Exact Solutions to the Sine-Gordon Equation

EXA CT SOLUTIONS TO THE SINE-GORDON EQUA TION T unca y Aktosun Departmen t of Mathematics Univ ersit y of T exas at Arlington Arlington, TX 76019-0408, USA F rancesco Demon tis and Cornelis v an der Mee Dipartimen to di Matematica e Informatica Univ ersit` a di Cagliari Viale Merello 92 09123 Cagliar i , Italy Abstract : A systematic metho d is presen ted t o pro vide v arious equi v al en t sol ution for- m ul as for exact solutions to the s ine-Gordon e quation. Suc h solutions are analytic i n the spatial v ariable x and the temp oral v ariable t, and they are exp onen t ially asymptot i c to in teger m ul tiples of 2 π as x → ±∞ . The solution form ulas are expressed explicit ly in terms of a real triplet of constant matrices. The metho d presen t ed is gen eralizable to other i n- tegrable evolution equations where the in vers e scatteri ng transform is applied vi a the use of a Marchen k o integral equation. By expressing the kernel of that Marc henk o equatio n as a matrix exp onen t i al in terms of the matrix triplet and b y exploiting the separabilit y of that kern el, an exact solution formula to t he Marchen k o equation is derived, yielding v arious equiv alen t exact sol ution form ulas f or the sine-Gordon equati on. Mathemat ics Sub ject Cla ssification (2000): 37K15 35Q51 35Q53 Keyw ords: Sine-Gordon equation, exact solutio ns, explicit solutions 1 1. INTR ODUCTION Our goal in this paper is to derive, in terms of a triplet of c onstan t matrices, explicit form ula s for exact solutions to the sine-Gordon equation u xt = sin u, (1.1) where u is real v al ued and the subscripts denote the part ial deriv ativ es with resp ect to t he spatial co ordinate x and the temp oral co ordinate t. Under the t r a nsformation x 7→ ax + t a , t 7→ ax − t a , where a is a p ositive constan t, ( 1.1) i s transformed in to the al ternate form u xx − u tt = sin u, (1.2) and hence our ex plicit form ulas can easily b e mo dified to obtain expli ci t solutio ns to ( 1 .2) as well. Let us note that o ne can omit a mu ltiple of 2 π from a n y solutio n to ( 1 .1). W e are interested in sol utions to (1.1) where u x ( x, t ) → 0 as x → ±∞ for each fixed t, and hence without a n y loss of generality we wil l normalize our solutions so that u ( x, t ) → 0 as x → + ∞ . The sine-Gordon equation arises in appli cations as div erse as the description of surfaces of constan t mean curv ature [10 ,16], one-dimensional crystal dislo cation theory [17,23,40,41 ], magnetic flux propagation in Josephson junctions (gaps b etw een tw o sup erconductors) [29,31], condensation o f c harge densit y w av es [11,22 ,36], w a v e propagatio n in ferromag- netic materials [19,2 7,30], excitat ion of phonon mo des [35], and propagation of deforma- tions along the DN A double helix [18,26, 38,43]. The l i terature on exact solut i ons t o (1.1 ) is lar g e, and w e wi ll men tion only a few and refer the reader to those references and further o nes t herein. F or a p ositive constan t a, by substituting u ( x, t ) = 4 tan − 1  U ( ax + a − 1 t ) V ( ax − a − 1 t )  , (1.3) 2 in to ( 1 .1) and solving the resulting part ial differen t ial equat i ons for U and V , Steuerw a ld [42] has catalogued ma n y exa ct solutions to the sine-Gordon equation in terms of ellipti c functions. Some of these sol uti ons, including the one-solit on soluti o n, tw o-soliton solutions mo deling a soliton-solit on and soliton-antisoliton col lision, and the breather solution, can b e written in terms of elementary functions [25,3 7 ] , whil e the n -soliton solutions can b e expressed as i n (1.3) where U and V are certai n determinants [ 34,39,45] . The same sepa- ration techn ique can also b e used to find exact solutions to the sine-Gordon equation on finite ( x + t )-in terv als [13]. Solutions to the sine-Gordon equati on with initial data sp ecified on in v arian t algebraic manifolds of conserved q uan ti ties can b e written explicitl y in terms of Jacobi theta functions [24 ]. The ordered exp onen t ial integrals app earing in suc h solu- tions can b e ev aluated explicitly [9,28] . Let us also mention that some exact solutions to the sine-Gordon equati ons can b e obtai ned via the Darb oux or B¨ ac klund t ra nsformations [21,37] from already known ex act sol utions. The sine-Gordon equati on w as the fourth nonlinear partial differen tial equation whose initial-v alue problem was disco v ered [ 2,3] to b e solv able b y the inv erse scatt eri ng transform metho d. This metho d asso ciates (1 . 1) wit h the first-order system of ordinary differen t ial equations      dξ dx = − iλξ − 1 2 u x ( x, t ) η , dη dx = 1 2 u x ( x, t ) ξ + iλη, (1.4) where u x app ears in the co efficien ts as a p otential. By exploiting the one-to-one corresp on- dence b etw een u x and the corresp onding scattering data for ( 1.4), t he i n verse scat t ering transform metho d determines the time ev olution u ( x, 0) 7→ u ( x, t ) for (1.1) with the help of the solutions t o the direct and inv erse scattering problems for (1.4). The direct scattering problem for (1.4) amoun ts to finding the scattering co efficien ts (rela t ed t o the asymptot ics of scattering soluti ons to (1.4) as x → ±∞ ) when u ( x, t ) is k nown for a ll x. On the other hand, the i n verse scattering problem consists of finding u ( x, t ) from an appropriate set of scattering data for (1.4) . 3 In this pap er we pro vide seve ral, but equiv alen t, explicit formulas for exact soluti o ns to (1.1). The key i dea to obtain suc h expli cit formulas is to express the ke rnel of a related Marc henko i n t egra l equation arising in the i n verse scatt ering problem for (1.4 ) in terms of a real triplet ( A , B , C ) of constan t matrices and b y using matrix exp onen tials. Suc h explicit form ul a s provide a compact and concise w a y t o express our exact solutions, which can equiv alently b e expressed i n terms of exp o nen tial , trigonometri c (sine and cosine), and p olynomial functions of x and t. Thi s can b e done by “unpacking” matrix exp onen t ials in our ex plicit form ulas. As the matrix size i ncreases, t he unpac ked expressions b ecome very long. How eve r, suc h expressions can b e ev aluated explicitly for any matri x size either by hand or by using a symboli c soft w are pack age suc h as Mathematica. One of the p ow erful features of our met ho d comes from the fact t hat our concise and compact expl i cit solution form ula s are v alid for an y mat rix size i n the mat rix exp onen tials in v olv ed. In some other a v a ilable metho ds, exact solutio ns are attempted in terms of elemen tary functions without the use of matr i x ex p onen tia ls, and hence exact solutions pro duced by suc h other metho ds will b e relatively simple and w e cannot exp ect those metho ds to pro duce our solutions when t he matrix size is large. Our metho d is generalizable and applicable to obtain similar explicit form ulas for ex act solutions t o other in tegrable nonlinear part ial differen tial equati o ns, where a Marc henk o in tegral equati on is used to solve a related i nv erse scattering problem. W e refer the reader to [5-7,14,15 ] , where similar ideas are used to obtain explicit form ulas for exact solutions to the Kortew eg-de V ries equation on the half line and to the fo cusing nonli near Sch r¨ odinger equation and its m a t rix generalizations. In our metho d, wi th t he help o f the matrix tripl et a nd matri x exp o nen tial s, we easily establish the separability of the kernel of t he relev an t Marc henko integral equation and th us solve it exactly b y using li near algebra. W e then obtain our exact sol utions to t he sine-Gordon equation b y a simple i ntegration of t he sol ution t o t he Marc henko equation. 