Exceptional discretisations of the sine-Gordon equation

Exceptional discretisations of the sine-Gordon equation I.V. Barashenk o v ∗ and T.C. v an Heerden † Dep artment of Mathematics , University of Ca p e T own, R ondeb osch 7701, South Afric a (Dated: No v em b er 10, 2018) Abstract Recen tly , the m etho d of one-dimen sional maps was in tro d uced as a means of generating excep- tional discretisations of the φ 4 -theories, i.e., discrete φ 4 -mo dels wh ic h s u pp ort kinks centred at a con tin uous range of p ositions relativ e to the lattice. In this pap er, w e employ this metho d to obtain exceptional discretisations of the sin e-Gordon equation (i.e. exceptional F renk el-Kon toro v a chains). W e also use one-dimensional maps to construct a d iscrete sine-Gordon equation su pp orting k in ks mo ving with arbitrary v elo cities without emitting radiation. P ACS num b ers: 0 5 .45.Yv ∗ Electronic a ddr ess: Igo r.Bara shenko v@uct.a c.za,igor@o dette.mth.uct.ac.za † Electronic address : tv anheerden@g mail.com 1 I. INTR ODUCT ION Discrete analo gs o f nonlinear ev olution equations hav e b een the sub ject of intense inv es- tigation ov er the last 15 y ears. A great deal of insigh t has b een gained in to the prop erties of the discretised nonlinear Sc hr¨ odinger, Landau-Lifsc hitz, Kortewe g- de V ries, φ 4 - and other equations whic h were originally in tro duced in the con text of contin uous nonlinear media. As for the discrete sine-Gordon equation that w e study in this pap er, it precede d the ap- p earance of its con tinuum counterpart in the ph ysics literature. The equation dates bac k to 1938 when it w as prop osed b y Y ak o v F renk el and T at y ana Kon torov a to mo del stationary and mov ing crystal dislo catio ns [1]. The orig ina l F renk el-Kon toro v a model consisted of a c hain of harmonically coupled atoms in a spatially p erio dic p oten tial: ¨ θ n = 1 h 2 ( θ n +1 − 2 θ n + θ n − 1 ) − sin θ n . (1) Here θ n is the p osition of the n -th atom in the chain, 1 /h 2 is a coupling constant, a nd the ov erdots indicate differen tiation with resp ect to time: ¨ θ = d 2 θ /dt 2 . An alternativ e in terpretation of Eq.(1) is t ha t of a c hain of torsionally coupled p endula, with θ n b eing the angle the n -th p endulum mak es with the v ertical. Finally , Eq.(1) can b e seen simply as a discretisation of the sine-Gordon equation ¨ θ = θ xx − sin θ, (2) whic h was conceiv ed f o r the n umerical sim ulation o f this partial differential equation. Since its original inception a s the F renk el-Kontoro v a mo del, the discrete sine-Gordon equation (1) has reapp eared in a gr eat n um b er of ph ysical conte xts, including domain w alls in ferro - and an tiferromagnetic crystals, c harge- densit y w a v es in solids, crow dions in metals, v ortices in array s of Josephson junctions, incommensurate structures in metals a nd insu- lators, and nonlinear excitations in h ydrogen-b onded molecules. (See [2, 3] for review and references.) The equation has also b een generalised in a v ariet y of w a ys. In the presen t pap er w e study , systematically , tw o classes of suc h generalisations. In the mo dels of the first class the main part of the inters ite coupling is still harmonic, as in the original F renk el- Kon torov a mo del, but in addition there is an anharmonic part of in teraction arising from the mo dified p erio dic p oten tial. These types of mo dels a re o f interes t primarily in the station- ary case where they define nontrivial systems of statistical mec hanics (systems with con v ex 2 in teractions) [3, 4]. The stationa ry discretisations in this class ha v e the form 1 h 2 ( θ n +1 − 2 θ n + θ n − 1 ) = f ( θ n − 1 , θ n , θ n +1 ) , (3) where the function f ( not necessarily a p erio dic function) reduces to sin θ in the contin uum limit: f ( θ n − 1 , θ n , θ n +1 ) → sin θ n as θ n − 1 , θ n +1 → θ n . The other class consists of all-p erio dic discretisations: 1 h 2 sin( θ n +1 − 2 θ n + θ n − 1 ) = f ( θ n − 1 , θ n , θ n +1 ) . (4) Here f is a p erio dic function of eac h of it s three a rgumen ts, suc h tha t f ( θ n , θ n , θ n ) = sin θ n . Equations of this type g ov ern arra ys of electric dip oles or magnetic spins in whic h the in ter- actions b etw een the neigh b ouring elemen ts a re c haracterised by the trigonometric functions of the corr esp onding ang les. These mo dels ar e commonly referred t o as the “sine lattices” [5]. In the stationary case, examples o f the sine la ttices include the usual and the c hiral one- dimensional X Y mo del in the magnetic field [3 , 6]. In the time-dep endent setting, the sine lattices w ere used to mo del the rotational dynamics of meth yl groups in 4-meth yl-pyridine [7], C H 2 units in crystalline p oly eth ylene [8] and bases in a DNA macromolecule [8, 9]; to study conforma t io nal defects in p olymer crystals [10 ] a nd nonlinear w av es in c hains of electric dip oles [1 1]. In addition to (3) and (4), w e also consider some o ther discretis atio