Soliton and Periodic solutions of the Short Pulse Model Equation

Soliton and P erio dic Solutions of the Short Pulse Mo del Equati on Y oshimasa Matsuno ∗ Division of Applie d Mathematic al Scienc e, Gr aduate Scho ol of Scienc e and En g i n e ering, Y amaguchi University, Ub e, Y amaguchi 755-8611, Jap an ABSTRACT The short pulse (SP) eq uatio n is a no ve l mo del equation desc ribing the propagation of ultra -short optical pulses in nonlinear media. This a rticle re- views some recen t results ab out the SP equation. In particular, w e fo cus our attention on its exact solutions. By using a newly deve lop ed metho d of solution, w e deriv e m ultisoliton solutions as w ell as 1-and 2-phase p erio dic solutions and in v estigate their prop erties. 1 I NTR ODUCTION In this article, w e address t he follo wing short pulse (SP) mo del equation u xt = u + 1 6 ( u 3 ) xx , (1 . 1) where u = u ( x, t ) represen ts the magnitude of the electric field and subscripts x and t app ended to u denote partial differen tiatio n. The SP equation w as prop osed as a mo del nonlinear equation describing the propag ation of ultr a - short o ptical pulses in nonlinear media [1]. It is an alternat ive mo del equation to the cubic nonlinear Sc hr¨ odinger (NLS) equation. The ba sic assumption in deriving the NLS equation is a slo wly v arying amplitude appro ximation. Hence, as discussed in the contex t of self-fo cusing of ultra-short pulses in non- linear media [2, 3], its v alidit y w ould b e violated if the pulse width b ecomes v ery short. A recen t n umerical analysis rev eals that as t he pulse length short- ens, the SP equation b ecomes a b etter appro ximation to the solutio n of the Maxw ell equation when compared with t he prediction of the nonlinear NLS equation [4]. Although the ma t hematical structure of the NLS equation has b een studied extensiv ely , only a few results are known fo r t he SP equation. Here, w e describ e some recen t results asso ciated with the SP equation. In particular, we fo cus o ur atten tion on a n exact metho d of solution, soliton and p erio dic solutio ns a nd their prop erties. ∗ E-mail addr ess : ma tsuno@yamaguc hi-u.ac .jp 1 This a rticle is orga nized as follows : In Sec. 2, we deriv e the SP equation starting with Maxwe ll equations f or the electric and magnetic fields. In Sec. 3, an exact metho d of solution is dev elop ed fo r the SP equation whic h trans- forms the SP equation to the in tegrable sine-Gordon (sG) equation through a ho dograph transformation. In Sec. 4, the soliton solutions ar e constructed whic h include the multiloo p soliton and m ultibreather solutions. Subsequen tly , the in teraction pro cess of solitons is describ ed in detail. In Sec. 5, the exact metho d is applied t o obta ining 1- a nd 2-phase p erio dic solutions. Some prop- erties of the solutions are discussed as w ell a s their long- w av e limit. In Sec. 6, an alternativ e metho d of solution is introduced whic h enables us t o construct a more general class of p erio dic solutions. Then, the 1- and 2-phase solutions are exemplified. Section 7 is dev oted to conclusion. 2 SH OR T PULSE EQUA TION 2.1 Basic equations The electric and magnetic fields E and H as w ell as the the electric a nd magnetic flux densities D and B are gov erned by the follo wing set of equations div D = ρ, div B = 0 , r ot E = − ∂ B ∂ t , r ot H = j + ∂ D ∂ t , (2 . 