4 Our metho d easily handles complicatio ns arising from the presence of non-simple p oles of the transmission coefficien t in the related linear system (1.4) . Deali ng wi th n on-simple p oles wi thout the use o f matrix exp onen tials is very complicated, and this issue has also b een a problem [33, 44] in solvi ng other integrable nonlinear partial differen t i al equati o ns suc h as the nonlinear Sc hr¨ odinger equation. Our pap er is org a nized as follo ws. In Section 2 we establ ish our notation, in tro duce the relev an t Marc henk o in tegral equation, and mention ho w a solution to the sine-Gordon equation is obtained from the solution t o the Marc henk o equation b y using the i n verse scattering transform me tho d. In S ection 3 w e outline the solution t o the Marc henko in tegral equation when its kern el i s represen ted in terms of a tri pl et of matri ces ( A, B , C ) and thus we derive tw o soluti o n form ulas for exact solutions to the sine-Gordon equation. In Sections 4 and 5 we sho w t hat our explicit soluti on form ulas hold when the input mat rix triplets come from a larger family; we sho w that our solutio n form ulas in the more general case can b e obtained b y constructing tw o a uxiliary constan t matrices Q and N satisfying the res p ective Ly apuno v equations giv en in Section 4, or equiv al en tl y by constructing an a ux iliary constan t matrix P satisfying the Sylvester equation giv en i n Section 5. In Section 4 we also show that t he matrix tri plet ( A, B , C ) used as input to construct our exact soluti o ns to the sine-Gordon equation can b e chos en in v arious equiv alen t w a y s and w e pro v e that o ur exact solutions are analyt ic on the xt -pla ne. In Section 5 we also explore the relationship betw een the Ly apuno v equations and the S ylvester equat i on and show ho w their solutions are related to eac h other in a simple but interesting wa y . In that section w e also sho w that the t w o solution form ulas derived in Section 3 are equiv alen t. In Section 6 w e sho w t hat those t w o equiv alent solutio n form ulas can b e represen ted in other eq uiv alen t forms. In Section 7 we ev aluate the square of the spatial deriv ative of our solutio ns to (1.1) b y pro v iding some explicit form ulas in terms of the matrix triplet ( A, B , C ) , and w e ev aluate the a symptotics of our exact solutio ns as x → −∞ for each fixed t. In Section 8 w e sh o w that the refle ction co efficien ts asso ciated with suc h solutions are zero, and w e 5 also ev aluate expl i citly the corresp onding t ransmission co efficient. Fi nally , in Section 9 w e pro v ide some sp ecific examples of our exact soluti o ns and their snapshots. Let us remark on the loga rithm and inv erse tangen t functions we use throughout our pap er. The log function w e use i s the principal branc h of the complex-v alued logari thm function and it has its branch cut al ong the negative real axi s while log(1) = 0 . The t an − 1 function we use is the single-v alued branc h relat ed to t he principal branc h of the lo garithm as tan − 1 z = 1 2 i log  1 + iz 1 − iz  , log z = 2 i tan − 1  i (1 − z ) 1 + z  , (1.5) and its branc h cut is ( − i ∞ , − i ] ∪ [ i, + i ∞ ) . F or any square matrix M not having eigen v alues on that branch cut, we define tan − 1 M : = 1 2 π i Z γ dz [tan − 1 z ]( z I − M ) − 1 , (1.6) where t he con tour γ encircles each eigen v alue of M exactly once in the p ositive directi on and av oids the branc h cut of tan − 1 z . If a l l eigenv alues of M ha ve mo dulus less than 1 , w e then ha v e the fami liar series expansion tan − 1 M := M − 1 3 M 3 + 1 5 M 5 − 1 7 M 7 + . . . . F or real-v alued h ( x ) that v anishes as x → + ∞ , the function tan − 1 ( h ( x )) alw a y s has range ( − π / 2 , π / 2) when x v alues are restricted to ( x 0 , + ∞ ) for some large x 0 v alue; our tan − 1 ( h ( x )) is the con tin uous extension of that piece from x ∈ ( x 0 , + ∞ ) to x ∈ ( −∞ , + ∞ ) . 2. PRELIMINARIES In this section we briefly review the scattering and in v erse scattering theory for (1. 4 ) b y intro ducing the scattering co efficien ts and a Marchen k o in tegral equation associat ed with (1.4). W e assume that u i s real v alued and that u x is integrable in x for each fixed t. W e also men ti on ho w a soluti o n t o the sine-Gordon equation is o bt a ined from the solutio n 6 to the M arc henk o eq uation. W e refer the reader to the generic references suc h as [1,4,25,32] for the details. Tw o li nearly indep enden t sol ut i ons to ( 1 .4) known as the Jost solutions from the l eft and from the righ t, denoted b y ψ ( λ, x, t ) and φ ( λ, x, t ) , resp ectively , are those sol utions satisfying the resp ectiv e spatial asymptot ics ψ ( λ, x, t ) = " 0 e iλx # + o (1) , x → + ∞ , (2.1) φ ( λ, x, t ) = " e − iλx 0 # + o (1) , x → −∞ . The scattering co efficients for (1.4), i.e. the transmission coefficient T , the righ t reflec- tion co efficien t R, and the left reflection co efficien t L, can b e defined through the spatial asymptotics ψ ( λ, x, t ) =      L ( λ, t ) e − iλx T ( λ ) e iλx T ( λ )      + o (1) , x → −∞ , (2.2) φ ( λ, x, t ) =      e − iλx T ( λ ) R ( λ, t ) e iλx T ( λ )      + o (1) , x → + ∞ , where T do es not dep end on t, and R and L dep end on t as R ( λ, t ) = R ( λ, 0) e − it/ (2 λ ) , L ( λ, t ) = L ( λ, 0) e it/ (2 λ ) . W e recall that a b ound st a te correspo nds to a square-in tegrable solution to ( 1.4) and suc h solutions can only o ccur a t the p oles of the meromor phic extension of T to the upp er half complex λ -plane denoted b y C + . Because u ( x, t ) is real v alued, suc h p oles can o ccur either on the p o si tive ima g inary axi s, or for eac h p ole not on t he p ositive imaginary ax is there corresp onds a po l e symmetrically lo cated with respect to the imaginary axis. F urthermore, suc h p oles are not necessarily simple. If u x is integrable in x for eac h fixed t and if the 7 transmission co efficien t T is con tin uous for real v alues of λ, it can b e pro v ed by elemen t a ry means that the num b er of suc h p oles and their multiplicities are finite. With the con ven tion u ( x, t ) → 0 as x → + ∞ , it is kno wn that u ( x, t ) in (1. 4) can b e determined as u ( x, t ) = − 4 Z ∞ x dr K ( r , r , t ) , (2.3) or eq uiv alen tl y w e hav e u x ( x, t ) = 4 K ( x, x, t ) , where K ( x, y , t ) i s the sol ution t o t he March enk o integral equatio n K ( x, y , t ) − Ω( x + y , t ) ∗ + Z ∞ x dv Z ∞ x dr K ( x, v , t ) Ω( v + r, t ) Ω( r + y , t ) ∗ = 0 , y > x, (2.4) where the a sterisk i s used to denote complex conjugatio n (wit hout taking the matrix transp ose) and Ω( y , t ) = 1 2 π Z ∞ −∞ dλ R ( λ, t ) e iλy + n X j =1 c j e iλ j y − it/ (2 λ j ) , (2.5) pro v ided the p oles λ j of the transmission co efficien t are all simple. The i nv erse scattering tra nsform pro cedure can b e summarized v ia t he following diag ram: u ( x, 0) − − − − → u x ( x, 0) direct scattering at t =0 − − − − − − − − − − − − − − − → { R ( λ, 0) , { λ j , c j }} sine-Gordon solutio n   y   y time ev olution u ( x, t ) ← − − − − u x ( x, t ) ← − − − − − − − − − − − − − inv erse scattering a t t { R ( λ, t ) , { λ j , c j e − it/ (2 λ j ) }} W e note that in general t he summation term in (2.5 ) is m uc h more complicated, and the expression w e ha v e provided for it in (2.5) is v al i d only when the transmission co efficien t T has simple p o l es at λ j with j = 1 , . . . , n on C + . In case of b ound states wi th nonsimple p ol es, it is unkno wn to us if the normi ng constan t s with the appropriate time dep endence hav e eve r b een presen ted i n the literature. Extending our previo us results for 8 the nonlinear Sc hr¨ odinger equati on [6,7,1 2,14] t o the si ne-Gordon equati on, i t is p ossible to obtain t he normi ng constants with appropriate dep endence on t he parameter t in the most general case, whether the b o und-state p oles o ccur on the p ositive i maginary axis or o ccur pairwise lo cat ed symmetrically with resp ect to the p ositive imaginary axis, and whether an y such p o l es are simple or hav e m ultiplicities. I n fact, in Section 8 we presen t the norming constan ts and their prop er time dep endence on t as we ll as the most general form of the summation term that should app ear i n (2. 