Soliton solutions for Q3
We construct N-soliton solutions to the equation called Q3 in the recent Adler-Bobenko-Suris classification. An essential ingredient in the construction is the relationship of $(Q3)_{\delta=0}$ to the equation proposed by Nijhoff, Quispel and Capel i…
Authors: James Atkinson, Jarmo Hietarinta, Frank Nijhoff
Soliton solutio ns for Q3 James A tkinson 1 , Jarmo Hietarin ta 2 and F rank Nijhoff 1 1 Department of Applied Mathematics, Universit y of Leeds, Leeds LS2 9J T, UK 2 Department of Physics, Universit y of T ur ku, FIN-2001 4 T urku, FINLAND Abstract. W e constr uct N-soliton solutions to the equation called Q3 in the recent Adler-Bob enko-Suris cla ssification. An essential ing redient in the construction is the relationship of (Q3) δ =0 to the equation prop osed b y Nijhoff, Quisp el and Cape l in 198 3 (the NQC equation). This latter equation has tw o extra par ameters, a nd dep ending on their sign ch oices we get a 4- to -1 rela tionship fro m NQC to (Q3) δ =0 . This leads to a four-term background s olution, and then to a 1-so lito n solution using a B¨ acklund transformatio n. Using the 1SS as a guide allows us to g et the N-so liton solution in terms of the ta u-function o f the Hirota-Miwa equation. 1. In tro duction In tegrable lattice equations ha v e a lo ng history , going bac k to pioneering w or k in the 1970’s, [1, 2], with subseque n t dev elopmen t of systematic approac hes in the early 1980’s, [3, 4, 5], cf. also the review [6 ]. It has long b een kno wn, in fact it is implicit in these constructions, that the emerging examples p ossess the prop erty of “m ultidimensional consistency ”. By this w e mean the prop erty that the partial difference equation describing the lattice systems can b e consisten tly em b edded in a multidime nsional space, with in principle an infinit y of latt ice v ariables. Most succinctly t his prop ert y w as describ ed in recen t y ears in [7], in a con text where it was explicitly used to ac hiev e m ulti-dimensional reductions, and it w as subseq uen tly re-appraised in [8]. This m ultidimensional consistency is a v ery natural prop erty , it is the precise analogue of the w ell kno wn ex istence of c ompatible higher t ime-flows in hierarc hies of soliton systems, and as suc h it w as quite w ell understo o d in the ear lier publications mentioned. In [9] the prop erty of “consistenc y around a cub e” (CA C) was used to classify partial difference equations defined on an elemen tary square o f a tw o-dimensional lattice. Remark ably , within certain restrictiv e conditions (symmetry a nd the “tetrahedron condition”), a full list of scalar quadrilatera l lattice equations could b e established, and this list is surprisingly short. (In a more recen t pap er [10] the classification statemen t w as prov en under sligh tly les s restrictiv e conditions.) The list of CA C-integrable lattice equations in [9] con ta ins some in teresting examples, but the equation at the top of the list (denoted as Q4 in [9]) was actually found ear lier by V. Adler, cf. [11]. This equation, which w e refer to as A d ler’s e quation , is in fact an integrable discretization of the famous Kric hev er-Novik o v (KN) equation, Soliton solutions for Q3 2 [12, 1 3]. F or Adler’s equation a Lax pair was established in [1 4] on the basis of its m ultidimensional consistency , and in [15] this equation w as sho wn to b e a maste r equation among v arious w ell-kno wn in tegrable systems asso ciated with an elliptic curv e. The first solutions fo r Adler’s equation w ere established in a recen t pa p er [16], cf. a lso [17] for a sligh t generalisation o f those results. Ho wev er, so far little is known ab o ut the algebraic