m constitutes a BT b et w een the equations p 1 ( u e u + b u b e u ) − q 1 ( u b u + e u b e u ) = 2( p 2 − q 2 ) , p 2 1 ( v + b v )( e v + b e v ) − q 2 1 ( v + e v )( b v + b e v ) = p 2 2 q 2 1 − q 2 2 p 2 1 . (21) B¨ acklund tr ansformations for inte gr able lattic e e quations 8 The equations (21 ) are m ultidimensionally cons istent if and only if the comp onen ts of the lattice para meters are connected by the relations a + bp 2 1 + cp 2 = 0 , a + bq 2 1 + cq 2 = 0 , (22) for some constan ts a, b and c not all equal to zero. (The solution of (22) yields the fifth and eigh th en tries in table 3.) On the other hand, when (16 ) constitutes an auto-BT, so that Q ∗ = Q , w e hav e found no coun terexamples t o the conjecture that the equation defined b y Q is m ultidimensionally consis tent. 6. Concluding remarks The B¨ ac klund transformations (BTs) giv en in this article establish new relationships b et we en equations within the classification o f Adler, Bob enk o and Suris (ABS) [3]. Alternativ e auto-BTs t urn out to establish a kind o f dualit y b et w een some pairs of equations. T ransformations connecting o t her pairs of equations are of practical significance, for example allowing for soliton solutions to b e found for one equation from t hose of the other (cf. [17]). New BTs ha v e also b een established for integrable lattice eq uatio ns wh ich lie outside the classification of ABS. In pa r t icular w e giv e a BT b et w een systems of rank-2 where only a few examples of multidime nsionally consisten t equations are kno wn. Ac knowledgm ents The author w as supp or t ed b y the UK Engineering and Ph ysical Sciences Researc h Council (EPSR C) and is indebted to F ra nk Nijhoff fo r his contin ued guidance. 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