5). When u is real v alued, it is k no w n that for real λ w e hav e R ( − λ, t ) = R ( λ, t ) ∗ , L ( − λ, t ) = L ( λ, t ) ∗ , T ( − λ ) = T ( λ ) ∗ . Because u is real v alued, as we verify in Section 3, b oth the k ernel Ω( y , t ) and the sol ution K ( x, y , t ) i n (2.4) are also real v alued, i .e. Ω( y , t ) ∗ = Ω( y , t ) , (2.6) K ( x, y , t ) ∗ = K ( x, y , t ) . (2.7) 3. EXPLICIT SOL UT IONS TO THE SINE-GORDON EQUA TION Our goal in this section is to obtain some exact solutions to the sine-Gordon equati on in terms of a triplet of constant mat rices. F oll o w i ng the main idea o f [6,7] w e will replace the su mmation term i n (2.5) b y a compact expression in terms of a matrix triplet ( A, B , C ) , i.e. we will replace Ω( y , t ) when R = 0 b y Ω( y , t ) = C e − Ay − A − 1 t/ 2 B , (3.1) where A, B , C are real and constant mat rices of sizes p × p, p × 1 , and 1 × p, resp ectively , for some p ositive in teger p. 9 Recall that any rational function f ( λ ) t hat v anishes as λ → ∞ in the complex λ -plane has a matrix realizatio n in terms of three constan t matrices A, B , C as f ( λ ) = − iC ( λI − iA ) − 1 B , (3.2) where I is the p × p identit y matrix, A has size p × p, B has size p × 1 , and C has size 1 × p for some p. W e wil l refer to ( A, B , C ) as a matri x triplet of size p. It is p ossible to pad A, B , C with zeros or it may b e p ossible t o c hange t hem and i ncrease o r decrease the v alue of p wit hout chan ging f ( λ ) . The smallest p ositive in t eger p yielding f ( λ ) gi v es us a “minimal” reali zation for f ( λ ) , and it is k nown [8] that a minimal realization is unique up to a simi larity transformation. Th us, without any loss of generalit y we can alwa ys assume that our tri plet ( A, B , C ) corresp onds to a minimal realization, and w e wil l refer to suc h a triplet as a minimal tri plet. N ote that the p ol es o f f ( λ ) corresp ond to the eigen v alues of the matrix ( iA ) . By tak i ng the F ourier transform o f b oth sides of (3. 2 ), where the F ourier transform is defined as ˆ f ( y ) : = 1 2 π Z ∞ −∞ dλ f ( λ ) e iλy , w e o btain ˆ f ( y ) = C e − Ay B . (3.3) W e note that under t he similari ty transformation ( A, B , C ) 7→ ( S − 1 AS, S − 1 B , C S ) for some in v ertible ma t rix S, t he quan tities f ( λ ) and ˆ f ( y ) remain unc hanged. Comparing (3 . 1) and (3.3) we see t hat they are closely related t o each other. As men ti oned earlier, without loss of any generalit y w e assume that the real tri pl et ( A, B , C ) in (3.1) corresp onds to a minimal realization in (3.2). F or the time being, w e will also assume that all eigen v alues of A in (3.1) hav e p ositi v e real parts. How eve r, i n later sections w e will relax the latter assumption and choose our tri plet i n a less restrictive w a y , i.e. i n the admissible class A defined in Section 4. Let us use a dagger t o denote the mat rix adjoin t ( complex conjugation and ma - trix transp ose). Although the adjoin t and the tra nsp ose ar e equal to eac h other for real 10 matrices, w e will con tin ue to use the dagger notation even for the matrix transpo se of real mat rices so that w e can utili ze the previ ous r el a ted results in [5,6 ] obtained for t he Zakharov-Sh abat system and the nonli near Schr¨ odinger eq uation. Si nce Ω app earing i n (3.1) is a scal a r we hav e Ω † = Ω ∗ ; th us, we get Ω( y , t ) ∗ = B † e − A † y − ( A † ) − 1 t/ 2 C † . W e note that when Ω is given by ( 3.1), the Marc henko eq uation is exa ctly solv able by using linear algebra. This follows from the separability prop erty of the ke rnel, i.e. Ω( x + y , t ) = C e − Ax e − Ay − A − 1 t/ 2 B , (3.4) indicating t he separability in x and y ; th us, (3.4) all o ws us to try a solution to (2 . 4) in t he form K ( x, y , t ) = H ( x, t ) e − A † y − ( A † ) − 1 t/ 2 C † . (3.5) Using (3.5) in ( 2.4) we get H ( x, t ) Γ( x, t ) = B † e − A † x , (3.6) or eq uiv alen tl y H ( x, t ) = B † e − A † x Γ( x, t ) − 1 , (3.7) where w e hav e defined Γ( x, t ) := I + e − A † x − ( A † ) − 1 t/ 2 Qe − 2 Ax − A − 1 t/ 2 N e − A † x , (3.8) with the constan t p × p matrices Q and N defined as Q : = Z ∞ 0 ds e − A † s C † C e − As , N := Z ∞ 0 dr e − Ar B B † e − A † r . (3.9) It is seen from (3.9) that Q and N are selfadjoin t , i.e. Q = Q † , N = N † . (3.10) 11 In fact, since the t r i plet ( A , B , C ) is r eal , the ma t rices Q a nd N are also real and hence they are symmet r i c matrices. Using (3. 7) in (3.5) we obtain K ( x, y , t ) = B † e − A † x Γ( x, t ) − 1 e − A † y − ( A † ) − 1 t/ 2 C † , (3.11) or eq uiv alen tl y K ( x, y , t ) = B † F ( x, t ) − 1 e − A † ( y − x ) C † , (3.12) where w e hav e defined F ( x, t ) := e β † + Q e − β N , (3.13) with the quan tity β defined as β := 2 A x + 1 2 A − 1 t. (3.14) F rom (2.3) and (3.12) we see that u ( x, t ) = − 4 Z ∞ x dr B † F ( r , t ) − 1 C † . (3.15) The pro cedure describ ed in (3.4 ) -(3.15) is exa ctly the same pro cedure used in [5, 6] with t he onl y difference of using A − 1 t/ 2 i n the matrix exp onential in (3.15) instead of 4 iA 2 t used i n [5,6] . How ever, suc h a difference do es not affect the sol ution to the Marc henk o in tegral equatio n at all t hank s to the fact that A a nd A − 1 comm ut e with eac h other. In fact, t he solution to the March enk o equation is obtained the same wa y if one replaces A − 1 / 2 by an y function of the matrix A b ecause suc h a matrix function comm utes with A. W e wil l later prov e t hat F ( x, t ) given in (3.13) i s in v ertible on the en tire xt -plane and that F ( x, t ) − 1 → 0 exponentially as x → ±∞ and hence u ( x, t ) given in (3.15 ) is w ell defined on the entire xt -plane. W e note that, as a result of (2.6), the solution K ( x, y , t ) t o the Marc henk o equati on ( 2.4) is real and hence (2.7) is sat i sfied. Hence, from (2 . 3) we see that u ( x, t ) is real v alued, and b y taking the adjoin t of b oth sides o f (3. 15) we get u ( x, t ) = − 4 Z ∞ x dr C [ F ( r , t ) † ] − 1 B . (3.16) 12 Instead of using (2.6) at the last stage, l et us instead use i t from the ve ry b eginning when w e solve t he Marc henk o equation (2. 4). Replacing Ω ∗ b y Ω in t he tw o o ccurrences in ( 2.4), w e can solve (2.4) in a simi lar wa y as in (3. 