structure b ehind Adler’s equation, and since this equation has lattice parameters whic h lie o n an elliptic curve , the underlying structure is exp ected to b e in teresting but rather complicated. Th us, b efore tac kling the Adle r’s equation, it is of in terest to study some of the other e xamples emerging from the list o f [9], in order to see ho w the underlying structure of suc h equations can b e rev ealed. Here w e fo cus on the equation denoted as Q3 in [9] and whic h is just b elow Q4 in the hierarc hy . It is written as: o p (1 − o q 2 )( u b u + e u b e u ) − o q (1 − o p 2 )( u e u + b u b e u ) − ( o p 2 − o q 2 )( b u e u + u b e u ) = δ 2 ( o p 2 − o q 2 )(1 − o p 2 )(1 − o q 2 ) / (4 o p o q ) . (1.1) In (1.1) we hav e adopted the con v ention of represen ting shifts in the r ectangula r lattice with tildes and hats: The corners of an elemen ta ry plaquette on a rectangular lattice are th us u ≡ u n,m , e u ≡ u n +1 ,m , b u ≡ u n,m +1 , b e u ≡ u n +1 ,m +1 . The lattic e p ar ameters o p, o q in (1.1) a r e asso ciated with the n, m directions in the lattice, resp ectiv ely , (they can b e thought o f a s measuring the grid size in these directions) while δ is a global parameter. In this pap er w e construct a general family of N -soliton solutions o f Q3. The construction of these solutions is ba sed on the relatio nship of (1.1) to a lattice equation that first app eared in [3] (cf. also [4]) ab out 25 y ear s ago, and whic h is equiv alen t to the δ = 0 case o f Q3. The explicit 4-to - 1 relationship b etw een these tw o equations is explained in Section 2, and it requires the in tro duction o f a nother parametrisation whic h is mor e suitable fo r the solution. The N-soliton solution of Q3 with pa rameter δ (denoted b y (Q3) δ ) then a pp ears as a line ar com binatio n of four indep endent solutions of the δ = 0 case , (Q3) 0 , with four arbitrary co efficien ts sub ject to one single relation. W e ha v e obtained this result in t w o differen t w ay s, one of whic h employs a no ve l Miura transformation (explained in the App endix), whic h allo ws us to deriv e soliton solutions for (Q3) δ from the kno wn solutions of (Q3) 0 of [3, 4]. W e will not presen t this a pproac h here, because it requires quite a bit of a notational mac hinery for whic h w e lack space in this note, and this approach will b e published elsewhere, [18]. Alternativ ely , w e can use the 4- to-1 corresp ondence which app ears through the in tro duction of some additional lattice shifts asso ciated with the parametrisation giv en in Section 2 , t o obtain a general “seed” solution for the relev an t B¨ ac klund transformations. This allo ws us in Section 3 to obta in the 1-soliton solutio n in a form whic h suggests a τ -function desc ription. W e will presen t the N -soliton solution of Q3 in Hiro t a form in Sec. 4, and sho w how the solution is related to a set of discrete Hirota-Miw a equations in a four- dimensional la ttice. Soliton solutions for Q3 3 2. The basic lattice equations The sp ecial case δ = 0 of (1.1), app eared for first time in [3], (cf. also [4]) in the form 1 + ( p − a ) s − ( p + b ) e s 1 + ( q − a ) s − ( q + b ) b s = 1 + ( q − b ) e s − ( q + a ) b e s 1 + ( p − b ) b s − ( p + a ) b e s . (2.1) W e call t his the NQC equation, follo wing [19]. T o bring the equation (2.1) to the f o rm of (Q3 ) 0 w e p erform the transformatio n ( a + b 6 = 0): u n,m = τ n σ m s n,m − 1 a + b , (2.2) where τ := q ( p + a )( p + b ) ( p − a )( p − b ) , σ := q ( q + a )( q + b ) ( q − a )( q − b ) . (2.3) This yields (Q 3) 0 with the parametrization P ( u b u + e u b e u ) − Q ( u e u + b u b e u ) − ( p 2 − q 2 )( b u e u + u b e u ) = 0 , (2.4) where the