1) where ρ and j are the electric charge and curren t densities, resp ectiv ely . W e consider the one-dimensional propag ation of the wa ve so tha t we can put E = E 3 ( x, t ) e 3 , H = H 2 ( x, t ) e 2 , (2 . 2) where e 2 and e 3 are unit ve ctors p erp endicular to the x axis. W e also a ssum e the follo wing relatio ns D = ǫ 0 E + P , B = µ 0 H , (2 . 3) where P is the induced electric@p olarization, ǫ 0 is the v acuum p ermittivit y and µ is the v acuum p ermeability . In view of (2.2), Eqs. (2.1) and the first equation of (2.3) are simplified to ∂ H 2 ∂ x = ∂ D 3 ∂ t , ∂ E 3 ∂ x = µ 0 ∂ H 2 ∂ t , (2 . 4) D 3 = ǫ 0 E 3 + P 3 , (2 . 5) resp ectiv ely . Com bining (2.4 ) and (2.5), w e obtain the equation for E 3 . It reads in form E xx − 1 c 2 E tt = P tt , (2 . 6) 2 where E = E 3 , P = µ 0 P 3 and c 2 = ( ǫ 0 µ 0 ) − 1 . The p olarization P can b e split in to the linear part P lin and the nonlinear part P nl and it may b e written in the form P = P lin + P nl = Z ∞ −∞ χ (1) ( t − τ ) E ( x, τ ) dτ + Z ∞ −∞ Z ∞ −∞ Z ∞ −∞ χ (3) ( t − τ 1 , t − τ 2 , t − τ 3 ) E ( x, τ 1 ) E ( x, τ 2 ) E ( x, τ 3 ) dτ 1 dτ 2 dτ 3 , (2 . 7) where χ (1) and χ (3) are the susce ptibilities. If w e consider the pr o pagation of ligh t with the w av elength b et w een 1600nm and 300 0nm, t hen the F ourier transform ˆ χ (1) of χ (1) is found to b e w ell appro ximated by the relation ˆ χ (1) ≃ ˆ χ (1) 0 − ˆ χ (1) 2 λ 2 [1]. It follows from this and the relat io n ω = 2 π c/ λ that the linear equation for (2.6) written in F o urier transformed form b ecomes ˆ E xx + 1 + ˆ χ (1) 0 c 2 ω 2 ˆ E − (2 π ) 2 ˆ χ (1) 2 ˆ E = 0 . (2 . 8) As for the nonlinear term in Eq. ( 2 .6), w e a ssume that the instantaneous con tribution is dominan t for the short a nd small amplitude pulses. Under this situation, w e can set χ (3) ( t − τ 1 , t − τ 2 , t − τ 3 ) = χ 3 δ ( t − τ 1 ) δ ( t − τ 2 ) δ ( t − τ 3 ) where χ 3 is a constan t. If w e in tro duce this relation into the nonlinear term on the rig ht-hand side of (2.7), we obtain P nl = χ 3 E 3 , which combined with (2.8), yields a single nonlinear w av e equation for E E xx − 1 c 2 1 E tt = 1 c 2 2 E + χ 3 ( E 3 ) tt , (2 . 9) where c 1 = c/ q 1 + ˆ χ (1) 0 and c 2 = 1 / (2 π q ˆ χ (1) 2 ). 2.2 Perturbation analysis Equation (2 .9) describ es the in teractions b et wee n the left and righ t mo ving pulses. Since the pulses are v ery short, the interaction b et w een them w ould giv e rise to a higher-order effect on the ev olution o f the w av es. Consequen tly , w e ma y a ddress o nly the righ t moving pulses, for instance. W e use the m ultiple scale metho d to deriv e the approximate equation b y expanding E as E ( x, t ) = ǫE 0 ( φ, X ) + ǫ 2 E 1 ( φ, X ) + · · · , (2 . 10) where ǫ is a small parameter whic h measures the shortness of the pulse relativ e to the time scale determined by the resonance, a nd φ and X are the scaled 3 v ariables defined by φ = t − x c 1 ǫ , X = ǫx. (2 . 11) If we intro duce (2.10) with (2.11) in to Eq. (2.9), w e obtain, at the order O ( ǫ ), the follo wing par t ial differen tial equation (PDE) for E 0 : − 2 c 1 ∂ 2 E 0 ∂ φ∂ X = 1 c 2 2 E 0 + χ 3 ∂ 2 E 3 0 ∂ φ 2 . (2 . 12) After an a ppro priate c hange of the v ariables, w e a r r iv e at the no r ma lized form of the SP equation (1.1). 