ns whic h ha v e the sp ecial pro p ert y o f exhibiting exact solutions. In most ph ysical applications of t he sine-Gordon theory , b o th contin uum and discre te, the cen tral role is play ed b y its solitary-w av e solution, called a kink. The kink represen ts a dislo cation in the crystal, a 2 π - t wist wa v e in the c hain of p endula, a nd a quan tum of magnetic flux in a lo ng Josephson junction. In the con tin uum mo del (2), the kink solution is a v ailable explicitly; in particular, t he stationary kink has the expression θ ( x ) = 4 arctan [exp ( x − x (0) )] . (5) The stat io nary kink (5) dep ends on a single parameter, the p osition of its cen ter x (0) , whic h can b e v a ried con tin uously: −∞ < x (0) < ∞ . In g eneric discretisations of the sine-Gordon theory , how ev er, the kink can only b e cen tred on a lattice site o r strictly midw a y b etw een t w o neighbouring sites [12, 13, 14, 15]. Mathematically , this is a conseque nce of the br eaking 3 of the translation symmetry of t he con tin uum mo del. The phys ical interpretation is that the discrete kink can only remain stationary when placed at a minim um or a maxim um of the so-called P eierls-Nabarro barrier, a p erio dic p oten tial induced by the discretisation of equation (2) [12, 13, 14, 15, 16 ]. Sp eigh t and W ard [17] we re the first to realise tha t the breaking of the t r a nslation sym- metry do es not necessarily preclude the exis tence of a one-parameter fa mily of stationary discrete kinks with an arbitrary cen tring relativ e to the lattice. In other w ords, despite not b eing translation inv ariant, the la t t ice equation may supp ort a kink solution whic h de- p ends on a con tinuous translation parameter. This “sp ontaneous symmetry restoration” is a nongeneric phenomenon whic h ma y only o ccur in isolated, or exc eptional , discretisations of the sine-Gordon mo del. Ph ysically , it implies that the discretisation do es no t induce the P eierls-Nabarro barrier, or that the barrier is transparent to kinks. Flac h, Zolotar yuk and Kladk o ha ve disco v ered a similar phenomenon in a class of discrete Klein- G ordon system s with nonlinearities o f a sp ecial fo r m [18]. The classification of t he exceptional discrete sine-Gordon equations, i.e., equations sup- p orting families of stationary kinks with a contin uously v a riable p osition relativ e to the lattice, is of fundamental interes t. Firstly , the discrete kinks tend to b e more mobile in exceptional discrete mo dels. There are some isolated v elo cities at whic h kinks in the ex- ceptional mo dels ma y slide , i.e. tra v el without losing energy to radiation [19]. F or other v elo cities, t he mo ving kinks do radiate but the amplitude of r a diation is m uc h smaller than in generic systems [17, 19, 2 0, 21]. In addition, the collisions of kinks w ere rep orted to b e more elastic in the exc eptional mo dels [2 2]. W e also sho w in this pap er tha t some ex- ceptional systems admit time-dep enden t ve rsions which supp o rt sliding kinks with arbitr ary v elo cities. F urthermore, there are indications [23] that all excep tiona l discretisations p ossess a conserv ation la w: they conserv e either energy or mo mentum. Therefore, exceptional dis- crete mo dels app ear to b e “b etter” approximations of t he partial-differen tial equation ( 2 ) — at least a s far as the kink solutions a r e concerned — as they preserv e imp ortan t prop erties of the con tinuum mo del. The ob jectiv e of the prese nt pap er is to iden tify exceptional F renk el-Ko n torov a mo dels within the families (3) and (4). W e are not go ing to a t t empt a complete classification here; instead, we fo cus on identifying simple particular cases whic h ma y b e of pr a ctical use in future. W e also construct t wo discrete mo dels with ex act (stationary and mov ing) kink 4 solutions. An outline of t he rest of the pap er is a s fo llo ws. In the next section (section I I), we presen t the metho d of o ne- dimensional maps as applied to discrete sine-Gordon equations. In section I I I, the metho d is used to iden tify simple exceptional discretisations of the form (3) in v olving ratios of trig onometric and linear functions. The symmetric maps found in this section ha v e one f urther use; in section IV w e utilise t hem to construct purely trig onometric discretisations [o f the f o rm (4)]. In the subsequen t sections w e presen t discrete sine-Gordon equations with exa ct stationary (section V) and moving (section VI) kink solutions. Finally , sev eral concluding remarks are made in section VI I whic h summarises the results of this study . I I. THE METH OD OF ONE-DIMENSI ONAL MAPS T o deriv e an exceptional discrete sine-Gordon mo del, Sp eight and W ard used the Bog o- moln y energy-minimality a r gumen t [17]. The energy minimalit y requiremen t has naturally led them to consider a one- dimensional map rather than t he o r iginal, second-order, dif- ference equation. In the follow-up w ork [24], Sp eight