4 ) -(3.15) and obtain K ( x, y , t ) = C E ( x, t ) − 1 e − A ( y − x ) B , (3.17) where w e hav e defined E ( x, t ) := e β + P e − β P , (3.18) with β as in (3.14) and the constan t matrix P gi v en b y P := Z ∞ 0 ds e − As B C e − As . (3.19) Th us, from (2.3) and (3.17) w e obtain u ( x, t ) = − 4 Z ∞ x dr C E ( r, t ) − 1 B . (3.2 0 ) W e will show in Section 5 that the tw o ex pl i cit solutions t o the sine-Gordon equation giv en b y ( 3.15) a nd (3.20) are iden t i cal b y pro ving that E ( x, t ) = F ( x, t ) † . (3.21) 4. EXA CT SOLUTIONS USING THE L Y APUNO V EQUA TIONS In Section 3 we hav e derived (3.1 5) and (3.20) b y assuming that we start wit h a real minimal tri plet ( A, B , C ) where the eig env alues of A h a v e p ositi v e real part s. In this s ection w e show that the explici t formula (3.15) for exact solutions to the sine-Gordon equati o n remains v alid if the matri x triplet ( A , B , C ) used t o construct suc h solutions is chos en in a larger class. Star t ing with a more arbitrary tripl et we wi ll construct t he matrix F given in (3.13), where the auxili a ry matrices Q and N are no longer given by (3. 9) but obtained b y uniq uely sol ving the r esp ecti ve Lyapuno v eq uations A † Q + QA = C † C, (4.1) 13 AN + N A † = B B † . (4.2) Man y of the p ro ofs in this section are similar to those obtained earlier for the nonlinear Sc hr¨ odinger equat ion [5 , 6] and hence w e will refer the reader to those references for the details of some of the pro ofs. Definit ion 4.1 W e s ay that the triplet ( A, B , C ) of size p b elongs to the admissi b le class A if the fol lowing c ondi tions ar e met: (i) The matric es A, B , and C ar e al l r e al value d. (ii) The triplet ( A , B , C ) c orr esp onds to a mi ni mal r e alization for f ( λ ) when that triplet is use d on the ri ght hand si de o f (3.2). (iii) None of the eigenv a lues of A ar e pur ely imagi n ar y and no two ei genvalues of A c an o c cur symmetric al ly with r es p e ct to the imaginary axis i n the c omplex λ -p lane. W e note that, since A is real v alued, the condition stated in (ii i) is equiv alen t to the condition that zero is not an eigen v alue of A a nd that no tw o eigen v alues of A are lo cated symmetrically with resp ect to the origin in t he complex plane. Equiv alen tly , (iii) can b e stated as A and ( − A ) not having an y common eigenv alues. W e w i ll say that a triplet is admissible if it b elongs to the admissible class A . Starting wi t h a triplet ( A, B , C ) i n the admissible class A , we wil l o btain ex act solu- tions to the sine-Gordon equation as follows: (a) Using A, B , C as input, construct the auxil iary ma t rices Q and N by solvi ng the re- sp ectiv e Ly apunov equatio ns (4 .1) and (4.2). As t he next theorem sho ws, the s olutions to (4.1) and (4.2) are unique and can b e obtai ned as Q = 1 2 π Z γ dλ ( λI + iA † ) − 1 C † C ( λI − iA ) − 1 , (4.3) N = 1 2 π Z γ dλ ( λI − iA ) − 1 B B † ( λI + iA † ) − 1 , (4.4) 14 where γ is an y p o si t iv ely oriented simple closed contour enclosing all eigen v al ues of ( iA ) and leaving o ut all eigen v alues of ( − iA † ) . If all eigenv alues of A hav e p o sitive real parts, then Q and N can also b e ev aluated as in (3.9) . (b) Using t he auxiliary matrices Q and N and the t ri plet ( A, B , C ) , form the matrix F ( x, t ) as in (3.13 ) and obtai n the scalar u ( x , t ) as in (3.15) , whic h b ecomes a solutio n to (1.1). Theorem 4.2 Consi der any triplet ( A, B , C ) b elonging to the admissible class A des crib e d in Definition 4.1. Then: (i) The Lyapunov e quations (4.1) and (4.2) ar e uniquely solvable, and their s olutions ar e given by (4.3 ) and (4 . 4), r e s p e ctively. (ii) The c onstant matric es Q and N given in ( 4 . 3) and (4.4), r esp e cti vely, ar e selfadjoint; i.e. Q † = Q a nd N † = N . In fact, si nc e the tri plet ( A, B , C ) is r e al, the matric es Q and N ar e also r e al. F urthermor e, b oth Q a nd N ar e inverti ble. (iii) The r es ulting matrix F ( x, t ) forme d as in (3 . 13) i s r e al value d and i nvertible o n the entir e xt -plane, and the function u ( x, t ) define d in (3.15) is a solution to the sine- Gor don e quation everywher e on the xt -plane. Mor e over, u ( x, t ) is a na lytic on the entir e xt -plane and u x ( x, t ) de c ays to zer o exp onential ly as x → ±∞ at e ach fi xe d t ∈ R . PR OOF: The pro of of (i) foll o ws from Theorem 4. 1 of Section 4.1 of [20] . It is directly seen from (4.1 ) that Q † is also a solutio n whenever Q is a soluti on, and hence the uniqueness of the soluti on assures Q = Q † . Simila rly , a s a result of the realness of the triplet ( A, B , C ) , one can sho w that Q ∗ is also a solution to ( 4.1) and hence Q = Q ∗ . The selfadjointnes s and realness of N are establi shed t he same wa y . The i n vertibility o f Q and N is a result of the minimality of the tri pl et ( A, B , C ) and a pro of can b e found in the pro ofs of Theorems 3.2 and 3. 3 of [5] b y replacing (2.2) of [5] with ( 3.13) in the curren t pap er, completi ng the 15 pro of of (ii ). F rom (3 .13) and (3.14) it is seen that the realness of t he triplet ( A, B , C ) and of Q and N implies the realness of F . The pro of of the inv ertibili t y of F is simi lar to the pro of of Prop osition 4.1 (a) of [5] and t he rest of the pro of of ( iii) i s o btained as in Theorem 3.2 ( d) and (e) of [5]. W e will sa y that tw o triplets ( A , B , C ) and ( ˜ A, ˜ B , ˜ C ) are equiv al ent if they l ead t o the same u ( x, t ) given i n ( 3 .15). The next result sho ws that t w o a dmi ssible tri plets a re closely related t o each other and can al wa ys b e transformed in to each other. Theorem 4.3 F or any admissible triplet ( ˜ A, ˜ B , ˜ C ) , ther e c orr e s p onds an e quivalent ad- missible triplet ( A, B , C ) in such a way that al l ei genvalues o f A have p ositi ve r e al p arts . PR OOF: The pro of is simila r to the pro of of Theorem 3.2 of [5], where the triplet ( ˜ A, ˜ B , ˜ C ) is expressed explicitly when one starts with the t riplet ( A , B , C ) . Below we provide the explicit form ulas of constructing ( A, B , C ) b y sta rt ing wit h ( ˜ A, ˜ B , ˜ C ); i.e., b y pro viding the in v erse transformation form ulas for those give n in [5]. Without lo ss of an y generality , w e can assume that ( ˜ A, ˜ B , ˜ C ) has the form ˜ A = " ˜ A 1 0 0 ˜ A 2 # , ˜ B = " ˜ B 1 ˜ B 2 # , ˜ C = [ ˜ C 1 ˜ C 2 ] , where all eigenv alues o f ˜ A 1 ha ve p ositive real part s a nd all eigenv alues of ˜ A 2 ha ve negative real part s, and for some 0 ≤ q ≤ p, the sizes of the mat rices ˜ A 1 , ˜ A 2 , ˜ B 1 , ˜ B 2 , ˜ C 1 , ˜ C 2 are q × q , ( p − q ) × ( p − q ) , q × 1 , ( p − q ) × 1 , 1 × q, and 1 × ( p − q ) , resp ectiv ely . W e first construct the matrices ˜ Q and ˜ N b y solving the resp ective Ly apuno v equations ( ˜ Q ˜ A + ˜ A † ˜ Q = ˜ C † ˜ C , ˜ A ˜ N + ˜ N ˜ A † = ˜ B ˜ B † . W riting ˜ Q and ˜ N in blo ck matrix forms of appropriate sizes as ˜ Q = " ˜ Q 1 ˜ Q 2 ˜ Q 3 ˜ Q 4 # , ˜ N = " ˜ N 1 ˜ N 2 ˜ N 3 ˜ N 4 # , (4.5) and, for appropriate blo c k matrix sizes, b y letting A = " A 1 0 0 A 2 # , B = " B 1 B 2 # , C = [ C 1 C 2 ] , (4.6) 16 w e o btain A 1 = ˜ A 1 , A 2 = − ˜ A † 2 , B 1 = ˜ B 1 − ˜ N 2 ˜ N − 1 4 ˜ B 2 , B 2 = ˜ N − 1 4 ˜ B 2 , (4.7) C 1 = ˜ C 1 − ˜ C 2 ˜ Q − 1 4 ˜ Q 3 , C 2 = ˜ C 2 ˜ Q − 1 4 , (4.8) yielding ( A, B , C ) b y starti ng wit h ( ˜ A, ˜ B , ˜ C ) . When the triplet ( A, B , C ) is decomp osed a s i n (4.6), let us decomp ose the corre- sp onding soluti ons Q a nd N to the resp ective Lyapuno v equations (4.1) and (4.2), in an analogous manner to (4.5), as Q = " Q 1 Q 2 Q 3 Q 4 # , N = " N 1 N 2 N 3 N 4 # . (4.9) The relationship b etw een (4.5 ) and (4.9) is summarized in the foll o wing theorem. Theorem 4.4 Under the tr ansformation ( A, B , C ) 7→ ( ˜ A, ˜ B , ˜ C ) s p e cifie d in The o r em 4. 