2.3 Remarks 1. The SP equation has b een derive d for the fir st time in an attempt to con- struct in tegrable differen tial e quations associated with pseudospherical sur- faces [5]. Sc h¨ afer and W ay ne rederiv ed it star t ing from Maxw ell’s equ at ions of electric field in the fib er as describ ed in this section. See also a prior work due to Alterman and Ra uch who discuss the breakdo wn of the slowly v arying en v elop e approximation and p erform an asymptotic a na lysis fo r a new ty p e of nonlinear ev olution equation [6]. 2. The integrabilit y of the SP equation has b een established fro m v arious mathematical p oin ts o f view [5, 7-1 0 ]. 3. There exist sev era l analogous equations to the SP equation whic h ha ve b een pro v en to b e completely in tegrable. W e write one of them in the form u xt = αu + 1 2 (1 − β ) u 2 x − uu xx . (2 . 13) When β = 2, Eq. (2.1 3) b ecomes the short-w av e mo del for the Camassa- Holm equation while when β = 4, it reduces to the the short-wa ve mo del for the Degasp eris-Pro cesi equation and the V akhnenk o equation. The general m ultisoliton solutions of t hese eq uatio ns hav e been obtained in parametric forms [11]. 3 ME THOD OF E XA CT SOLUTION 3.1 Reduction to the sine-Gordon equation Here w e dev elop an ana lytical metho d f or solving the SP equation whic h emplo ys the ho dograph transformat ion to reduce it to the completely inte - grable sG equation [12]. W e first introduce the new dep enden t v ariable r r 2 = 1 + u 2 x , (3 . 1) 4 to transform the SP equation (1.1) in to the form of conserv ation law r t =  1 2 u 2 r  x . (3 . 2) W e then define the ho dograph transformation ( x, t ) → ( y , τ ) b y means of dy = r dx + 1 2 u 2 r dt, dτ = dt, (3 . 3 a ) or equiv alently ∂ ∂ x = r ∂ ∂ y , ∂ ∂ t = ∂ ∂ τ + 1 2 u 2 r ∂ ∂ y . (3 . 3 b ) In terms of the new v ariables y and τ , (3.1 ) and (3.2) are recast into r 2 = 1 + r 2 u 2 y , (3 . 4) r τ = r 2 uu y , (3 . 5) resp ectiv ely . F urthermore, w e define the v ar ia ble φ by u y = sin φ, φ = φ ( y , τ ) . (3 . 6) Inserting (3.6) in to (3.4) giv es 1 r = cos φ. (3 . 7) It follow s from (3.5)- (3.7) that u = φ τ . (3 . 8) Finally , if w e substitute (3.8) in to (3.6), we find that φ ob eys the following sG equation φ y τ = sin φ. (3 . 9) This form of the sG equation will b e used to construct soliton solutions of the SP equation. F o r the p erio dic solutions, it is appro priate to introduce the t wo indep enden t phase v aria bles ξ and η according to ξ = ay + τ a + ξ 0 , (3 . 10 a ) 5 η = ay − τ a + η 0 , (3 . 10 b ) where a ( 6 = 0) , ξ 0 and η 0 are a rbitrary constan ts. In terms of the new v aria bles, the sG equation (3.9) is transformed to φ ξ ξ − φ ηη = sin φ, φ = φ ( ξ , η ) . (3 . 11) 3.2 Pa rametric representation of the solution The solutio n u has a parametric represen tation giv en b y (3.8). T o b e more sp ecific u ( y , τ ) = φ τ . (3 . 12) T o obtain the par ametric represen tation of the co ordinate x , w e note from (3.3b) that the in v erse mapping ( y , τ ) → ( x, t ) is go ve rned b y the system of linear PDE for x = x ( y , τ ) x y = 1 r , x τ = − 1 2 u 2 . (3 . 13) Since the integrabilit y of the ab o ve system of equations is assured automat i- cally b y Eq. (3 .5 ), w e are able to in tegrat e ( 3.13) immediately to obtain x ( y , τ ) = Z cos φ dy + c, (3 . 