Direct "Delay" Reductions of the Toda Equation

DIRECT “DELA Y” REDUCTIONS OF THE TOD A EQUA TION NALINI JOSHI Abstract. A new di r ect metho d of obtaining reductions of the T oda equation is described. W e find a c anonical and c omplete class of all possible reductions under certain assumptions. The r esulting equations are ordinary differential- difference equations, s ometimes r eferred to as delay -different ial equations. The represen tativ e equation of t his class is h ypothesized t o be a new version of one of the classical P ainlev ´ e equations. The Lax pair asso ci ated to this equation is obtained, al so by r eduction. 1. I ntr oduction Many pa per s hav e b een written o n the T o da equation  u t = u ( v − v ) v t = 2 ( u 2 − u 2 ) (1.1) where u = u n ( t ) = u ( n, t ), v = v n ( t ) = v ( n, t ), u t = ∂ u ( n, t ) /∂ t , u = u ( n + 1 , t ), u = u ( n − 1 , t ), v = v ( n + 1 , t ), v = v ( n − 1 , t ). The familiar exp onential form of the T oda equation d 2 Q n dt 2 = ex p  − ( Q n − Q n − 1 )  − ex p  − ( Q n +1 − Q n )  (1.2) can b e o btained through the change o f v ar ia bles u n = 1 2 exp  − ( Q n − Q n − 1 ) / 2  (1.3) v n = − 1 2 dQ n − 1 dt (1.4) In either case , the solutions de p end o n tw o indep enden t v ariables ( n, t ). Equations that in volve iterates in one independent v ariable a nd der iv atives in another ar e referred to as differential-difference equations. Below, w e a lso use this term for equations in which itera tes and deriv a tiv es in the same v ariable app ear. The T o da equation is als o an example of a “la ttice mo del” . Such la ttice mo dels app ear in many physical settings, ra nging from the study o f ther malization in metals to the study o f c e llular neur al netw orks a nd o ptical lattices. F or these a pplications, it is essential to understand the wide v ariety of so lutions po ssible. Reductions provide one metho d of extending our knowledge of the s pace of known solutions. W e fo cus o n reductions to equatio ns inv olving only one indep endent v ariable H η = K ( H, H , H ) , H : R 7→ R (1.5) Date : 28 Octob er 2008. 2000 Mathematics Subje ct Classific ation. 37K10,34K17. The research rep orted in this pap er was supp orted by the Austr al ian Researc h Council Dis- co v ery Pr ogram grant #D P0559019. P A CS: 02.30.Ik, 02.03.Ks. 1 2 NALINI JOSHI where H η = dH /dη , H = H ( η , t ), H = H ( η , t ). Examples of suc h reductions of the T oda equation w e re obtained thro ugh Lie symmetry analysis b y Levi and Win ternitz [4]. 1.1. Bac kground. Symmetry reductions of differential-difference e q uations have bee n studied b y man y author s. A comprehensive r eview can be found in [4]. E arly results on symmetry reductions kept n fixed while a llowing u and t to b e deformed by a method based on the classical appr o ach for differen tial equatio ns that w a s developed by Sophus Lie. An in teresting v aria tion on this standard a ppr oach was considered in [5] b y allowing n to deform contin uously in a ddition to u a nd t . The resulting equatio ns arising as reductions are ordinar y differen tial- difference equations of the form (1.5). In [5], Quisp el et al found an equation of the form (1.5) a s a reduction