lattice parameters hav e no w b ecome p oints p = ( p, P ) and q = ( q , Q ), resp ectiv ely , on the (Jacobi) elliptic curv e: P 2 = ( p 2 − a 2 )( p 2 − b 2 ) , Q 2 = ( q 2 − a 2 )( q 2 − b 2 ) . (2.5) In this para metrizatio n the δ 6 = 0 vers ion of (1.1) reads P ( u b u + e u b e u ) − Q ( u e u + b u b e u ) − ( p 2 − q 2 )( b u e u + u b e u ) = δ 2 ( p 2 − q 2 ) 4 P Q . (2.6) Note that the original parametrization of Q3 in (1.1 ) is subtly diffe ren t from (2.4) and they can b e related b y the identific ations: o p 2 = p 2 − b 2 p 2 − a 2 , o q 2 = q 2 − b 2 q 2 − a 2 , P = ( b 2 − a 2 ) o p 1 − o p 2 , Q = ( b 2 − a 2 ) o q 1 − o q 2 . (2.7) In fact, the tw o different parametrizations presen ted in (1.1) and (2 .6) are just differen t w ays to pa r a metrize the equation α ( u b u + e u b e u ) − β ( u e u + b u b e u ) − γ ( b u e u + u b e u ) = δ , with the constrain ts arising from C A C (related to dependency on lattice parameters associated with different directions o f the cub e). It is imp orta nt f or later to observ e that the parametrizatio ns of P , Q, o p, o q are in v ariant under the sign c hange of a and/or b while the NQC equation itself is not. This means that there are in fact four differ ent versions of NQC (corresp onding to these sign ch anges) and t hey all prov ide a different solution to (Q3) 0 , tho ug h the transformation (2.2) with corresp onding sign c hanges in it and (2.3). W e will us e this 4-to-1 corresp o ndence later to construct solutions to Q3. It w a s sho wn in [3] that (2.1) it in terp olates thr o ugh differen t c ho ices of the auxiliary parameters a , b b et w een v arious lattice equations “ o f KdV t yp e”, and hence could b e though t o f as an in t erp olating equation. W e can iden tify , e.g., the f o llo wing sub cases of (2.1) app earing in the list of [9] (up to gauge transforma t io ns, where necessary): Soliton solutions for Q3 4 a = b = 0 ⇒ (Q 1 ) 0 , a = 0 , b → ∞ ⇒ (H 3 ) 0 and a, b → ∞ ⇒ H 1 , whic h are r esp ectiv ely the lattice Sc hw arzian KdV equation, t he lattice p otential mo dified K dV equation and the latt ice p oten tial KdV equation. F or all these sub cases N -soliton solutions can b e giv en in closed f o rm, whic h follow s as an immediate application of the direct linearisation sc heme elab orated in [3, 4]. 3. Bac kground and one-soliton solutions 3.1. Se e d or b ackgr ound solution f o r Q3 The trivial solution to (2.1) is s n,m ≡ 0 and f r om this it fo llows that u n,m = cτ n σ m is a bac kground or “seed” solution for (Q3) 0 . F urthermore, a s discusse d b efore, by changing the signs of a and/or b w e get other seed solutions f o r (Q3 ) 0 , namely u ++ n,m = Aτ n σ m , u −− n,m = B τ − n σ − m , u + − n,m = C ˙ τ n ˙ σ m , u − + n,m = D ˙ τ − n ˙ σ − m , (3.1) where τ and σ w ere defined in (2.3) and ˙ τ := q ( p + a )( p − b ) ( p − a )( p + b ) , ˙ σ := q ( q + a )( q − b ) ( q − a )( q + b ) . (3.2) Starting with one suc h seed solution for ( Q 3) 0 one can use a Miura transformation (see App endix) to deriv e a solution for (Q3) δ . The result turns o ut to b e a linear com bination of three of the ab ov e terms, and leads us to try a linear com bination of all four terms, that is u 0 S S n,m = Aτ n σ m + B τ − n σ − m + C ˙ τ n ˙ σ m + D ˙ τ − n ˙ σ − m . (3.3) It is easy to v erify that this is indeed a solution of (Q3) 0 pro vided that AB ( a + b ) 2 − C D ( a − b ) 2 = 0 , (3.4) and, p erhaps surprisingly , that (3.3) is also a solution of (Q3) δ , provide d that AB ( a + b ) 2 − C D ( a − b ) 2 = − δ 2 16 ab . (3.5) F rom this last result w e see, that when δ 6 = 0 one cannot use any single seed giv en in (3.1) as a starting solution, and for a “germinating seed” (in the terminology of [16]) one needs at least the pair ++ , −− or the pair + − , − +. 