14) where c is an in tegration constan t. 3.3 Criterion for the single-v alued solutions As will b e demonstrated later, most of the parametric solutions (3.12) and (3.14) b ecome multiv alued functions for b oth t he s olit o n and p erio dic solutions. The single-v alued f unctions are particularly useful in application to the real ph ysical problem suc h as the propagation of nonlinear short pulses in an optical fib er. A criterion f o r single-v alued functions may b e obtained simply by requiring that u x exhibits no singularities. It follows from (3.3 b), (3.6) and (3.7) that u x = tan φ . Th us, if − π 2 < φ < π 2 , (mo d π ) , ( − √ 2 + 1 < tan φ 4 < √ 2 − 1) . (3 . 15) then the parametric solutions (3.12) and (3.14) would b ecome single-v a lued functions for all v alues o f x and t . 3.4 Remark 6 The reduction of the SP equation to the sG equation has also b een estab- lished through the c hain of transformations [8]. 4 SOLITON SOLUTIONS 4.1 Pa rametric representation of the N -soliton solution In the context o f t he sG mo del, the soliton solutions are called kinks or breathers. These solutions are reduced from the solito n solutions by sp ecifying the parameters suc h as the amplitude and the phase. Here w e presen t the parametric represen t a tion for the N -solito n solution of the SP equation [12]. The general N -soliton solution of the sG equation can b e written in a compact form as [13] φ = 2i ln f ′ f , ( 4 . 1) with f = X µ =0 , 1 exp " N X j =1 µ j  ξ j + π 2 i  + X 1 ≤ j 0 and ξ 10 is a real constan t. If w e introduce a new v ariable X ≡ x + c 1 t − x 10 with c 1 = 1 /p 2 1 and x 10 = − ξ 10 /p 1 , then we can parameterize x ( y , t ) by a single v aria ble ξ 1 . T o b e more sp ecific, it reads X = ξ 1 p 1 − 2 p 1 tanh ξ 1 + d 1 . (4 . 11) It follow s from (4.10 a) and (4 .11) that du dX = sinh ξ 1 2 − cosh 2 ξ 1 . (4 . 12) 9 W e see fr o m (4.12) that du/dX c hanges sign three times and go es infinit y at ξ 1 = ± cosh − 1 2. Thu s, t he parametric solutoin exhibits singularities whic h has a fo rm of single lo op. The multi-v alued feature of the solution is also confirmed b y applying the criterion (4.8) . In fa ct, it follows from (4 .9) that Re f = 1 and Im f = e ξ 1 and hence (4.8 ) cannot b e satisfied for arbitr a ry v alues of y and t . Figure 1 sho ws a t ypical profile of the 1- lo op soliton solution with the para meters p 1 = 1 , d 1 = 0. The lo op soliton propagates to t he left (i.e., negativ e x direction) at a constant v elo cit y c 1 . If w e define the amplitude A 1 of the lo op soliton b y 2 / p 1 (maxim um v alue of u ), then c 1 = A 2 1 / 4. Th us, the large lo op soliton mo ve s more ra pidly than the small lo op soliton, indicating the t ypical solito nic b ehavior. 4.2.2 N -lo op soliton solution The g eneral N -lo op soliton solutio n of the SP equation arises from ( 4 .5) and (4.6) b y taking t he parameters p j ( j = 1 , 2 , ..., N ) p ositiv e a nd ξ j 0 ( j = 1 , 2 , ..., N ) real. W e first inv estigate the asymptotic b ehav ior of the solution for large time and show that it is r epres ented by a sup erp osition of N -lo o p solitons. The pro cedure for deriving the large t ime asymptptics can b e p er- formed straigh tforwardly by in v estigating the b eha vior of the τ -functions giv en b y (4.2). Hence, we omit the detail and describe o nly the result. T o this end, w e put c j = 1 /p 2 j and order the magnitude of the v elo cit y of each lo op solito n as c 1 > c 2 > ... > c N . W e observ e the in teraction of N lo op solitons in a mo ving frame with a constan t v elo cit y c n . W e tak e the limit t → −∞ with the phase v ariable ξ n b eing fixed. W e then find the following asymptotic form of u and x : u ∼ 2 p n sec h  ξ n + δ ( − ) n  , (4 . 