of the Kac-v an Mo erb eke or V olterra eq ua tion. They show ed that their reduced equa- tion becomes the classical first P ainlev´ e equation [3] in a contin uum limit. This equation was describ ed as a “delay-differential” 1 version o f a Painlev´ e equa tio n by Grammaticos et al. [2 ]. Other such equatio ns were pro pos ed by Grammaticos et al. [2] as delay Painlev ´ e eq uations b y using the criterion of the singularit y con- finemen t metho d. In [4], o ther or dinary differential-difference equatio ns were found as reductions of the T oda equa tion. W e show that these are contained in our r e- sults. Moreov er, we s how that o ur results are co mplete under the assumptions g iven below. 1.2. Direct Me tho d. In 1989, Clar kson and Krusk al [1] used a “ direct a ppr oach”to find reductions of the B oussinesq equation and found new reductions which were not captured by the classica l Lie symmetry appr oach. This direct a pproach was later shown to be r elated to “non-classical”sy mmetries. W e dev e lop a direct approach to finding reductions of different ial-difference eq ua tions. The mos t general form for a reduction is u ( n, t ) = U ( n, t, H ( η ) , G ( η )) , v ( n, t ) = V ( n, t, H ( η ) , G ( η )) , η = η ( n, t ) , (1.6) where H a nd G form a coupled system o f equatio ns of the form (1.5). F or Equations (1.1), it turns out to b e sufficient to take the ansa tz u ( n, t ) = a ( n, t ) + b ( n, t ) H ( η ) (1.7a) v ( n, t ) = c ( n, t ) + d ( n, t ) G ( η ) (1.7b) where η = η ( n, t ). Central to our argument are the following rules (stated for a , b and H for concis eness, but they apply also to c , d a nd G ) Rule 1: If a ( n, t ) = a 0 ( n, t ) + b ( n, t ) Γ( η ), then we ca n tak e Γ ≡ 0 w.l.o.g . by substituting H ( η ) 7→ H ( η ) − Γ( η ). Rule 2: if b ( n, t ) has the for m b ( n, t ) = b 0 ( n, t ) Γ( η ), then we can take Γ ≡ 1 w.l.o.g. by substituting H ( η ) 7→ H ( η ) / Γ( η ). Rule 3: If η ( n, t ) is determined b y an equation of the for m Γ( η ) = η 0 ( n, t ), wher e Γ is in vertible, then we ca n tak e Γ( η ) = η w.l.o.g . by s ubstituting η 7→ Γ − 1 ( η ). 1 W e note that this reduced equation cont ains b oth r etarded H ( η − 1) and adv anced terms H ( η + 1), whilst the usual terminology li mits the usage of the term “delay” to equat ions con taining only retarded terms such as H ( η − 1). DIRECT “DELA Y” REDUCTIONS OF THE TODA EQUA TION 3 Definition 1.1. Given non-z e ro, differentiable a nd inv ertible functions, Γ( η ), w e refer to the transformations H ( η ) 7→ H ( η ) − Γ( η ), H ( η ) 7→ H ( η ) / Γ( η ), and η 7→ Γ − 1 ( η ) as the r e duction tr ansformations on (1.7). Our main r esult is given in Prop osition 2.1 b elow. 1.3. Outl ine of Res ul ts. In this pap er, we obtain direct re ductio ns o f Equations (1.1). The details of our direct method are giv en in Section 2. In Section 3, we show tha t the ans¨ atze (1.7) in fact re present the ge ne r al case and ca n b e assumed without loss of generality . A contin uum limit o f the resulting differ en tial-difference equations is given in Section 4. In Section 5, we find co rresp onding reduction of the Lax pair for the T o da equa tion. Finally , a conclusio n r ounds off the pap