utilised the energy-minimising map (the Bogomoln y map) to pro v e the existenc e of a one-parameter fa mily of kinks for their discretisation of the φ 4 -theory , i.e. the exceptionality of their φ 4 -mo del. A further insight was due to Kevrekidis [25]. Inspired b y Herbst and Ablow itz’ results on the discrete nonlinear Schr¨ odinger equation [26], Kevrekidis reform ulated the exceptionalit y of a statio na ry discrete Klein-Gordon equation as the existenc e of a t w o- p oin t in v arian t. He also provide d tw o phenomenological recip es o f construction o f stationa ry disc retisations with suc h in v arian ts. Th us, the existence of a t w o- p oint in v arian t replaced the energy minimalit y requiremen t as the crucial prop ert y of exceptional discretisations. The univers ality of one-dimensional maps as generators of translationally-inv arian t fami- lies of solutions has b een fully realised in Ref.[2 7]. Instead of trying to iden tify discretisations exhibiting a tw o-p oint inv ariant, the authors of [27] prop osed to generate exceptional dis- cretisations departing from a p ostulated map. (That is, the t w o- p oint inv ariant has no w b ecome a starting p oin t rather than the final ob jectiv e of the analysis.) In this wa y , the classification of exceptional discretisations has b een reduced to the classification of one- dimensional maps. This will b e our approac h in this pap er as w ell. 5 W e start b y considering a discrete sine-Gordon equation of the form (3) and assume that the corresp onding stat io nary equation, 1 h 2 ( θ n +1 − 2 θ n + θ n − 1 ) = f ( θ n − 1 , θ n , θ n +1 ) , (6) has a solution of the form θ n = g ( nh ), where the con tinuous function g ( x ) is defined for −∞ < x < ∞ and is monoto nically grow ing, with g ( −∞ ) = 0 a nd g ( ∞ ) = 2 π . Since n do es not app ear in equation (6) explicitly , fr o m the existence of the ab ov e solution it follow s that equation (6) also has a whole family of solutions θ n = g ( nh − x (0) ), with a n y real x (0) , a nd therefore, that t he mo del (6) is exceptional. F or eac h x (0) , the solutio n θ n = g ( nh − x (0) ) represen ts a discrete kink; if we in terpret v alues x n = nh as p ositions of the la t t ice sites on the x -axis, the kink θ n app ears cen tred on the p oin t x = x (0) . It is imp ortant to emphasise that we do not need to kno w an explicit form of g ; all w e need to kno w is t ha t a function with these prop erties exists (for example, as an implicit function). As g ( x ) is a monoto nically g r owing function, w e can in v ert it to obtain θ n +1 = g ( g − 1 ( θ n ) + h ) ≡ F ( θ n ). Sinc e g ( x ) is defined for all real x , the function F ( θ n ) is defined for an y θ n . Th us the fact that the discretisation is exceptional implies that the kink solutio n θ n satisfies a one-dimensional map [27]. The opp o site is a lso true. Namely , assume Eq.(6) results from the iteration of a one-dimensional map θ n +1 = F ( θ n ) (in a similar w a y as a second-order differen tial equation can b e deriv ed b y differen tiating a first-order one). I n additio n, let the function F b e suc h that F ( θ ) > θ for an y θ b et w een 0 a nd 2 π , whereas F (0) = 0 and F (2 π ) = 2 π . A simple cob w ebbing arg ument shows then that for an y θ 0 within the range 0 < θ 0 < 2 π , the map generates a discrete kink solution ..., θ − 1 , θ 0 , θ 1 , ... . Therefore we ha v e a one-parameter family of kinks and so Eq.(6) represen ts an exceptional discretisation. This o bserv ation implies that w e can find exceptional discretisations of the sine-Gordon equation b y considering a one-dimensional map θ n +1 − θ n = hH ( θ n +1 , θ n ) , (7) where t he function H satisfies sev eral requiremen ts. First of all, it should satisfy the condi- tion H ( θ, θ ) = 2 sin θ 2 (8) whic h ensures that the map (7) reduces to equation θ x = 2 sin θ 2 (9) 6 in the con tinuum limit (where θ n +1 − θ n → hθ x ). Equation (9) is the Bogomoln y equation for the stationary contin uum sine-Gordon theory: the sine-Gordon equation θ xx = sin θ follo ws from Eq.(9) b y differen tiation, while its kink solution (5) is sim ultaneously a solution of Eq.(9). Therefore, the condition (8) selects maps whic h generate discretisations of t he sine-Gordon rather than some other equation. Our second requiremen t is that H ( θ n , θ n +1 ) should b e b ounded and p ositiv e for all pair s of θ n and θ n +1 with sufficien tly small v alue of the difference | θ n +1 − θ n | (where 0 < θ n , θ n +1 < 2 π ). Using this pro p erty o f H , assuming that h is sufficien tly small, and inv oking the implicit function theorem, w e can sho w that Eq.(7) defines, fo r any 0 < θ n < 2 π , a function θ n +1 = F ( θ n ), with θ n +1 > θ n . Thus Eq.