13 a ) x ∼ y − 2 p n tanh  ξ n + δ ( − ) n  − 4 N X j = n +1 1 p j − 2 p n + c. (4 . 13 b ) where δ ( − ) n = N X j = n +1 ln  p n − p j p n + p j  2 . (4 . 13 c ) The corresp onding asymptotic forms for t → + ∞ are giv en b y u ∼ 2 p n sec h  ξ n + δ (+) n  , (4 . 14 a ) x ∼ y − 2 p n tanh  ξ n + δ (+) n  − 4 n − 1 X j =1 1 p j − 2 p n + c, (4 . 14 b ) 10 with δ (+) n = n − 1 X j =1 ln  p n − p j p n + p j  2 . (4 . 14 c ) Let x c b e the cen ter p osition of the n th lo op soliton in the ( x, t ) co ordinat e system. It then follow s from (4.13) and (4.14) that x c + c n t − x n 0 ∼ − δ ( − ) n p n − 4 N X j = n +1 1 p j + d n , ( t → −∞ ) (4 . 15 a ) x c + c n t − x n 0 ∼ − δ (+) n p n − 4 n − 1 X j =1 1 p j + d n , ( t → + ∞ ) , (4 . 15 b ) where x n 0 = − ξ n 0 /p n and d n = c − 2 / p n are phase constan ts. In view of the fact that all the lo op solitons pro pagate to the left, w e can define the phase shift of the n th lo op soliton as ∆ n = x c ( t → −∞ ) − x c ( t → + ∞ ) . (4 . 16) This quan tity is ev a luated using (4.13c), (4.14c) and (4.15 ) to giv e ∆ n = 1 p n ( n − 1 X j =1 ln  p n − p j p n + p j  2 − N X j = n +1 ln  p n − p j p n + p j  2 ) +4 n − 1 X j =1 1 p j − N X j = n +1 1 p j ! , ( n = 1 , 2 , ..., N ) . (4 . 17) Note that the first term on the righ t-hand side of (4 .1 7) coincides with the form ula for t he phase shift arising from the in teraction of N kinks of the sG equation. On the other hand, the second term arises due to the coo rdinate transformation (3.3). The latter changes the ch ar acteristics of the in teraction pro cess of lo op solitons substan tially when compared with those of the sG kinks. 4.2.3 2-lo op soliton solution The τ -functions f and f ′ for the 2-lo op solito n solution are written as f = 1 + ie ξ 1 + ie ξ 2 − γ e ξ 1 + ξ 2 , ( 4 . 18 a ) and f ′ = f ∗ with γ =  p 1 − p 2 p 1 + p 2  2 . (4 . 18 b ) 11 - 20 0 20 x - 5 - 2.5 0 2.5 5 t 0 5 u - 20 0 20 x Fig. 2 : The in teraction of t w o lo op solitons. 0 5 10 15 20 x 0 1 2 3 4 5 u Fig. 3 : The profile of the 2-lo op soliton solution. The parametric represen tatio n of the solution is then giv en by (4.5), (4.6) and (4.18). It reads u ( y , t ) = 2 √ γ p 1 p 2 ( p 1 + p 2 ) cosh ψ 1 cosh ψ 2 + ( p 1 − p 2 ) sinh ψ 1 sinh ψ 2 cosh 2 ψ 1 + γ sinh 2 ψ 2 , (4 . 19 a ) x ( y , t ) = y + 1 p 1 p 2 ( p 1 − p 2 ) sinh 2 ψ 1 − γ ( p 1 + p 2 ) sinh 2 ψ 2 cosh 2 ψ 1 + γ sinh 2 ψ 2 − 2( p 1 + p 2 ) p 1 p 2 + c, (4 . 19 b ) 12 where w e hav e put ψ 1 = 1 2 ( ξ 1 − ξ 2 ) , ψ 2 = 1 2 ( ξ 1 + ξ 2 ) + 1 2 ln γ , (4 . 19 c ) for simplicit y . The p ositiv e parameters p 1 and p 2 are a ssum ed to satisfy the condition p 2 > p 1 . Figure 2 shows the interaction of tw o lo o p solitons with the parameters giv en by p 1 = 0 . 5 , p 2 = 1 . 0 , c = ξ 10 = ξ 20 = 0. Figure 3 sho ws the profile of t he 2-lo op soliton solution at t = − 5 with the same para meters as those of Fig. 2. F o r the 2-lo o p solito n case, formulas (4.17) for the phase shift are written as ∆ 1 = − 1 p 1 ln  p 1 − p 2 p 1 + p 2  2 − 4 p 2 , (4 . 20 a ) ∆ 2 = 1 p 2 ln  p 1 − p 2 p 1 + p 2  2 + 4 p 1 . (4 . 