er. 2. D irect Reduction of the Toda Equa tion Prop osition 2.1. Supp ose t hat the ans¨ atze (1.7 ) hold. Then the only p ossible non- line ar se c ond-or der r e duction of Equation (1.1) of the form (1.5) that is unique u p to r e duction tr ansformations of H , G and η is given by − c 0 H + H η = H  G − G  (2.1a) p 0 − c 0 G + G η = 2  H 2 − H 2  (2.1b) wher e the r e duction is given by η ( n, t ) = ν ( n ) + σ ( t ) , ν ( n ) b eing an arbitr ary func- tion, with σ ( t ) =  1 c 0 log( c 0 t + c 1 ) + c 2 if c 0 6 = 0 a 0 t + a 1 otherwise (2.2) wher e c j , j = 0 , . . . , 2 , a 0 , a 1 ar e c onst ants and the re ductions of u , v ar e given by the fol lowing two r esp e ctive c ases (1) Case c 0 6 = 0 : u ( n, t ) = ± 1 c 0 t + c 1 H ( η ) (2.3a) v ( n, t ) = − p 0 c 0 1 ( c 0 t + c 1 ) + c 3 + 1 c 0 t + c 1 G ( η ) (2.3b) (2) Case c 0 = 0 : u ( n, t ) = ± a 0 H ( η ) (2.4a) v ( n, t ) = p 0 a 2 0 t + p 1 + a 0 G ( η ) (2.4b) with p 0 , p 1 b eing c onst ants. Pr o of. Under the ans¨ atze (1.7), the T oda equation beco mes a t + b t H + b η t H η = ( a + b H )  c − c + d G − d G  (2.5a) c t + d t G + d η t G η = 2  a 2 − a 2 + 2 a b H − 2 a bH + b 2 H 2 − b 2 H 2  (2.5b) In the following we indicate g eneric functions of η (which are assumed to b e differ- ent iable, non-zer o and inv er tible) by the notation Γ j ( η ). Since w e seek non-linear re duce d equations of the for m (1.5), we requir e the terms H G and H G to be pre s en t in the reduced equation (2.5a). Therefore, we require that bη t Γ 1 ( η ) = b d (2.6a) d Γ 2 ( η ) = d (2.6b) 4 NALINI JOSHI Consider Equation (2.6a), whic h implies d = η t Γ 1 ( η ). How ever, b y Rule 2, this implies Γ 1 ( η ) ≡ 1 a nd, therefor e , d = η t w.l.o.g. Using this in Equation (2.6b), we obtain η t = η t Γ 2 ( η ). How ever, by ta k ing a change of v aria ble s η = Ω  ξ ( n, t )  , where Ω ξ = Ω ξ Γ 2  Ω( ξ )  , we can take η t = η t , w.l.o.g. This implies that η t = σ ′ ( t ), fo r s ome differen tiable function σ ( t ), a nd therefore, η ( n, t ) = ν ( n ) + σ ( t ), where ν ( n ) is an ar bitrary function. Now consider the second equation (2 .5b). Requiring that the terms H 2 and G η bo th remain in the r e duced equatio n, we find that we must hav e b 2 = dη t Γ 3 ( η ) ⇒ b 2 = ( σ ′ ( t )) 2 , w . l . o . g . where we hav e used Rule 2 once ag ain. Therefore, we hav e b = ± σ ′ ( t ). Now consider the linear ter ms in Equation (2.5a ). In par ticular, if we require that terms linear in G remain in the r educed equa tio n, we must hav e a d = bη t Γ 4 ( η ) ⇒ a = ± σ ′ ( t )Γ 4 ( η ) = b Γ 4 ( η ) ⇒ a ≡ 0 , w . l . o . g . by an application of Rule 1. If, on the other hand, the linea r term in H on the left side o f the equa tion remains in the r e duced equatio n, then b t = bd Γ 5 ( η ) ⇒ σ ′′ = ( σ ′ ) 2 Γ 5 ( η ) . How ever, since σ only depe nds on t , while η also dep ends on n , this can only hold if Γ 5 is iden tically constant. Le tting this constan t b e − c 0 , we find σ ′′ ( t ) = − c 0 ( σ ′ ( t )) 2 . W e integrate this ODE for σ ( t ) to find σ ( t ) =  1 c 0 log( c 0 t + c 1 ) + c 2 if c 0 6 = 0 a 0 t + a 1 if c 0 = 0 where c j , j = 0 , 1 , 2 and a 0 , a 1 are ar bitrary constants. Finally , if the linear