3, the quantities Q, N , F , E app e a r i ng in (4.1), (4.2) , (3.13), (3.18) , r esp e ctively, ar e tr ans- forme d as ( Q, N , F , E ) 7→ ( ˜ Q, ˜ N , ˜ F , ˜ E ) , wher e ˜ Q = " Q 1 − Q 2 Q − 1 4 Q 3 − Q 2 Q − 1 4 − Q − 1 4 Q 3 − Q − 1 4 # , N = " N 1 − N 2 N − 1 4 N 3 − N 2 N − 1 4 − N − 1 4 N 3 − N − 1 4 # , ˜ F = " I − Q 2 Q − 1 4 0 − Q − 1 4 # F " I 0 − N − 1 4 N 3 − N − 1 4 # , (4.10) ˜ E = " I − N 2 N − 1 4 0 − N − 1 4 # E " I 0 − Q − 1 4 Q 3 − Q − 1 4 # . (4.11) PR OOF: The pro of can b e obt a i ned i n a similar manner to the pro of of Theorem 3.2 of [5] by using ˜ A = " A 1 0 0 − A † 2 # , ˜ B = " I − N 2 N − 1 4 0 − N − 1 4 # B , ˜ C = C " I 0 − Q − 1 4 Q 3 − Q − 1 4 # , 17 corresp onding to the transformatio n sp ecified in ( 4.7) and ( 4.8). As the following theorem sho ws, for an admissible t riplet ( A, B , C ) , there is no loss of generalit y i n a ssuming that all ei g en v alues of A hav e p osit iv e real part s and B has a sp ecial form consisting of zeros and ones. Theorem 4.5 F or any admis sible triplet ( ˜ A, ˜ B , ˜ C ) , ther e c orr esp ond a s p e cial admissible triplet ( A, B , C ) , wher e A is in a Jor dan c anoni c al f orm with e ach Jor dan b lo ck c ontaini ng a disti nct eigenvalue havi ng a p ositi ve r e al p art, the entri es of B c onsist of zer os and ones, and C has c onstant r e al entri e s . Mor e sp e cific al ly, for s o me appr opriate p os itive inte ger m we have A =     A 1 0 · · · 0 0 A 2 · · · 0 . . . . . . . . . . . . 0 0 · · · A m     , B =     B 1 B 2 . . . B m     , C = [ C 1 C 2 · · · C m ] , (4.12) wher e in the c ase of a r e al (p osi tive) eig e nvalue ω j of A j the c orr esp onding blo cks ar e given by C j := [ c j n j · · · c j 2 c j 1 ] , (4.13) A j :=         ω j − 1 0 · · · 0 0 0 ω j − 1 · · · 0 0 0 0 ω j · · · 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 · · · ω j − 1 0 0 0 · · · 0 ω j         , B j :=     0 . . . 0 1     , (4.14) with A j having siz e n j × n j , B j size n j × 1 , C j size 1 × n j , and the c onstant c j n j is nonzer o. In the c ase of c omplex eigenvalues, which must app e ar in p airs as α j ± iβ j with α j > 0 , the c orr esp onding b lo ck s ar e given b y C j := [ γ j n j ǫ j n j . . . γ j 1 ǫ j 1 ] , (4.15) A j :=         Λ j − I 2 0 . . . 0 0 0 Λ j − I 2 . . . 0 0 0 0 Λ j . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . Λ j − I 2 0 0 0 . . . 0 Λ j         , B j :=     0 . . . 0 1     , (4.16) 18 wher e γ j s and ǫ j s for s = 1 , . . . , n j ar e r e al c onstants with ( γ 2 j n j + ǫ 2 j n j ) > 0 , I 2 denotes the 2 × 2 unit matrix, e ach c olumn ve c tor B j has 2 n j c omp onents, e ach A j has size 2 n j × 2 n j , and e ach 2 × 2 matrix Λ j is define d as Λ j := " α j β j − β j α j # . (4.17) PR OOF: The real triplet ( A, B , C ) can b e c hosen as describ ed in Section 3 of [7]. 5. EXA CT SOLUTIONS USING THE SYL VESTER EQUA TION In Section 3, starting from a minima l t riplet ( A, B , C ) wit h all eigen v a lues of A ha ving p ositive real part s, w e ha v e obtai ned the exact sol ution form ula ( 3 .20) to the sine-Gordon equation b y constructing the matrix E ( x, t ) in (3.18) with the help of the auxi liary matrix P in (3 . 19). In this section we sho w that t he explicit form ula (3. 20) for exact soluti ons to the sine-Gordon equati on remains v alid i f the ma trix tri plet ( A , B , C ) used to construct suc h solutions comes from a larger class, namely from the admissible cl a ss A sp ecified in Definition 4.1. Starting with any triplet ( A, B , C ) i n the admissible class A , w e obtain exact solutions to the sine-Gordon equation as follows: (a) Using A, B , C as input, construct the auxiliary matrix P b y solving the Sylv ester equation AP + P A = B C . (5.1) The unique solution to (5. 1) can b e obtai ned as P = 1 2 π Z γ dλ ( λI − iA ) − 1 B C ( λI + iA ) − 1 , (5.2) where γ is an y p o si t iv ely oriented simple closed contour enclosing all eigen v al ues of ( iA ) and lea ving out all eigen v al ues of ( − iA ) . If all eigen v alues of A ha ve p ositive real parts, then P can b e ev aluated as in ( 3.19). 19 (b) Using t he a uxiliary mat rix P and the t riplet ( A, B , C ) , form the ma t rix E ( x, t ) a s in (3.18) and then form the scalar u ( x, t ) as i n (3. 20). Theorem 5.1 Consi der any triplet ( A, B , C ) b elonging to the admissible class A des crib e d in Definition 4. 1. Then, the Sylvester e quation (5 . 1) is uniquely solvable, and its s o lution is given by (5.2). F urthermor e, th a t solution is r e al value d. PR OOF: T he uniq ue solv abil it y of ( 5.1) is already kno wn [20]. F or the b enefit of the reader w e o utline t he steps b elo w. F rom (5 .1) we get − ( λI − iA ) P + P ( λI + iA ) = iB C, or eq uiv alen tl y − P ( λI + iA ) − 1 + ( λI − iA ) − 1 P = i ( λI − iA ) − 1 B C ( λI + iA ) − 1 . (5.3) Dividing b oth sides of (5.3) b y (2 π ) and then i ntegrating al ong γ , and using 1 2 π i Z γ dλ ( λI − iA ) − 1 = I , 1 2 π i Z γ dλ ( λI + iA ) − 1 = 0 , w e obtai n (5.2) a s the unique sol ution to (5. 1 ). Since the admissible tripl et ( A, B , C ) is real, b y t aking complex conjugate of b ot h sides of (5. 1) we see that P ∗ also solv es (5. 1 ). F rom the uniqueness o f the solut i on to ( 5.1), it then foll o ws the P ∗ = P . Next we sho w that, for an y tripl et ( A, B , C ) in our admissible class A , there is a close relationship b etw een t he matrix P given in ( 5.2) and the matrices Q and N app eari ng in (4.3) and (4.4), resp ectiv ely . Theorem 5.2 L et the triplet ( A, B , C ) of size p b elong to the admiss ible class sp e c ifie d in Definition 4.1. Then the s olution P to the Sylvester e q uation (5.1 ) and the solutions Q and N to the r esp e cti ve Lyapunov e q uations (4.1 ) and (4.2) satisf y N Q = P 2 . (5.4) 20 PR OOF: Note t hat (5. 4) is v ali d when the matrix A in the t riplet i s diag o nal. T o see this, note t hat t he use of the triplet ( A, B , C ) wi th A = diag { a 1 , · · · , a p } , B =   b 1 . . . b p   , C = [ c 1 · · · c p ] , in ( 4.1), (4.2), and ( 5.1) y ields P j k = b j c k a j + a k , Q j k = c j c k a j + a k , N j k = b j b k a j + a k , where the subscript j k denotes the ( j, k ) en t ry of the relev an t ma trix. Hence, ( N Q ) j k = p X s =1 b j b s c s c k ( a j + a s )( a s + a k ) , ( P 2 ) j k = p X s =1 b j c s b s c k ( a j + a s )( a s + a k ) , establishing ( 5.4). Next, let us assume that A is not diag onal but diagonalizable through a real -v alued i n vertible matrix S so that ˜ A = S − 1 AS and ˜ A i s diag o nal. Then, under the transformation ( A, B , C ) 7→ ( ˜ A, ˜ B , ˜ C ) = ( S − 1 AS, S − 1 B , C S ) , w e g et ( Q, N , P ) 7→ ( ˜ Q, ˜ N , ˜ P ) = ( S † QS, S − 1 N ( S † ) − 1 , S − 1 P S ) , where ˜ Q, ˜ N , and ˜ P satisfy ( 4.1), (4.2), and (5.1), resp ectively , when ( A, B , C ) is replaced with ( ˜ A, ˜ B , ˜ C ) in those three equati ons. W e note that ( ˜ A, ˜ B , ˜ C ) is an a dmi ssible t riplet when ( A, B , C ) is admissible b ecause the eigen v al ues of A and ˜ A coincide. Since ˜ A is diagonal, we al ready ha v e ˜ N ˜ Q = ˜ P 2 , which easily reduces to N Q = P 2 give n in (5. 4). In case A is not diagonali zable, we pro ceed as follows. There exists a sequence o f admissible triplets ( A k , B , C ) con verging to ( A, B , C ) as k → + ∞ suc h that each A k is diagonalizable. Let the t r i plet ( Q k , N k , P k ) corresp ond t o the sol utions to (4.1), ( 4.2), a nd (5.1) , resp ec- tively , when ( A, B , C ) is replaced with ( A k , B , C ) i n those three eq uations. W e then hav e N k Q k = P 2 k , and hence ( Q k , N k , P k ) → ( Q , N , P ) yields (5.4) . Not e that w e ha v e used the stability of soluti o ns t o (4.1) , (4.2 ), and (5. 