20 b ) Figure 4 plots p 1 ∆ 1 and p 1 ∆ 2 as a function of s ( ≡ p 1 /p 2 ). Th us, the la rge lo op soliton alw a ys exhibits a p ositiv e phase shift whereas the small one exhibits a p ositiv e phase shift for 0 < s < s c and a negative pha se shift for s c < s < 1 where s c is a solution o f the transcenden tal equation ∆ 2 = 0 and is given b y s c = 0 . 834 . 0 0.2 0.4 0.6 0.8 1 s - 6 - 4 - 2 0 2 4 6 Phase Shift Fig. 4 : The phase shift p 1 ∆ 1 and p 1 ∆ 2 as a function of s . The solid (broken) line represen ts the phase shift o f the large (small) lo op soliton. 13 4.2.4 Lo op-an t iloop soliton solution The solution represen ting the interaction of a lo op soliton and an antiloop soliton arises if w e c ho ose p 1 > 0 and p 2 < 0 with p 1 < | p 2 | in t he 2-soliton τ -function (4.2). The parametric solution t a k es exactly the same fo rm as that of the 2-lo op solito n solution (4.19). - 20 0 20 x - 5 - 2.5 0 2.5 5 t - 5 0 5 u - 20 0 20 x Fig. 5 : The in teraction of a lo op soliton and an an tilo op soliton. 0 5 10 15 20 25 x - 2 - 1 0 1 2 3 4 u Fig. 6 : The profile of the lo op-antiloop soliton solution. 14 The formu las f o r the phase shift a re giv en b y ∆ 1 = 1 p 1 ln  p 1 − p 2 p 1 + p 2  2 + 4 p 2 , (4 . 21 a ) ∆ 2 = 1 p 2 ln  p 1 − p 2 p 1 + p 2  2 + 4 p 1 . (4 . 21 b ) Notice that the form ula for ∆ 1 is altered when compared with (4.2 0a). F ig ure 5 sho ws the in teraction of a loop soliton and an antiloo p soliton with the parameters giv en b y p 1 = 0 . 5 , p 2 = − 1 . 0 , c = ξ 10 = ξ 20 = 0 and Figure 6 shows the profile of the solution at t = − 5. 4.3 Breather solutions 4.3.1 1-breather solution The breather solution of the sG equation is the b ound state o f the kink and an tikink solutions. Under certain condition, the breather solution is sho wn to yield a nonsingular o scillating pulse solution of the SP equation, whic h we shall term the breather solution as we ll. The 1-breather solution of the SP equation is deriv ed if we put N = 2 in (4.2 ) and sp ecify the parameters as p 1 = a + i b, p 2 = a − i b, (4 . 22 a ) ξ 10 = λ + i µ, ξ 20 = λ − i µ, (4 . 22 b ) where a and b are p ositiv e constan ts a nd λ and µ are r eal constan ts. Then, (4.2) giv es f = 1 + ie ξ 1 + ie ξ ∗ 1 +  b a  2 e ξ 1 + ξ ∗ 1 , (4 . 23 a ) and f ′ = f ∗ where ξ 1 = θ + i χ with θ = a  y + 1 a 2 + b 2 t  + λ, (4 . 23 b ) χ = b  y − 1 a 2 + b 2 t  + µ. ( 4 . 23 c ) Substituting (4.23) in to (4.5) and ( 4.6), w e obtain the fo llo wing parametric represen tat io n of the solution u ( y , t ) = 4 ab a 2 + b 2 b sin χ cosh  θ + ln b a  − a cos χ sinh  θ + ln b a  b 2 cosh 2  θ + ln b a  + a 2 cos 2 χ , (4 . 24 a ) 15 x ( y , t ) = y − 2 ab a 2 + b 2 a sin 2 χ + b sinh 2  θ + ln b a  b 2 cosh 2  θ + ln b a  + a 2 cos 2 χ − 4 a a 2 + b 2 + c. (4 . 24 b ) Note that u has t wo differen t phase v ariables θ and χ . The phase θ c harac- terizes the en v elop e of the breather whereas the phase χ gov erns the in ternal oscillation. In general, the solution (4.24) w ould exhibit singularities. Unlike the lo op soliton solutions, the solution b ecomes a no nsingular function of x and t if w e imp ose a conditio n for the parameters a and b . T o see this, w e apply the criterion (4.8) to the τ - function (4.2) to obtain − √ 2 + 1 < a b cos χ cosh  θ + ln b a  < √ 2 − 1 . (4 . 25 a ) This inequalit y m ust b e satisfied for a ny v alue of θ and χ . Since a > 0 and b > 0, w e see from (4.25) that the condition imp osed on the parameters turns out to b e 0 < a/b < √ 2 − 1 . (4 . 