term in H on the right side o f E quation (2.5a) remains, then b ( c − c ) = b η t (Γ 6 − Γ 6 ) where we hav e taken the lib erty of writing the arbitrar y function of η on the right as the difference of another such function. This leads to c − c = σ ′ (Γ 6 − Γ 6 ) ⇒ c = σ ′ Γ 6 ( η ) + γ ( t ) , where γ ( t ) is an ar bitrary function of t . How ever, since σ ′ = d , we now hav e c = d Γ 6 ( η ) + γ ( t ). By Rule 2, we can take Γ 6 ≡ 0 and, therefor e, c = γ ( t ) w.l.o.g . The re duced equations (2.5 a-2.5b) are now − c 0 H + H η = H  G − G  γ ′ ( t ) ( σ ′ ( t )) 2 − c 0 G + G η = 2  H 2 − H 2  Since this equa tion can only co n tain co efficients tha t are functions of η , we must hav e γ ′ = p 0 ( σ ′ ) 2 , where p 0 is a constant. That is, γ =  − p 0 c 0 1 ( c 0 t + c 1 ) + c 3 , if c 0 6 = 0 p 0 a 2 0 t + p 1 if c 0 = 0 where c 3 and p 1 are constants.  DIRECT “DELA Y” REDUCTIONS OF THE TODA EQUA TION 5 R emark 2 .1 . Our reduced equations (2.1) form a system o f differential difference equations, which evolv e s on a sequence of doma ins containing p oints P =  η 0 , η 0 , η 0 , . . .  , where if η 0 = ν ( n ) + σ ( t ), then η 0 = ν ( n + 1) + σ ( t ). In the interior of any domain in n , where the mapping ν ( n ) 7→ ν ( n + 1 ) is defined, we get a semi-infinite c hain of p oints P a nd a corr esp onding se q uence o f domains on w hich these iterates ar e defined. Since ν ( n ) is an arbitrary function, w e hav e an infinite-dimensional family of reductions. R emark 2.2 . W e no te that tw o cases of non-linear r eductions w ere found by Levi and Winternitz [4]. In these cas es, the new indep enden t v ariable, called y in their pap er, is given resp ectively by (A) y = t ex p( − α n ) or (B) y = t − α n , wher e α is a constant. Cas e (A), after taking log y as a new v ariable is the sub-ca s e c 0 = 1, c 1 = 0 , c 2 = 0 of the ab ov e r esult. Cas e (B), is the sub-case c 0 = 0 , a 0 = 1 , a 1 = 0 of the a bove result. In b o th case s , ν ( n ) = − α n . 3. G eneraliza tion of A ns ¨ atze Here w e show how th e ans¨ atze (1 .7) represent the gener al case. Consider the general reduction u ( n, t ) = c  n, t, H ( η ( n, t )) , G ( η ( n, t ))  (3.1a) v ( n, t ) = d  n, t, H ( η ( n, t )) , G ( η ( n, t ))  (3.1b) Under these transforma tio ns, the T o da equation b ecomes c t + c H H η η t + c G G η η t = c ( d − d ) (3.2a) d t + d H H η η t + d G G η η t = 2 ( c 2 − c 2 ) (3.2b) F or the r educed equation to each contain the non-linear terms in H , G , we req uire d = d Γ 1 ( η , H, G ) (3.3a) c 2 = c 2 Γ 2 ( η , H, G ) (3.3b) Rewriting Γ 1 and Γ 2 in these e q uations appro priately , we can sum ea ch to g et c = e ( t ) Γ 3 ( η , H, G ) , d = f ( t ) Γ 4 ( η , H, G ) . Therefore, c H η t = e ( t ) Γ 4 ( η , H, G ) η t = e ( t ) Γ 3 ( η , H, G ) f ( t )Γ 4 ( η , H, G ) ⇒ c H = g ( n, t ) Γ 5 ( η , H, G ) ⇒ c = h ( n, t ) Γ 6 ( η , H, G ) + k ( n, t ) . Similarly , we find d = r ( n, t ) Γ 7 ( η , H, G ) + s ( n, t ) . By defining Γ 6 and Γ 7 to b e new v ariables , if necessa ry , we repro duce the ans¨ atze assumed ear lier as (1.7). 