(7) will giv e rise to an ex c eptional discretisation of the sine-Gordon equation. This discretisation results from squaring b oth sides of (7) and subtracting the square of its bac k-iterated cop y , θ n − θ n − 1 = hH ( θ n , θ n − 1 ) . This yields [27 ] θ n +1 − 2 θ n + θ n − 1 h 2 = H 2 ( θ n +1 , θ n ) − H 2 ( θ n , θ n − 1 ) θ n +1 − θ n − 1 . (10) If H is symmetric [i.e. in v arian t under the p ermutation of its ar g umen ts: H ( x, y ) = H ( y , x )], the nume rat or in (10) v anishes whenev er the denominator equals zero and hence the right- hand side of (10) is nonsingular. Th us the classification of exceptional discretisations reduces to the classification of all symmetric functions H ( x, y ) with the a b ov e prop erties. The next section summarises results of this analysis. I I I. RA TIONAL-TRIGONOMET RIC DISCRETI SA TIONS A. H 2 ( x, y ) = F ( x ) + F ( y ) The simplest p osibilit y is to let H 2 ( x, y ) = F ( x ) + F ( y ). F rom H ( x, x ) = 2 sin( x/ 2) it follo ws that F ( x ) = 2 sin 2 ( x/ 2) and so H 2 ( x, y ) = 2 sin 2 x 2 + 2 sin 2 y 2 . (11) This function g ives rise to one of Kevrekidis’ discretizations [25 ]: θ n +1 − 2 θ n + θ n − 1 h 2 = − cos θ n +1 − cos θ n − 1 θ n +1 − θ n − 1 , (12a) 7 or, equiv alently , θ n +1 − 2 θ n + θ n − 1 h 2 = 2 sin [( θ n +1 − θ n − 1 ) / 2] θ n +1 − θ n − 1 sin  θ n +1 + θ n − 1 2  . (12b) B. H 2 ( x, y ) = F ( x + y ) Letting H 2 ( x, y ) = F ( x + y ) and substituting into t he con tin uum limit condition, yields H 2 ( x, y ) = 4 sin 2 x + y 4 . (13) This symmetric function generates the discretisation of the f orm θ n +1 − 2 θ n + θ n − 1 h 2 = 4 sin [( θ n +1 − θ n − 1 ) / 4] θ n +1 − θ n − 1 sin  θ n +1 + 2 θ n + θ n − 1 4  . (14) C. H 2 ( x, y ) = F ( x ) F ( y ) Another simple p ossibilit y is to assume that H 2 ( x, y ) = F ( x ) F ( y ). F rom the con tin uum limit w e obtain F ( x ) = 2 sin( x/ 2) and so H 2 ( x, y ) = 4 sin x 2 sin y 2 . (15) By substituting into equation (10) we find the follow ing discretisation: θ n +1 − 2 θ n + θ n − 1 h 2 = 8 sin[( θ n +1 − θ n − 1 ) / 4] θ n +1 − θ n − 1 sin  θ n 2  cos  θ n +1 + θ n − 1 4  . (16) D. H 2 ( x, y ) = [ F ( x ) + F ( y )] 2 Considering the symmetric function of the form H 2 ( x, y ) = [ F ( x ) + F ( y )] 2 , w e obtain from the con tinuum limit: H 2 ( x, y ) =  sin x 2 + sin y 2  2 . (17) This giv es rise to the following discretisation: θ n +1 − 2 θ n + θ n − 1 h 2 = 2 sin [( θ n +1 − θ n − 1 ) / 4] θ n +1 − θ n − 1 cos  θ n +1 + θ n − 1 4  ×  sin θ n +1 2 + 2 sin θ n 2 + sin θ n − 1 2  . (18) 8 E. H 2 ( x, y ) = F ( x ) F ( y ) + G ( x ) G ( y ) A simple symmetric generalisation in v olving tw o functions of a single argumen t, sa y F ( x ) and G ( x ), is H 2 ( x, y ) = F ( x ) F ( y ) + G ( x ) G ( y ). Setting x = y yields F 2 ( x ) + G 2 ( x ) = 4 sin 2 x 2 . One p ossibilit y here is to assume that the functions F and G hav e the for m F ( x ) = 2 η ( x ) sin x 2 , G ( x ) = 2 ξ ( x ) sin x 2 , where η and ξ satisfy η 2 ( x ) + ξ 2 ( x ) = 1. The simplest trigonometric choice f or η and ξ is η ( x ) = sin( ax ) , ξ ( x ) = cos( ax ) , where a is a parameter. T aking, for instance, a = 1 2 , gives us the follo wing expression for H 2 ( x, y ): H 2 ( θ n +1 , θ n ) = 4 sin θ n +1 2 sin θ n 2 cos θ n +1 − θ n 2 . (19) The function (1 9) is ob viously p ositiv e for | θ n +1 − θ n | < π and therefore the resulting discretization, θ n +1 − 2 θ n + θ n − 1 h 2 = 4 sin [( θ n +1 − θ n − 1 ) / 2] θ n +1 − θ n − 1 sin  θ n 2  cos  θ n +1 − θ n + θ n − 1 2  , (20) is exceptional for sufficien tly small h . Another simple symmetric com bination of t w o functions of a single argumen t, is H 2 ( x, y ) = F ( x ) G ( y ) + F ( y ) G ( x ); ho we ve r, this H 2 giv es rise to the discretisation that w e hav e already identifie d, Eq.(12). F. More complex symmetric functions It is not difficult to construct more example s of symmetric functions H ( x, y ) , with in- creasing complexit y . One p ossibilit y is to t a k e H 2 ( x, y ) = N X n =1 F n ( x ) F n ( y ) , where F n ( x ) ( n = 1 , 2 , ..., N ) are appropria te trigo no metric f unctions. Another symmetric com bination is H 2 ( x, y ) = N Y n =1 F n ( x | n | ) + N Y n =1 F n ( x | n +1 | ) , where | n | ≡ n mo d 2, and x 1 = x , x 2 = y . 9 IV. PUREL Y TRIGON OMETRIC DISCRETISA T IONS Our original one-dimensional map (7) can b e mo dified to pro duce new, p erio dic, dis- cretizations. Instead o f (7), w e consider the ma p ℓ sin θ n +1 − θ n ℓ = hH ( θ n +1 , θ n ) , (21) where H ( θ n +1 , θ n ) is a t r ig onometric function of its argumen ts and ℓ is a p ositiv e in teger. Subtracting from t he square of (21) the square of its bac k-iterated cop y yields the discrete mo del ℓ 2 h 2 sin θ n +1 − 2 θ n + θ n − 1 ℓ = H 2 ( θ n +1 , θ n ) − H 2 ( θ n , θ n − 1 ) sin[( θ n +1 − θ n − 1 ) /ℓ ] . (22) As in Eq.(7 ) , w e assume that H ( θ n +1 , θ n ) is p ositiv e, symmetric and has the con tin uum limit H ( θ, θ ) = 2 sin( θ / 2). F or sufficien tly small h and | θ n +1 − θ n | , Eq.