1). In fact, t hat st a bility directly fol lo ws from 21 the uniq ue solv abili t y of the matrix equations (4. 1), (4.2), (5.1) and the fact that t heir unique solv ability is preserved under a small p erturbation o f A. Theorem 5.3 L et the triplet ( A, B , C ) b elong to the admissible clas s sp e cifie d in Defini tion 4.1. The n, the s olution P to the Sylvester e quation ( 5 . 1) and the solutions Q and N to the r e s p e ctive Lyapunov e quations (4. 1) and (4 . 2) sati sfy N ( A † ) j Q = P A j P , j = 0 , ± 1 , ± 2 , . . . . (5.5) PR OOF: Under the transformation ( A, B , C ) 7→ ( ˜ A, ˜ B , ˜ C ) = ( A, A j B , C ) , w e g et ( Q, N , P ) 7→ ( ˜ Q, ˜ N , ˜ P ) = ( Q, A j N ( A † ) j , A j P ) , where ˜ Q, ˜ N , and ˜ P satisfy ( 4.1), (4.2), and (5.1), resp ectively , when ( A, B , C ) is replaced with ( ˜ A, ˜ B , ˜ C ) in those t hree equat i ons. Since ( ˜ A, ˜ B , ˜ C ) is also admissible, (5.4 ) i mpl i es that ˜ N ˜ Q = ˜ P 2 , whic h y ields ( 5 .5) after a minor simplificati on. Next, given an y admissible t r i plet ( A, B , C ) , w e prov e that t he corresp onding soluti on P to (5. 1) i s in v ertible and that the matri x E ( x, t ) given in (3 .18) is i n vertible and that (3.21) holds everywhere on the xt -plane. Theorem 5.4 L et the triplet ( A, B , C ) b elong to the admissible clas s sp e cifie d in Defini tion 4.1, a nd let the matric es Q , N , P b e the c orr esp onding solutions to (4. 1), (4.2) , and (5.2), r esp e ctively. Then: (i) The matrix P is inv ertible. (ii) The matric es F and E gi ven in (3.13) and (3.18 ) , r esp e ctively, ar e r e al valu e d and satisfy (3.21 ) . (iii) The matri x E ( x, t ) i s i nvertible on the entir e xt -p lane. 22 PR OOF: The in ve rtibility of P follows from (5.4) and the fact that b oth Q and N are in v ert ible, as stated in Theorem 4.2 ( i i); th us, (i) is established. T o prov e (ii) we pro ceed as fol lo ws. The real-v aluedness of F has already been established in Theorem 4.2 (iii). F rom (3. 18) i t is seen that the real-v aluedness of the tr i plet ( A, B , C ) and of P implies that E is real v alued. F rom (3.1 3), (3. 14), a nd (3.18) w e see that (3. 21) holds if and o nl y if we hav e N e − β † Q = P e − β P , (5.6) where w e hav e already used N † = N and Q † = Q, as established in Theorem 4.2 (ii). Since (5.5) implies N ( − β † ) j Q = P ( − β ) j P , j = 0 , 1 , 2 , . . . , w e see that (5.6) holds. Ha v ing established (3.21) , the inv ertibili t y of E ( x, t ) on the en ti re xt -plane follo ws from the i n vertibilit y of F ( x, t ) , which has been establi shed in Theorem 4. 2 (iii). Next, we show that t he ex plicit form ulas (3.15), ( 3.16), and (3. 20) are a l l equiv alen t to eac h other. Theorem 5.5 Consi der any triplet ( A, B , C ) b elonging to the admissible class A des crib e d in Definition 4.1. Then: (i) The explici t f ormulas (3. 15 ) , (3.16), and ( 3.20) yield e q ui v alent exact solutions to the sine-Gor don e quation (1 . 1) everywher e on the enti r e x t -plane. (ii) The e quivalent solution u ( x, t ) given in (3.15), (3.1 6 ) , and (3.20) is analytic on the entir e xt -plane, and u x ( x, t ) de c a ys to zer o exp onential ly as x → ±∞ at e ach fixe d t ∈ R . PR OOF: Because u ( x, t ) i s real and scal a r v alued, we al ready ha v e the equiv al ence of (3.15) and (3. 16). The equiv alence of (3.16) and ( 3.20) follows from ( 3.21). W e then hav e (ii) as a consequence o f Theorem 4.2 ( iii). 23 6. FUR TH ER EQUIV ALENT F ORMS F OR EXACT SOLUTIONS In Theorem 5.5 w e ha v e shown that t he exact sol utions given b y t he explicit formulas (3.15), ( 3 . 16), and (3.2 0) are equiv al en t. In t his section w e show that our ex act solutions can b e writ ten in v ario us ot her eq uiv alen t forms. W e first presen t t w o prop osi t ions that will b e useful in later sections. Prop osition 6.1 If ( A , B , C ) is admis sible, then the quantities F − 1 and E − 1 , ap p e aring in (3.13) and (3.18 ) , r esp e ctively, vanish exp one ntial ly as x → ±∞ . PR OOF: It is sufficient to give the pro of w hen t he ei gen v alues of A ha v e all p ositive real parts b ecause, as seen from (4. 1 0) and (4.11) , the same result also holds when some or all eigen v alues of A hav e negative real parts. When the eigenv alues of A ha v e p ositive real parts, from (3.13) w e g et F − 1 = e − β † / 2 [ I + e − β † / 2 Qe − β N e − β † / 2 ] − 1 e − β † / 2 , (6.1) where the in v ertibility of Q and N is guaran teed by Theorem 4.2 (i i). Hence, (6.1) implies that F − 1 → 0 ex p onen ti ally as x → + ∞ . F ro m (3.21) and the real ness of E and F we also get E − 1 → 0 ex p onen ti ally as x → + ∞ . T o obtain the asymptotics as x → −∞ , w e pro ceed as follows. F rom (3.13) w e obtain Q − 1 F N − 1 = e − β / 2 [ I + e β / 2 Q − 1 e β † N − 1 e β / 2 ] e − β / 2 , and hence F − 1 = N − 1 e β / 2 [ I + e β / 2 Q − 1 e β † N − 1 e β / 2 ] − 1 e β / 2 Q − 1 , and th us F − 1 → 0 exp onen tial ly as x → −∞ . F rom (3.21) and the realness of E and F w e a lso get E − 1 → 0 exp onen tia l ly as x → −∞ . Prop osition 6.2 The quantity E ( x, t ) de fine d i n (3.18) satisfies E x = 2 AE − 2 B C e − β P , E e − β P = P e − β E , e β P − 1 E = P + e β P − 1 e β . (6.2) 24 If ( A, B , C ) i s ad mi ssible and a l l eigenvalues of A hav e p osi tive r e al p arts, then E − 1 P e − β → P − 1 exp onential ly as x → −∞ . PR OOF: W e obtai n the first equality (6.2) b y taking the x -deriv at iv e of ( 3.18) and b y using (5. 1). The second equali t y can b e verified directl y b y using (3. 1 8) i n it. The third equality is obtai ned b y a direct premultiplication from ( 3 .18). The limi t as x → −∞ is seen from the l ast eq ual it y in (6.2) with the help of ( 3 .14). Let us start with a triplet ( A, B , C ) of size p b elonging to the admissible class specified in Definiti on 4.1. Letting M ( x, t ) := e − β / 2 P e − β / 2 , (6.3) where β as in (3.1 4 ) a nd P is the unique solution to the Sylvester equation (5.1) , we can write (3.18) also a s E ( x, t ) = e β / 2 Λ e β / 2 , ( 6.4) where w e hav e defined Λ( x, t ) := I + [ M ( x, t )] 2 . (6.5) Using (5.1) in ( 6.3), w e see that the x -deriv a t iv e of M ( x, t ) is given b y M x ( x, t ) = − e − β / 2 B C e − β / 2 . (6.6) Prop osition 6.3 The eig e nv alues o f the matrix M define d in (6.3) c annot o c cur on the imaginary axis in the c omplex plane. F urthermor e, the matric es ( I − iM ) and ( I + iM ) ar e i nvertible on the entir e xt -p lane. PR OOF: F rom (6.4) and (6.5) w e see that ( I − iM ) ( I + iM ) = e − β / 2 E e − β / 2 , and by Theorem 5.4 ( i ii) the matrix E i s i nv ertible on the entire xt -plane. Thu s, b oth ( I − iM ) and ( I + iM ) are inv ertible, and consequen tly M cannot ha v e eigen v alues ± i. F or 25 an y real, nonzero c, consider the transformatio n ( A , B , C ) 7→ ( A, cB , cC ) of an admissible triple ( A, B , C ) . The resulting triple i s a l so admissible, and as seen from ( 5.1) and (6.3 ) we ha ve ( P , M , I + M 2 ) 7→ ( c 2 P , c 2 M , I + c 4 M 2 ) . Thus, M cannot ha ve any purely imaginary eigen v alues. Since P is known to b e inv ertible by Theorem 5 . 