6 NALINI JOSHI 4. Con tinuum Limits T o find con tinuum limits, it is useful to first conv ert the system o f equations (2.1) to a single scalar equa tion. Lemma 4.1. D efi ne r ( η ) = 2 log  2 H ( η )  . Then r ( η ) satisfies r ηη = c 0  r η − 2 c 0  +  exp( r ) − 2 exp( r ) + exp( r )  (4.1) Pr o of. The pro of is by direct c a lculation. r ( η ) = 2 log(2 H ) ⇒ r η = 2 H η H = 2  c 0 + G − G  ⇒ r ηη = 2 c 0  G − G  + 4  H 2 − H 2  − 2  H 2 − H 2  which gives the de s ired result up on using H 2 = ex p( r ) / 4.  Assume for simplicity tha t η = ν 0 + η . Then a contin uum limit of E quation (4 .1) is obtained by as s uming r = ǫ 2 w ( z ) , z = ǫ η , ν 2 0 = 1 , c 0 = ǫ 4 K as ǫ → 0. Then w ( z ) satisfies w z z + 6 w 2 + α z + β = 0 (4.2) where α and β ar e integration constants. Note that this equation is a sca led a nd translated version of the classica l fir st Painlev ´ e equatio n. 5. L ax p air W e a pply the reduction found in § 2 to the Lax pair for the T o da equation given by W adati and T o da [7]: u ψ + u ψ + v ψ = λ ψ (5.1a) ψ t = u ψ − u ψ (5.1b) Using the results of Pro po sition 2.1 for the case c 0 6 = 0, w e find from Equation (5.1a) ± σ ′ H ψ + ± σ ′ H ψ +   − p 0 c 0 σ ′ + c 3  + σ ′ G  ψ = λ ψ (5.2) In order for this to be a reduced equation containing coefficients tha t are only functions of η , we requir e c 3 = 0 a nd need to c ho ose a definite sign in b = ± σ ′ , say the p ositive sig n. F ur thermore, we define a new spectr al v ariable ζ = λ/σ ′ . Then we obtain for φ ( η , ζ ) = ψ ( n, t ), ψ t = ζ t φ ζ + σ ′ φ η = σ ′ ( c 0 ζ φ ζ + φ η ) . Hence we get the reduced Lax pair H φ + H φ +  G − p 0 c 0  φ = ζ φ (5.3a) c 0 ζ φ ζ + φ η = H φ − H φ (5.3b) W e note that the character of the Lax pair has changed from a spec tr al problem to a mo no dr omy problem, since now deriv atives in ζ also app ear in the linear problem. By differe n tiating Equation (5 .3a) in tw o differe nt wa ys, once with r esp ect to ζ and once with r esp e ct to η , a nd using (5.3 b) to replace φ ζ , while using (5.3a) to replace DIRECT “DELA Y” REDUCTIONS OF THE TODA EQUA TION 7 φ , we ca n show that the compatibility co nditions for the linea r system (5.3) a r e precisely Equatio ns (2.1). R emark 5.1 . W e note that the classica l first Painlev´ e equation has no explicit solutions expressible in terms of pr eviously known functions. There is no reason to belie v e that Equations (2.1) should p osse ss any suc h explicit solutions. How ev er, the reduced equations (2 .1) do hav e the solution H ≡ 0, G ≡ 0, if p 0 = 0. The corres p onding so lution of the Lax pair (5.3) is φ ≡ 0. These r e s ults underly the solv ability of th e re duce d equation and the reduced Lax pair. Local existence of solutions to res pective initial v alue problems ca n b e inferred from the standard existence theore ms for differential-difference equatio ns. 6. Con cluding Remarks W e hav e developed a new direct metho d of obtaining reductions of differential- difference equations a nd applied this metho d to the T oda e q uation (1.1).The com- pleteness of o ur results in the cla ss of “ reduction transfor mations” (see Definition 1.1) is shown in tw o steps. First, in Section 2, we found the most gener a l p ossible reduction sub ject to the ans¨ atze (1.7). Second, in Section 3, w e showed that the ans¨ atze assumed in fact