(21 ) defines an implicit function θ n +1 = F ( θ n ), with θ n +1 > θ n . Consequen tly , the discretisation (22) is exceptional. The discretisations (22) are different from those in (10) in that ev ery term in (22) is p erio dic in eac h of its three argumen ts, θ n − 1 , θ n and θ n +1 . The mo dels of the form (2 2) find their applications in the description of coupled chains of elemen ts where eac h elemen t is c haracterised b y a p erio dic v ariable (an angle) and the coupling of elemen ts do es not violate this p erio dicit y . One example is giv en b y the Sp eigh t-W a r d discretisation [1 7] 4 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  = sin  θ n +1 + 2 θ n + θ n − 1 4  , (23) whic h has recen tly b een sho wn to describ e c hains o f electric dip o les constrained to rotate in the plane con taining the c hain [11]. Another example is the o ne-dimensional c hiral X Y mo del sin( χ n +1 − χ n − γ ) − sin( χ n − χ n − 1 − γ ) = K sin( pχ n ) ( p = 1 , 2 , ... ) , (24a) or, equiv alently , ℓ h 2 sin  θ n +1 − 2 θ n + θ n − 1 ℓ  cos  θ n +1 − θ n − 1 ℓ − γ  = sin θ n , (24b) where θ n = pχ n , ℓ = 2 p and h 2 = pK . The c hiral X Y model (24) describ es arrays of spins with the nearest-neigh b our interactions in an external magnetic field [3, 6]. It is used to mo del helimagnetic materials, discotic a nd ferro electric smectic liquid crystals, crystalline p olymers, thin magnetic films a nd Josephson junction a rra ys. 10 The classification of discretisations of the fo rm (22) reduces to the classification of all p ossible symmetric functions H ( x, y ) — the task completed in section I I I ab ov e. Eac h func- tion H 2 ( x, y ) iden tified in section I I I g iv es rise to a n um b er of purely p erio dic discretisation of the form (22), with v arious ℓ ; that is, eac h rat io nal-trigonometric exceptional mo del (10) has a set of purely trigo no metric coun terparts (22). W e will restrict ourselv es to the simplest represen tativ e(s) of these sets b y c ho osing appropria te v alue(s) of ℓ . The resulting mo dels can b e summarised as follows . Pic king the symmetric function (11) of the section I I I A and letting ℓ = 2, giv es rise to a v ery simple exceptional discretisation 2 h 2 sin  θ n +1 − 2 θ n + θ n − 1 2  = sin  θ n +1 + θ n − 1 2  . (25) If, instead, w e to o k ℓ = 4, w e w ould o btain a slightly more complicated mo del: 4 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  = sin  θ n +1 + θ n − 1 2  cos  θ n +1 − θ n − 1 4  . (26) Finally , if we “extend” the symmetric function (1 1) b y adding a term that v anishes in the con tin uum limit, H 2 ( x, y ) = 2 sin 2 x 2 + 2 sin 2 y 2 + 2 sin 2 x − y 2 , then, k eeping ℓ = 4, we will arriv e a t a (still reasonably simple) exceptional mo del 2 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  = sin  θ n +1 − θ n + θ n − 1 2  cos  θ n +1 − θ n − 1 4  cos θ 2 . (27) Next, c ho osing the symmetric function (13) of the section I I I B a nd letting ℓ = 4 yields Sp eigh t and W ard’s mo del, Eq.(23). On t he other hand, ta king the symmetric function (15) of section I II C a nd letting ℓ = 4, giv es the discretisation 4 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  = 2 sin  θ n 2  cos  θ n +1 + θ n − 1 4  . (28) Eq.(28) r educes to Eq.(23) with the lattice spacing constant ˜ h = h (1 + h 2 / 4) − 1 / 2 , if w e use an iden tit y 2 sin  θ n 2  cos  θ n +1 + θ n − 1 4  = sin  θ n +1 + 2 θ n + θ n − 1 4  − sin  θ n +1 − 2 θ n + θ n − 1 4  . (29) Pic king the symmetric function of the form ( 1 7) from section I I I D, and letting ℓ = 4 , giv es an exceptional discretisation 4 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  = 1 2 cos  θ n +1 + θ n − 1 4   sin θ n +1 2 + 2 sin θ n 2 + sin θ n − 1 2  . (30) 11 Finally , c ho osing the symmetric function in the form (19) from section I I I E a nd letting ℓ = 2 pro duces an exceptional mo del 2 h 2 sin  θ n +1 − 2 θ n + θ n − 1 2  = 2 sin  θ n 2  cos  θ n +1 − θ n + θ n − 1 2  . (31) W riting the right-hand side as a sum of sines, we repro duce equation (25) with h r eplaced with ˜ h = h (1 + h 2 / 2) − 1 / 2 . The mo dels (25), (26), (27 ) and (30) constitute our list of new exceptional p erio dic discretisations of the sine-Gor do n equation. This list can b e generalised and extended in a v a r iety of w ays . F or example, w e can replace the sine function in (21) with tan[( θ n +1 − θ n ) /ℓ ] or, more generally , with sin[( θ n +1 − θ n ) /ℓ ] cos p [ m ( θ n +1 − θ n )] with arbitrary m and p . Also, w e can add a sum P A n sin 2 [ B n ( x − y )] with ar bit r a ry A n and B n to an y of the symmetric functions H 2 ( x, y ). Since w e a re mainly in terested in simple discretisations, w e are not pursuing these p ossibilities in our presen t work. V. DISCRETE SINE-GORDON EQ UA TION WITH EXACT KI NK SOLUTIONS In this section we consider one more exceptional discretisation of t he sine-Gordon equa- tion. In addition to admitting an arbitra ry centring relative to the lattice, t he kinks in this mo del are a v ailable in exact explicit form. W e start with what may seem to b e an unrelated map, φ n +1 − φ n = h (1 − φ n φ n +1 ) . W riting the square of this map as ( φ n +1 − φ n ) 2 1 − φ n +1 φ n = h 2 (1 − φ n φ n +1 ) , and subtracting its bac