3.2. 1-Soliton solution (Q3) 0 fr om BT Ha ving obtained a non t r ivial bac kground solution (3.3), w e can now pro ceed to construct soliton solutions start ing f rom this seed solution of the B¨ ac klund tr a nsformation (BT). F rom cubic consistency if follow s that we can imp ose in additio n to the la ttice equation (2.4) also the set of equations: P ( u u + e u e u ) − K ( u e u + u e u ) = ( p 2 − k 2 )( e uu + u e u + δ 2 4 P K ) , (3.6a) Q ( u u + b u b u ) − K ( u b u + u b u ) = ( q 2 − k 2 )( b uu + u b u + δ 2 4 QK ) , (3.6b) Soliton solutions for Q3 5 where K 2 = ( k 2 − a 2 )( k 2 − b 2 ). Since the “bar” -direction is fo r increasing n umber of solitons, w e searc h fo r a new u ( ≡ u 1 S S ) of the f orm: u = u θ + v , where u θ is the bar-shifted bac kground solution (3.3): ¯ u θ = Aτ n σ m κ + B τ − n σ − m κ − 1 + C ˙ τ n ˙ σ m ˙ κ + D ˙ τ − n ˙ σ − m ˙ κ − 1 , (3.7) where κ = q ( k + a )( k + b ) ( k − a )( k − b ) , ˙ κ = q ( k + a )( k − b ) ( k − a )( k + b ) . (3.8) (Note how ev er, that w e can absorb t he κ, ˙ κ factors in to A, B , C, D without c hanging t he relation (3.4) or (3.5 ).) F rom (3.6) o ne can then solv e e v = [( p 2 − k 2 ) e u θ + K e u θ ) − P u θ ] v − K v + [ P e u θ − K u θ − ( p 2 − k 2 ) u θ ] , b v = [( q 2 − k 2 ) b u θ + K b u θ ) − Qu θ ] v − K v + [ Q b u θ − K u θ − ( q 2 − k 2 ) u θ ] (3.9) where the exp ected δ 2 term in the nume rator disappears b y virtue o f the definitions (3.7,3.5). Then in tro ducing v = f /g we can linearise these equations a nd obtain e φ = Λ ( p 2 − k 2 ) e u θ + K e u θ − P u θ 0 − K P e u θ − K u θ − ( p 2 − k 2 ) u θ ! φ (3.10) and similarly for b φ , where φ = ( f , g ) T . The facto r Λ is determined by the condition that the shifts with e and b must comm ute, from whic h it follo ws w e should tak e Λ ∝ 1 /U θ , where (compare with (3.3) ) ( U θ ) nm = ( a + b ) Aτ n σ m − ( a + b ) B τ − n σ − m + ( a − b ) C ˙ τ n ˙ σ m − ( a − b ) D ˙ τ − n ˙ σ − m . (3.11) Then with some algebra w e obtain: φ n,m = ( p − k ) n ( q − k ) m ρ n,m ( U θ ) n,m / ( U θ ) 0 , 0 , 0 K 2 k [1 − ρ n,m ] / ( U θ ) 0 , 0 , 1 ! φ 0 , 0 , (3.12) where ρ n,m is the discrete ”plane-wa v e fa ctor”, defined by ρ n,m = p + k p − k n q + k q − k m ρ 0 , 0 . (3.13) F rom the result (3.12) w e can reconstruct v of the 1SS: v n,m = 2 k ( U θ ) nm ρ nm v 00 2 k ( U θ ) 00 + K v 00 (1 − ρ nm ) , (3.14) and finally u 1 S S nm = ¯ u θ + v . F or a more explicit form showin g the A, B , C , D dep endence, w e redefine the constan t ρ 0 , 0 in (3.13) so that the denominator b ecomes prop or tional to 1 + ρ nm . F urthermore, after scaling A, B , C , D so that ¯ u θ of (3.7) becomes u 0 S S of (3.3) w e can write the 1SS (3.14) as u 1 S S nm = h Aτ n σ m 1 + ρ nm ( a − k )( b − k ) ( a + k )( b + k ) + B τ − n σ − m 1 + ρ nm ( a + k )( b + k ) ( a − k )( b − k ) + C ˙ τ n ˙ σ m 1 + ρ nm ( a − k )( b + k ) ( a + k )( b − k ) + D ˙ τ − n ˙ σ − m 1 + ρ nm ( a + k )( b − k ) ( a − k )( b + k ) i / (1 + ρ nm ) . (3.15) Soliton solutions for Q3 6 4. N -soliton solutions and Hirota bilinear form W e will no w presen t the main result of the pa p er, whic h is a general N -soliton solution of Q3 . This solution can b e written in the fo llo wing for m: u nm = Aτ n σ m F ( n, m, α + 1 , β + 1) F ( n, m, α, β ) + B τ − n σ − m F ( n, m, α − 1 , β − 1) F ( n, m, α, β ) + C ˙ τ n ˙ σ m F ( n, m, α + 1 , β − 1) F ( n, m, α, β ) + D ˙ τ − n ˙ σ − m F ( n, m, α − 1 , β + 1) F ( n, m, α, β ) , (4.1) where the ta u-function F is giv en b y F ( n, m, α, β ) = X µ j ∈{ 0 , 1 } exp N X j =1 µ j η j + X 1 ≤ i
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