4 (i), as seen from (6.3) the matrix M is inv ertible on the en tire xt -plane and hence cannot hav e zero as its eigenv alue. Theorem 6.4 The s olution to the sine- Go r don e quation gi ven in the e quivalent f o r ms (3.15), (3.16), and (3.20 ) c an also b e written as u ( x, t ) = − 4T r[ t an − 1 M ( x, t )] , (6.7) u ( x, t ) = 2 i log  det( I + iM ( x , t )) det( I − iM ( x , t ))  , (6.8) u ( x, t ) = 4 tan − 1  i det( I + iM ( x, t )) − det( I − iM ( x, t )) det( I + iM ( x, t )) + det( I − iM ( x, t ))  , (6.9) wher e M is the matrix define d in (6.3) and T r denotes the matrix tr ac e (the sum of diagonal entries). PR OOF: Let us n ote that t he equiv alence of (6. 8) and ( 6.9) follows from the second equalit y in (1.5) b y using z = det( I + iM ) / det( I − iM ) there. T o sho w the equiv alence of (6. 7) and (6.8), we use the ma trix i den tity tan − 1 M = 1 2 i log  ( I + iM )( I − iM ) − 1  , whic h is closely related to the first iden tit y in (1. 5 ), and the matri x iden tity T r[log z ] = log det z , with the i n vertible matri x z = ( I + iM )( I − iM ) − 1 . Th us, w e hav e established the equiv- alence of (6.7), (6.8), and (6.9). W e will complete the pro of b y sho wi ng that (3.20) is equiv alen t t o (6.7) . U si ng the fact that for a ny m × n mat rix α and an y n × m matrix γ w e hav e T r[ αγ ] = T r[ γ α ] , (6.10) 26 from (6.4)-( 6.6) w e get − 4 C E − 1 B = 4T r[ M x ( I + M 2 ) − 1 ] . (6.11) By Prop ositio n 6. 1 we k no w t hat E − 1 v anishes exp onen tially as x → + ∞ . Hence, with the help of (6.11) w e see that we can wri te ( 3.20) as u ( x, t ) = 4T r  Z ∞ x dr M r ( r , t )[ I + M ( r , t ) M ( r , t )] − 1  , whic h yields (6.7). Theorem 6.5 The s olution to the sine- Go r don e quation gi ven in the e quivalent f o r ms (3.15), (3.16), (3. 2 0 ) , (6. 7 ) -(6.9) c an also b e wri tten as u ( x, t ) = − 4 p X j =1 tan − 1 κ j ( x, t ) , (6.12) wher e the sc alar functions κ j ( x, t ) c orr esp ond to the ei g e nvalues of the matrix M ( x, t ) define d in (6.3) and the r e p e ate d eigenvalues ar e al lowe d in the summation. PR OOF: A t a fixed ( x, t )-v alue, using the matri x iden tity T r[ M ( x, t ) s ] = p X j =1 [ κ j ( x, t )] s , s = 1 , 2 , 3 , . . . , for la rge | z | v alues in the complex z -plane we obtain T r[( z I − M ) − 1 ] = ∞ X s =0 z − s − 1 T r[ M s ] = ∞ X s =0 p X j =1 z − s − 1 κ s j = p X j =1 ( z − κ j ) − 1 , (6.13) where we dropp ed the arg uments o f M and κ j for simplicity . Cho osing the contour γ as in (1.6) so that eac h eigenv alue κ j ( x, t ) is encircled ex actly once in the p ositive direction, w e can extend (6.13) to z ∈ γ b y an analyti c contin uation wi th resp ect to z . Using (6. 12) in ( 1.6), w e t hen o bt a in 1 2 π i Z γ dz [tan − 1 z ] T r[( z I − M ) − 1 ] = p X j =1 1 2 π i Z γ dz [tan − 1 z ] ( z − κ j ) − 1 , 27 or eq uiv alen tl y T r[tan − 1 M ( x, t )] = p X j =1 tan − 1 κ j ( x, t ) , whic h yields (6.12) in view of (6.7). Let us not e t hat the equiv alence of ( 6.7)-(6.9) , and (6.12) implies that one can replace M by i ts Jordan canonical form in an y of those four expressions without c hanging t he v alue o f u ( x , t ) . This follows from the fact that u ( x, t ) in (6.8) remains unc hanged if M is replaced by it s Jo rdan canonical f orm and is confirmed in (6.12) b y the fact that the eigen v alues remai n unc hanged under a simila ri t y transformation on a matrix . The next result shows that we can write our explicit solution given in (6 .12) ye t another equiv alen t form, which is expressed in terms of the co efficien ts in the c haracteristic p olynomial of the matrix M ( x, t ) given in (6.8). Let t hat c haracteristic pol ynomial b e giv en b y det ( z I − M ( x, t )) = p Y j =1 [ z − κ j ( x, t )] = p X j =0 ( − 1) j σ j ( x, t ) z p − j , where the co efficien ts σ j ( x, t ) can b e writ t en in terms of the eigenv alues κ j ( x, t ) as σ 0 = 1 , σ 1 = p X j =1 κ j , σ 2 = p X 1 ≤ j 0 and c 6 = 0 , through the use of (3.19) and (6.3), yields P = h c 2 a i , M = h c 2 a e − 2 ax − t/ (2 a ) i , and hence from (6.7) w e get u ( x, t ) = − 4 tan − 1  c 2 a e − 2 ax − t/ (2 a )  . (9.1) If c > 0 , the solution in (9.1) is known as a “ k ink” [25]; it mo v es to the left wi th sp eed 1 / (4 a 2 ) and u ( x, t ) → − 2 π as x → −∞ . If c < 0 , the sol ut i on i n (9.1) is known as an “an tikink” [25]; it mov es t o t he left wi th sp eed 1 / (4 a 2 ) and u ( x, t ) → 2 π as x → −∞ . Example 9.2 The triplet ( A, B , C ) with A =  a b − b a  , B =  0 1  , C = [ c 2 c 1 ] , where a > 0 , b 6 = 0 , and c 2 6 = 0 , through the use of (3.19) , (6. 3 ), and (6.9) yields u ( x, t ) = − 4 tan − 1  n um den  , (9 . 2) where n um := 8 a 2 e aζ + [( ac 1 − bc 2 ) cos( bζ − ) − ( bc 1 + ac 2 ) sin( bζ − )] , den := b 2 ( c 2 1 + c 2 2 ) + 16 a 2 ( a 2 + b 2 ) e 2 aζ + , ζ ± := 2 x ± t 2( a 2 + b 2 ) . The solution i n (9.2 ) corresp onds to a “breather” [25] and u ( x , t ) → 0 as x → −∞ . F or example, the choice a = 1 , b = 2 , c 1 = 2 , c 2 = 1 simplifies (9. 2) to u ( x, t ) = 4 tan − 1  2 e 2 x + t/ 10 sin(4 x − t/ 5) 1 + 4 e 4 x + t/ 5  . Example 9.3 The triplet ( A, B , C ) with A =  a 1 0 0 a 2  , B =  1 1  , C = [ c 1 c 2 ] , 37 where a 1 and a 2 are disti nct p o si t iv e constants, and c 1 and c 2 are real nonzero constan ts, b y pro ceeding the same wa y as in t he previous ex a mple, y ields ( 9 .2) wit h n um := 2 ( a 1 + a 2 ) 2  a 1 c 2 e 2 a 1 x + t/ (2 a 1 ) + a 2 c 1 e 2 a 2 x + t/ (2 a 2 )  , den := − ( a 1 − a 2 ) 2 c 1 c 2 + 4 a 1 a 2 ( a 1 + a 2 ) 2 e ( a 1 + a 2 )(2 x + t/ (2 a 1 a 2 )) . (9.3 ) If ( c 1 c 2 ) < 0 t hen the quan tit y in (9.3 ) never b ecomes zero; the corresp onding solution is kno wn as a “soliton-antisoliton” [25] in t eraction. On the ot her hand, i f ( c 1 c 2 ) > 0 t hen the quan tit y in ( 9.3) b ecomes zero on a curv e on the xt -plane and the corresp onding solution is kno wn as a “solit on-soliton” [ 25] in teraction. F or example, the c hoice a = 1 , b = 2 , c 1 = ± 1 , c 2 = ∓ 1 yi elds u ( x, t ) = ± 4 tan − 1  18 e 2 x + t/ 2 − 36 e 4 x + t/ 4 ) 1 + 7 2 e 6 x +3 t/ 4  , with u ( x, t ) → 0 as x → −∞ . On the other hand, the c hoice a = 1 , b = 2 , c 1 = ± 1 , c 2 = ± 1 yi elds the sol ution u ( x, t ) = ∓ 4 tan − 1  18 e 2 x + t/ 2 + 36 e 4 x + t/ 4 ) − 1 + 72 e 6 x +3 t/ 4  , with u ( x, t ) → ∓ 4 π as x → −∞ . Example 9.4 The triplet ( A, B , C ) with A =   a − 1 0 0 a − 1 0 0 a   , B =   0 0 1   , C = [ c 3 c 2 c 1 ] , where a > 0 , and c 1 , c 2 , c 3 are real constants with c 3 6 = 0 , by pro ceeding t he same w a y as in t he previous ex a mple, y ields u ( x , t ) in the form of (9.2) , where n um := c 3 3 e − 4 ax − t/a + 32 g , den : = 4 ae − 2 ax − t/ (2 a ) [128 a 8 e 4 ax + t/a + h 1 + h 2 ] , g := ( 8 a 4 c 1 + 8 a 3 c 2 + 8 a 2 c 3 ) − (4 a 2 c 2 + 8 ac 3 ) t + c 3 t 2 + (16 a 4 c 2 + 16 a 3 c 3 ) x − 8 a 2 c 3 xt + 16 a 4 c 3 x 2 , 38 h 1 := (8 a 4 c 2 2 − 8 a 4 c 1 c 3 + 16 a 3 c 2 c 3 + 14 a 2 c 2 3 ) − (4 a 2 c 2 c 3 + 4 ac 2 3 ) t, h 2 := c 2 3 t 2 + (16 a 4 c 2 c 3 + 32 a 3 c 2 3 ) x − 8 a 2 c 2 3 tx + 1 6 a 4 c 2 3 x 2 . The c hoi ce a = 1 , c 1 = − 1 , c 2 = − 1 , c 3 = − 2 yi elds u ( x, t ) = − 4 tan − 1  e − 4 x − t + 8(16 − 10 t + t 2 + 24 x − 8 tx + 16 x 2 ) 2 e − 2 x − t/ 2 [32 e 4 x + t + 20 − 6 t + t 2 − 8 tx + 40 x + 16 x 2 ]  , with u ( x , t ) → 2 π as x → −∞ . On the other hand, t he c hoice a = 1 , c 1 = 0 , c 2 = 0 , c 3 = 1 yields u ( x, t ) = − 4 tan − 1  e − 4 x − t + 32(8 − 8 t + t 2 + 16 x − 8 tx + 16 x 2 ) 4 e − 2 x − t/ 2 [128 e 4 x + t + 14 − 4 t + t 2 − 8 tx + 32 x + 16 x 2 ]  , with u ( x, t ) → − 2 π as x → −∞ . Ac kno wl edgments . One of the a ut hors (T.A.) is greatl y indebted to the Unive rsit y of Cagli a ri for it s hospit a lity during a recent visit. This material is based in part up on w ork supp orted by the T exas Norman Hack erman Adv anced Researc h Program under Gran t no. 003656-004 6-2007, the U niv ersit y o f Cagliari, the It alian Ministry of Educa- tion and Researc h (MIUR) under PRIN grant no. 2006017542 -003, INdAM, and the Au- tonomous Region of Sardinia under gran t L.R.7/2007 “Promozio ne della ricerca scien ti fica e dell’ inno v azione tecnologica in Sardegna.” REFERENCES [1] M. J. Abl owitz and P . A. Clark son, Solitons, nonline ar evolution e quations a nd inverse sc attering, Camb ridge Univ. Press, Camb ridge, 1991 . [2] M. J. Ablowitz, D. J. Kaup, A. C. N ew ell , and H . Segur, Metho d for s olving the sine-Gor don e quation, Phys. Rev. Lett. 30 (1973), 1 262–1264. [3] M. J. Abl owitz, D. J. Kaup, A . C. N ewell, and H. Segur, The invers e sc attering tr ansform-F ourier analysi s for nonline ar pr oblems, Stud. Appl. Math. 53 (1974), 249–315. 