k-iterated cop y giv es φ n +1 − 2 φ n + φ n − 1 + φ n ( φ 2 n − φ n +1 φ n − 1 ) = − h 2 φ n (1 − φ n +1 φ n )(1 − φ n φ n − 1 ) . (32a) Equation (32a) with φ n = cos θ n 2 , (32b) that is, equation cos θ n +1 2 − 2 cos θ n 2 + cos θ n − 1 2 + cos θ n 2  cos 2 θ n 2 − cos θ n +1 2 cos θ n − 1 2  = − h 2 cos θ n 2  1 − cos θ n 2 cos θ n +1 2   1 − cos θ n 2 cos θ n − 1 2  , (33) 12 pro vides an exceptional discretisation o f the sine-Gordon equation. Indeed, the con tin uum limit of equation (3 2a) is φ xx (1 − φ 2 ) + φ φ 2 x = − φ (1 − φ 2 ) 2 , (34) whic h is nothing but the stationar y sine-Gordon equation θ xx = sin θ written in t erms of φ = cos( θ / 2). Equation (32a) has an exact kink solution φ n = tanh( k n − x (0) ) , tanh k = h, where x (0) is a t r anslation parameter whic h can b e c hosen arbitrarily . Applying the transfor- mation (32b) to φ n pro duces a n explicit kink solution of the discrete sine-Gordon equation (33): θ n = 4 arcta n[exp( k n − x (0) )] , tanh k = h. (35) W e are not a ware of any ph ysical sys tems represen ted by Eq.(33). This discrete mo del ma y find its uses, how ever, in n umerical simulations of the con tin uum sine-Gordon equation. Lik e other exceptional discretisations of t he sine-G o rdon equation, this mo del preserv es a n “effectiv e translatio n in v ariance” of the con tin uum equation. The f a ct that the station- ary discrete kinks of the mo del (3 3 ) are av ailable in exact explicit form is an additional computational adv an tage. VI. TRA VELLING KIN KS In this section w e show ho w the method of one-dimensional maps can b e used to construct mo ving kinks. The discretisation breaks the Loren tz inv ariance o f the con tin uum mo del (2) in the same w a y as it breaks it s translation symmetry; hence the mobility of the kink b ecomes a non trivial prop erty in the discrete case. As the kink mov es in the P eierls-Nabarro p oten tial, it excites resonan t ra dia tion and decelerates as a result of that [1 2 , 14, 2 0, 28, 29]. Surprisingly , some discrete mo dels exhibit isolated v alues o f the kink v elo city fo r which the kink can slide , i.e. tra v el without exp eriencing radiativ e friction [18, 1 9, 30, 31]. A p ertinen t question here is whether there are exceptional discretisations where the kink can slide with an arbitr ary v elo city . A discrete nonlinear Sch r¨ odinger equation with this 13 prop erty is w ell kno wn; it is the Ablo witz-Ladik mo del whose solitons are radiationless irresp ectiv e o f their v elo cities. On the other hand, no discrete K lein-Gordon equations whose kink v elo cities w ould al l b e sliding v elo cities ha ve b een found so far — neither in the F renke l- K on torov a class of mo dels nor among the discrete φ 4 -theories. In this section w e construct suc h a discrete sine-Gordon equation. Its kink solutions a re giv en b y explicit expressions , and, as will b ecome obv ious from these explicit fo rm ulas, all its kinks trav el without emitting radiation. W e start with a nonstationar y equation φ xx − φ tt − 2 φ φ 2 x − φ 2 t 1 + φ 2 = φ 1 − φ 2 1 + φ 2 , (36) whic h transforms in to the sine-Gor do n equation θ xx − θ tt = sin θ b y the substitution φ = tan( θ / 4). Our first observ a tion is that if φ ( x, t ) is a simultaneous solutio n of t w o first-o r der equations φ x = 1 √ 1 − v 2 φ (37) and φ t = − v √ 1 − v 2 φ, (38) then it also satisfies Eq.(36). In (37) and (38), v is a parameter; − 1 < v < 1. Note that Eq.(36) do es not contain v explicitly; hence finding a solution of equations ( 3 7) and (3 8 ) for al l v amounts to finding a one-parameter family of solutions to (36). Next, w e discretise Eq.(37) according to 1 h 2 ( φ n +1 − φ n ) 2 = 1 1 − v 2 φ n φ n +1 (39) and divide b oth sides b y the same expression to get 1 h 2 ( φ n +1 − φ n ) 2 (1 + φ 2 n +1 )(1 + φ 2 n ) = 1 1 − v 2 φ n φ n +1 (1 + φ 2 n +1 )(1 + φ 2 n ) . (40) W e a lso consider a discrete v ersion o f Eq.(38): ˙ φ n = − v √ 1 − v 2 φ n . (41) Subtracting Eq.(40 ) f rom its bac k-iterated cop y and replacing φ n with tan( θ n / 4), giv es 1 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  = 1 2 1 1 − v 2 sin  θ n 2  cos  θ n +1 + θ n − 1 4  . (42) 14 On the other hand, Eq.(41) yields ¨ φ n − 2 φ n ˙ φ 2 n 1 + φ 2 n = v 2 1 − v 2 φ n 1 − φ n − 1 φ n +1 1 + φ n − 1 φ n +1 , (43) where w e hav e used the relation φ n +1 φ n − 1 = φ 2 n whic h is straightforw ard from (39). Letting φ n = tan( θ n / 4) in (43), w e get ¨ θ n 4 cos  θ n +1 − θ n − 1 4  = 1 2 v 2 1 − v 2 sin  θ n 2  cos  θ n +1 + θ n − 1 4  . (44) Finally , subtracting (42) from (44) yields a discrete sine-Gordon equation cos  θ n +1 − θ n − 1 4  ¨ θ n 4 = 1 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  − 1 2 sin  θ n 2  cos  θ n +1 + θ n − 1 4  . (45) F o r any h , this equation has a n explicit mov ing kink solution whic h is a compatible solution of the first-order difference equation (39) a nd the first-order differen tial equation (41): θ n = 4 arctan  exp  k n − v t √ 1 − v 2  , (46) where k is defined by 2 sinh  k 2  = h √ 1 − v 2 (47) and v can tak e an y v alue b etw een − 1 and 1. As h → 0, the solution (4 6)-(47) t ends to the tra v elling kink solution of the contin uum sine-Gordon equation, θ n → 4 ar ctan  exp  x n − v t √ 1 − v 2  , x n = hn. Using an iden tity (29), equation (45) can b e cast in the form cos  θ n +1 − θ n − 1 4  ¨ θ n = 4 ˜ h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  − sin  θ n +1 + 2 θ n + θ n − 1 4  , (48) where ˜ h = h (1 + h 2 / 4) − 1 / 2 . Solution to (48) is g iv en b y the same Eq.