represent the general cas e. A c o n tin uum limit of the re- duced equatio ns is found to be related to the first Painlev´ e equa tio n in Section 4 . Finally , in Section 5, we showed that the reduced equatio ns inherit a linea r problem from the T o da e q uation. The reductions we have found through the direct metho d app ear to b e closely related to tho se obtained by the Lie-symmetry appro ach [4]. Ho wev er, we note that in the latter a pproach, the itera tions of the indep endent v a riable η ar e assumed to be the simple ones describ ed by η = η + α , or η = α η , where α is a constant. No such assumption is made in our approa ch. The reduced equations (2.1) allow a more general functional iteration of η to app ear. W e also note that neither a contin uum limit of these reductio ns o r their linear pr oblem has b een found b efore. The inherita nce of a linear problem through the reductio n indicates that Equa- tions (2.1) a re integrable. These r e s ults suggest that the sys tem (2.1) is analogo us to the well known reductions of completely integrable pa rtial differential equations, namely the c la ssical Painlev´ e equations . How ever, questions remain ope n on how close such an analo gy might b e. The solutions o f the Painlev ´ e equations hav e a characteristic complex ana lytic structure in the complex plane of their indep enden t v a riable. F o rmal La urent ex- pansions of the solutions of the T o da equation and one of its reductions has been discussed in [6 ]. How ever, while E quations (2.1) form a first-deg ree, sec ond-order system o f differ en tial equations in η , they form a second-deg ree, seco nd-order sys- tem when considered as a s ystem of difference equa tions. This introduces a multi- v aluedness in to the system when it is solved for the highest iterate of H , whic h makes any pro of of mer omorphicity o f the solutions slightly delica te. Moreov er, the simult aneous presence of iterates of φ a nd deriv atives of φ in η makes the linear system (5.3) a type o f mono dro my problem that has not b een studied b efore. References [1] P .A. Clarkson and M.D Krusk al, New sim ilarity reductions of the Boussi nesq equation, J. Math. Phys. 30 (1989) 220113. 8 NALINI JOSHI [2] B. Gramm aticos, A. Ramani and I.C. Mor ei r a, Delay-differen tial equations and the Painlev´ e transcenden ts, Physics A 196 (1993) 574590. [3] E. L. Ince, Or dinary D iffer ential Equations , Dov er, New Y ork, 1954. [4] D. Levi and P . Win ternitz, Contin uous symmetries of difference equations, J. Phys. A: Math. Gen. 39 (2006) R1R63. [5] G.R.W. Quisp el, H.W. Cap el and R. Sahadev an. Con tinuous symmetries of differen tial- difference equations: the Kac-v an Mo erb eke equation and Painlev ´ e reduction. Phys. L e tts. A 170 (1992) 379-383. [6] A. Ramani, B. Grammaticos and K.M. T amizhmani, Painlev´ e analysis and singularity con- finemen t: t he ultimate conjecture, J. Phys. A: Math. Ge n. 26 (1993) L53L58. [7] M. W adati and M. T o da, B¨ ac klund transformation f or the exp onen tial lattice, J. Phys. So c. Jap an 39 (1975) 1196–1203. School of Ma thematic s and St at istics F07, The Un iversity of S ydney, NSW 20 06 Aust ralia E-mail addr ess : nal ini@math s.usyd. edu.au