39 [4] M. J. Ablowitz and H. Segur, Soli tons and the inverse sc attering tr ansform , SIAM, Philadelphia, 1981. [5] T. Aktosun, T . Busse, F. Demon tis, and C. v an der Mee, Symmetries for exact solu- tions to the nonline ar Sc hr¨ odinger e quation, J. Ph ys. A 43 (2010), 025202 , 14 pp. [6] T. Akt osun, F. Demon tis, and C. v an der Mee, Exact solutions to the fo cusi ng non- line ar Schr¨ odinger e quati on , Inv erse Probl ems 23 (200 7), 217 1 –2195. [7] T. Aktosun and C. v an der Mee, Explici t s olutions to the Kortewe g - de V ries e q uation on the half-line, In v erse Problems 22 (2006), 2165–2 174. [8] H. B art, I. Goh b erg, and M. A. Kaasho ek, Minimal f a c torization of matrix and op er- ator functions, Birk h¨ auser, Basel, 1979. [9] E. D. Belokolos, Gener al formulae for solutions of ini tial and b oundary value p r oblems for sine-Gor do n e quati o n, Theoret. Math. Ph y s. 103 (1995) , 613–620. [10] E. Bour, Th´ eorie de la d´ ef ormation des surfac es , J. ´ Ecole Imp´ eriale Polytec h. 19 (1862), No. 39, 1–48. [11] P . B o wcock, E. Corrigan, and C. Zambon, Some asp e c ts of jump-def e cts in the quantum sine-Gor don mo del , J. High Energy Phys. 2005 (2005) , 023, 35 pp. [12] T. N. Busse, Gener ali z e d i nverse sc attering tr ansform for the nonline ar Schr¨ o d i nger e quation, Ph.D. thesis, Universit y of T exas at Arli ngton, 2008. [13] G. Costabile, R. D. P armen ti er, B. Sav o, D. W. McLaughlin, and A. C. Scott , Exact solutions of the si ne-Gor don e quati o n describi ng os cil lations in a long (but finite) Josephson junction, Appl. Phys. Lett. 32 ( 1978), 587–589. [14] F. Demontis, Dir e ct and inverse sc atteri ng of the matrix Zakhar ov-Shab at system, Ph.D. thesis, Universit y of Cagli ari, Italy , 2 0 07. [15] F. Demon tis and C. v an der Mee, Explicit solutions of the cubic matrix nonline ar 40 Schr¨ oding e r e quatio n, Inv erse Problems 24 (20 08), 025020, 16 pp. [16] L. P . Ei senhart, A tr e atise on the di ffe r ential ge ometry o f curves and s urf ac es, Dov er Publ., New Y ork, 1960. [17] F. C. F rank and J. H. v an der Merw e, One-dimensi onal di slo c ations. I. Static the ory, Pro c. Roy . Soc. London A 198 ( 1 949), 205–216 . [18] G. Gaeta, C. Reiss, M. Pe yrard, and T. Dauxois, Simple mo dels of non- line ar DNA dynamics, Riv. Nuov o Cimen to 17 (1994), 1–48. [19] R. N . Garifullin, L. A . Kalyakin, and M. A. Shamsutdino v, A uto-r esonanc e excitati on of a br e ather in we ak ferr omagnets, Comput. Math. Math. Ph y s. 47 (2007) , 1158– 1170. [20] I. Goh b erg, S. Goldberg, and M. A. Kaash o ek, Classes of line ar op er ators, V ol. I, Birkh¨ auser, Basel, 1990. [21] Chaohao Gu, H esheng Hu, and Zixia ng Zhou, Darb oux tr ansf ormations in inte gr able systems, Springer, Dordrec h t , 2005. [22] N. Jokela, E. Keski-V ak kuri, and J. M a jumder, Timelike b o undary s i ne-Gor don the ory and two-c omp onent plasma, Phys. Rev. D 77 (2008), 023 5 23, 6 pp. [23] A. Ko c hend¨ orfer and A. Seeger, The orie der V ersetzungen in ei ndimensionalen Atom - r eihen. I. P erio disc h ange or dnete V ersetzungen, Z. Phys. 127 (19 5 0), 533 – 550. [24] V. A. Kozel and V. R. Kotliarov, A lmost p erio di c solution of the e quation u tt − u xx + sin u = 0, Dok l. Ak ad. Nauk Uk rain. SSR Ser. A, 1976 (1976), 878–881. [25] G. L. Lamb, Jr., E lements of soliton the ory, Wiley , New Y ork, 1980. [26] E. Lennholm and M. H¨ ornquist, R evisi ti ng Salerno’s sine-Gor don mo del of DNA: active r e gions and r obustnes s, Phys. D 177 (2003) , 233–24 1. [27] K. M. Leung, D. W. Hone, D. L. Mill s, P . S. Riseb orough, and S. E. T rull inger, 41 Solitons in the line ar-c h a i n anti ferr omagnet, Phys. Rev. B 21 ( 1 980), 4 0 17–4026. [28] P . Mansfield, Solution of the initia l value pr oblem f or the sine- Gor don e quation using a Kac-Mo o dy algebr a, Commu n. Math. P h y s. 98 ( 1985), 525–537. [29] D. W. McLaughlin and A. C. Scott, Perturb ation analysis of fluxon dynamics, Ph ys. Rev. A 18 (1978), 1652–1 680. [30] H. J. Mikesk a, Solitons in a one-dimensiona l magnet with an e asy plane, J. Ph ys. C 11 (1977), L29–L32. [31] M. B. Mineev and V. V. Shmidt, R adiatio n fr om a vortex in a long Josephson junctio n plac e d in an alternating ele ctr o magne ti c field, Sov. Ph y s. JETP 52 (1980), 453–457. [32] S. P . No viko v , S. V. Manako v, L. B. Pit aevskii, and V. E. Zakharo v , The ory of solitons. The inverse sc attering metho d, P l en um P ress, New Y ork , 1984. [33] E. Olmedilla, Multiple p ole s olutions of the nonline ar Schr¨ odi nger e quation, Phys. D 25 (1987), 330–346. [34] C. P¨ opp e, Construction of solutions of the sin e - Gor don e quation by me ans of F r e d h o lm determinants, Ph ys. D 9 (198 3), 103–139. [35] N. R. Quin t ero and P . G. Kevreki dis, None quivalenc e of phonon mo des in the si ne - Gor don e quation, Ph ys. Rev. E 64 (20 01), 056 608, 4 pp. [36] M. Rice, A. R. Bishop, J. A . Krumhansl, and S. E. T rullinger, W e akly pinne d F r¨ olich char ge-densi ty-wave c ondensates : A new, nonline ar, curr ent- c arrying elementary ex- citation, Ph ys. Rev. Lett. 36 ( 1976), 432–435. [37] C. Rogers and W. K. Sch ief, B¨ acklund and Darb oux tr ansf ormations, Cam bridge Uni v . Press, Camb ridge, 2002. [38] M. Salerno, Discr ete mo del for DNA-pr omoter dynamics, Ph y s. Rev. A 44 (1991 ), 5292–5297. 42 [39] C. Sc hieb ol d, Solutions of the s i ne-Gor don e quations c oming in clusters, Rev. Mat. Complut. 15 (2002 ), 265–3 25. [40] A. Se eger, H. Donth, and A. Ko chen d¨ o rfer, The orie der V ersetzungen in eindimen- sionalen A tomr eihen. III. V ersetzungen, E igenb ewe gungen und i hr e We chselwirkung, Z. Ph ys. 134 (1953), 173–193. [41] A. Seeger and A. Ko c hend¨ orfer, The orie der V ersetzungen in ei nd i mensionalen A tom- r eihen. II. Belie b i g ange or dnete und b es chleunigte V ersetzungen, Z. Phys. 130 (195 1 ), 321–336. [42] R. Steuerw al d, ¨ Ub er ennep ersche Fl¨ achen und B¨ acklundsche T r ansformation, Abh. Ba y er. Ak ad. Wiss. (M ¨ unc hen) 40 (1936) , 1–105. [43] L. V. Y akushevic h, Nonline ar physi cs o f DNA, 2 nd ed., Wiley , Chi chester, 2004. [44] V. E. Zakharov and A. B . Shabat, Exact the ory of two-dimensi onal self-fo cusi ng and one-dimensiona l self-mo dulation of waves in nonline ar me dia, Sov. Phys. JETP 34 (1972), 62–69. [45] V. E. Zakharov, L. A. T akh t a dzh yan, and L. D. F addeev, Complete description of solutions of the “sine- Gor don” e q uation, Soviet Phys. Dokl. 19 (1 9 75), 82 4 –826. 43