(46) where k should no w b e defined b y sinh  k 2  = 1 √ 1 − v 2 ˜ h p 4 − ˜ h 2 . (49) Solution (46),(49) exists for any | v | < 1 and 0 < ˜ h < 2 . W e close this section b y noting that the stationary limit of Eq.(48) coincides with the stationary part of t he Sp eigh t-W ard mo del [17], cos − 1  θ n +1 − θ n − 1 4  ¨ θ n = 4 ˜ h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  − sin  θ n +1 + 2 θ n + θ n − 1 4  . (50) 15 (Here cos − 1 α should b e understo o d as 1 / cos α and not as a r ccos α .) The sim ulatio ns of Sp eigh t and W ard [17] ha v e demonstrated that the motion o f the kink in their Eq.(50 ) is accompanied b y a muc h w eak er radiation than the kink pro pa gation in a ty pical nonexcep- tional mo del. Now that we ha v e another time-dep enden t v ersion o f the same stat io nary mo del, in which the radiation is completely suppressed for all v elo cities, the low lev el of radiation fr om the moving Sp eight-W ard kink can b e explained simply b y the pro ximit y of their equation (50 ) to our mo del (4 8). VI I. CO NCLUDING REMARKS Results of this work can b e summarised as follo ws. • Using the metho d of one-dimensional maps, we ha v e deriv ed sev eral exceptional dis- cretisations of the sine-Gordon equation in v olving ratios of trig o nometric to linear functions: equations (14), (16), (18) and (20). All these exceptional mo dels are new. W e ha v e also reco v ered the excep tiona l system of Kevrekidis, Eq.(12), whic h w as orig- inally obtained within a differen t approac h [25]. • W e ha v e iden tified sev eral new purely-trigonometric exceptional discretisations, in particular equations (25), (26) and (30 ): ¨ θ n = 2 h 2 sin  θ n +1 − 2 θ n + θ n − 1 2  − sin  θ n +1 + θ n − 1 2  ; ¨ θ n = 4 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  − sin  θ n +1 + θ n − 1 2  cos  θ n +1 − θ n − 1 4  ; ¨ θ n = 4 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  − 1 2 cos  θ n +1 + θ n − 1 4   sin θ n +1 2 + 2 sin θ n 2 + sin θ n − 1 2  . • W e hav e deriv ed a new discretisation with exact explicit kink solutions, Eq.(33). • W e ha v e constructed a discrete sine-Gordon mo del whic h supp orts kinks trav elling with arbitrary velocities: ¨ θ n cos  θ n +1 − θ n − 1 4  = 4 h 2 sin  θ n +1 − 2 θ n + θ n − 1 4  − sin  θ n +1 + 2 θ n + θ n − 1 4  . (51) 16 The latter result deserv es an additional commen t. By analogy with the deriv a tion of the mo del (51), it is not difficult to construct discrete φ 4 theories supp orting sliding kinks with arbitrary ve lo cities. One suc h mo del has the form ¨ φ n 1 − φ n +1 φ n − 1 1 − φ 2 n = φ n +1 − 2 φ n + φ n − 1 h 2 + φ n 2 (1 − φ n +1 φ n − 1 ) . (52) (This is a time-dep enden t generalisation of the exceptional stationary φ 4 mo del derive d in [27].) The moving kink solution to Eq.(52 ) has the fo rm φ n = tanh  k n − v t 2 √ 1 − v 2  , 4 tanh 2 k 1 + ta nh 2 k = h 2 1 − v 2 . Another time-dep enden t discretisation of the φ 4 theory with sliding kinks is ¨ φ n 1 − φ n +1 φ n − 1 1 − φ 2 n = φ n +1 − 2 φ n + φ n − 1 h 2 + φ n 2  1 − φ n 2 ( φ n +1 + φ n − 1 )  . (53) (This is a time-dep enden t generalisation of the exceptional φ 4 mo del iden tified b y Bender and T o vbis [32] and Kevrekidis [2 5].) The sliding kink solutio n has t he form φ n = tanh k n − 1 2 p 1 + ta nh 2 k v t √ 1 − v 2 ! , 4 tanh 2 k = h 2 1 − v 2 . The a na logy b et w een equ atio n (51) and the Ablow itz-Ladik mo del is also worth commen t- ing up on. The Ablo witz-Ladik mo del is the only discrete nonlinear Sc hr¨ odinger equation whose solito ns can slide with an y c hosen v elo city . The absence o f the accompan ying radia- tion is usually explained by the integrabilit y of this equation. A similar b eha viour of kinks of Eq.(51) mak es one w onder whether the latter equation could also b e integrable. W e hav e tested the in tegrabilit y of Eq.(51) n umerically , b y sim ulating a collision of a kink and an an tikink. The scattering was found to b e inelastic: t he v elo cities of the kink and an tikink c hanged as a result of the collision, and significan t amount of r adiation w as detected. Con- sequen tly , we conclude that equation (5 1) is not in tegrable. This example demonstrates that, contrary to common b elief, the inte gra bilit y is not a prerequisite for the existence of a discrete soliton sliding at an arbitrar ily c hosen ve lo cit y . Finally , it is in teresting to compare tra v elling kink solutions of our mo del (51) with tra v elling kinks of another mo dification of the Sp eigh t-W a rd mo del prop osed by Zakrzewski [31]. Z akrzewski’s mo del is differen t f rom the Sp eigh t-W ard equation ( 50) in the presence of a factor (1 + α ˙ θ 2 n ) − 1 in fr o n t o f the left-hand side of (50), with α = const . The mo del has an 17 exact solution in the form of a sliding kink; how ev er, similarly to radiationless mo ving kinks in other systems [18, 19, 30], this kink can only slide with one particular v elo city whic h is determined b y the par a meters h and α . Unlik e this co dimension-1 solution, our sliding kinks (46), (49) ha ve co dimension 0 in the sense that they can mo v e with an arbit r a ry ve lo cit y indep enden t of h . Ac kno wledgments IB was supp orted b y the NRF of South Africa under grant 205372 3 . TvH w as supp orted b y the Natio nal Institute of Theoretical Phy sics. [1] Y. F r enk el and T. K ontoro v a, J. Phys. (USSR) 1 , 137 (1939) [2] P . Bak, Rep. Pr og. Phys. 45 587 (1982) ; L.M. Flor ´ ıa and J.J. Mazo, Adv ances in Physics 45 505 (1996); O.M. Braun and Y u .S . Kivshar, Phys. Rep . 306 1 (1998); O.M. Braun and Y u.S. Kivsh ar. The F renk el-Konto rov a mo del. Springer V erlag, New Y ork (2004 ) [3] R.B. Griffith s , in: F und amen tal Problems of S tatistical Mec hn ics VI I: p ro ceedings of the Sev ent h Internatio n al Summer School on F und amen tal Problems in S tatistica l Mec hanics, Alten burg, German y , Ju ne 18-30, 1989. Ed itor H. v an Beijeren. Elsevier Science Publish ers B.V., 1990, pages 69-110 [4] S. Aubry , Physica D 7 240 (1983); S. Aubry and P .Y. Le Daeron, Physica D 8 381 (1983) [5] S. T ak eno, J ourn. Phys. S o c. Jpn. 64 2380 (1995); S. T ak eno, M. P eyrard , Physica D 92 140 (1996); F. Zhang, Physica D 110 51 (1997) [6] A. Banerjea and P .L. T aylor, Phys. Rev. B 30 6489 (1984); W. Chou and R.B. Griffiths, Phys. R ev. B 34 6219 (1986); C.S.O. Y ok oi, L.H. T ang, and W. Chou, Phys. Rev. B 37 2173 (1988); J.J. Mazo, F. F alo and L.M. Flor ´ ıa, Phys. Rev. B 52 10433 (1995 ); 18 M. Monma and T. Horiguc hi, Physic a A 234 837 (1997); L. T rallori, Phys. Rev. B 57 5923 (1998); M. Momma and T . Horiguc hi, Ph ysica A 251 485 (1998); R. Balakrishnan and M. Meh ta, Ph ys. Rev. E 61 1312 (2000) [7] F. Fillaux and C .J. Carlile, Phys. Rev. B 42 5990 (1990); F. Fillaux, C .J. Carlile, G.J. Kearley , Ph ys. Rev. B 44 12280 (1991) [8] S. T ak eno and S. Homma, Jour n. Ph ys. So c. Jpn . 55 65 (1986) [9] S. T ak eno and S. Homma, Pr og. Theor. Phys. 77 548 (1987) [10] F. Zhang, M.A. Collins, Y.S. Kivsh ar, Ph ys. Rev. E 51 3774 (1995) [11] J .M. Sp eigh t and Y. Zolotaryuk, Nonlin earity 19 1365 (2006 ) [12] J .F. Cur r ie, S.E. T rullinger, A.R. Bishop, and J.A. Kr umhansl, Ph ys. Rev. B 15 5567 (1977) [13] M. P eyrard and M. Remoissenet, Phys. Rev. B 26 2886 (1982) [14] Y. Ishimori and T. Munak ata, J . Phys. S o c. Jpn. 51 3367 (1982); M. P eyrard and M.D. K rusk al, Ph ysica D 14 88 (1984) [15] R . Bo esc h, C.R. Willis, and M. El-Batanoun y , Phys. Rev. B 40 , 2284 (1989); T. Munak ata, Ph ys. Rev. A 45 1230 (1992 ); S. Flac h and K. Kladko, P hys. Rev. E 54 2912 (1996) [16] P . Stancioff, C. Willis, M. El-Batanoun y , and S. Bur d ic k, Phys. Rev. B 33 1912 (1986); R. Bo esc h and C.R. Willis, Phys. Rev. B 39 361 (1989); C.R. Willis and R. Bo esch, Phys. Rev. B 41 4570 (1990); S. Flac h and C.R. Willis, Ph ys. Rev. E 47 4447 (1993) [17] J .M. Sp eigh t and R.S. W ard, Nonlinearit y 7 475 (1994) [18] S . Flac h, Y. Zolotaryuk and K. Kladk o, Ph ys. Rev. E 59 6105 (1999) [19] O .F. Oxtoby , D.E. P elino vsky , and I.V. Barashenko v, Nonlinearit y 19 217 (2006) [20] S .V. Dmitriev, P .G. Kevr ekidis and N. Y oshik a wa , Journ. Ph ys. A: Math. Gen. 38 7617 (2005); S.V. Dmitriev, P .G. Kevrekidis, and N. Y oshik a w a, Journ. Phys. A: Math. Gen. 39 7217 (2006) [21] S .V. Dmitriev, P .G. Kevrekidis, N. Y oshik a w a, and D.J. F ran tzesk akis, Phys. Rev. E 74 046609 (2006 ) [22] I . Ro y , S .V. Dmitriev, P .G. Kevr ekidis, and A. S axena, Ph ys. Rev. E 76 026601 (2007) [23] S .V. Dmitriev, P .G. Kevrekidis, A. Khare, and A. Saxena, J . P h ys. A: Math. Theor. 40 6267 (2007 ) 19 [24] J .M. Sp eigh t, Nonlinearit y 10 1615 (1997) [25] P .G. Kevrekidis, Physica D 183 68 (2003) [26] B.M. Herbst and M.J. Ablo witz, J. Comp. Phys. 105 122 (1993) [27] I .V. Barashenk ov, O.F. Oxtoby and D.E. P elinovsky , Phys. Rev. E 72 035602(R) (2005) [28] C . Willis, M. El-Batanouny , and P . Stancioff, P hys. Rev. B 33 1904 (1986) [29] P .G. Kevrekidis, M.I. W einstein, Physic a D 142 113 (2000) [30] Y. Zolotaryuk, J.C. Eilb ec k, A.V. Sa vin, Physica D 108 81 (1997); A.V. Sa vin, Y. Zolotaryuk, J .C. Eilb ec k, Physic a D 138 267 (2000); A.R. Champn eys, Y u.S. Kivshar, Phys. Rev. E 61 2551 (2000); A.A. Aigner, A.R. C h ampneys, V.M. Rothos, P hysica D 186 148 (2003); G. Io oss, D.E. P elino vsky , Physica D 216 327 (2006) [31] W.J. Zakrzewski, Nonlinearity 8 517 (1995) [32] C .M. Bender and A. T o vbis, J . Math. Phys. 38 3700 (1997) 20