25 b ) 0 20 40 60 80 100 120 x -2 -1 0 1 2 u Fig. 7 : The profile of the nonsingular 1- breather solution. Figure 7 shows a profile of t he 1-breather solution at t = 0 with the pa- rameters a = 0 . 1 , b = 0 . 5 , c = 80 , λ = µ = 0. In this example, a/b = 0 . 2 so that there app ear no singularities as exp ected fr om the criterion (4.25b). F or completeness , it will b e instructive to presen t a singular breather solu- tion. Figure 8 sho ws an example of the singular solution with t he para meters a = 0 . 4 , b = 0 . 5 , c = 20 , λ = µ = 0. Obviously , the criterion for the nonsingular solution is violated. 16 0 5 10 15 20 25 30 x -3 -2 -1 0 1 2 3 u Fig. 8 : The profile of the singular 1-breather solution. 4.3.2 M-breather solution The g eneral M -breather solution is constructed from the M -breather solu- tion of the sG equation (4.1) and (4.2) with N = 2 M . Sp ecifically , w e set p 2 j − 1 = p ∗ 2 j ≡ a j + i b j , a j > 0 , b j > 0 , ( j = 1 , 2 , ..., M ) , (4 . 26 a ) ξ 2 j − 1 , 0 = ξ ∗ 2 j, 0 ≡ λ j + i µ j , ( j = 1 , 2 , ..., M ) , (4 . 26 b ) and write the phase v ariables ξ 2 j − 1 and ξ 2 j as ξ 2 j − 1 = θ j + i χ j , ( j = 1 , 2 , ..., M ) , (4 . 27 a ) ξ 2 j = θ j − i χ j , ( j = 1 , 2 , ..., M ) , (4 . 27 b ) with θ j = a j ( y + c j t ) + λ j , ( j = 1 , 2 , ..., M ) , (4 . 27 c ) χ j = b j ( y − c j t ) + µ j , ( j = 1 , 2 , ..., M ) , (4 . 27 d ) c j = 1 a 2 j + b 2 j , ( j = 1 , 2 , ..., M ) . (4 . 27 e ) The parametric solution (4.5) and (4.6) with (4 .26) and ( 4 .27) describ es m ulti- ple collisions of M breathers provide d that certain condition is imp osed on the parameters a j and b j ( j = 1 , 2 , ..., M ). In the presen t M -breather case, the sim- ple inequalit y lik e (4.25b) is still difficult to obtain. Ho we ver, as sho wn b elo w, 17 the M -breather solution splits in to M single breathers a s t → ±∞ . Hence, one can exp ect that the condition corresp onding to (4.25b) w ould b ecome 0 < M X j =1 a j b j < √ 2 − 1 . (4 . 28) It will b e demonstrated later tha t the 2-breather solution exists whose param- eters indeed satisfy the inequalit y (4.28). Let us now inv estigate the structure of the M -breather solution b y fo cusing on the a symptotic b eha vior for large time. T o this end, w e order t he magnitude of the v elo cit y of eac h breather as c 1 > c 2 > ... > c M . W e tak e the limit t → −∞ with θ n b eing fixed. Then, w e see t ha t u and x hav e the leading- order asymptotics u ( y , t ) ∼ 4 a n b n a 2 n + b 2 n G n F n , (4 . 29 a ) x ( y , t ) ∼ y − 2 a n b n a 2 n + b 2 n H n F n − 4 a n a 2 n + b 2 n + d, (4 . 29 b ) with F n = b 2 n cosh 2  θ n + α ( − ) n + ln b n a n  + a 2 n cos 2  χ n + β ( − ) n  , (4 . 29 c ) G n = b n sin  χ n + β ( − ) n  cosh  θ n + α ( − ) n + ln b n a n  − a n cos  χ n + β ( − ) n  sinh  θ n + α ( − ) n + ln b n a n  , (4 . 29 d ) H n = a n sin 2  χ n + β ( − ) n  + b n sinh 2  θ n + α ( − ) n + ln b n a n  , (4 . 29 e ) where the real parameters α ( − ) n and β ( − ) n are defined b y the relation 2 M X j =2 n +1 ln  p 2 n − 1 − p j p 2 n − 1 + p j  2 = α ( − ) n + i β ( − ) n . (4 . 30 a ) The explicit expressions of α ( − ) n and β ( − ) n in terms of a j and b j are calculated using (4.26a). They read α ( − ) n = M X j = n +1 ln { ( a n − a j ) 2 + ( b n − b j ) 2 }{ ( a n − a j ) 2 + ( b n + b j ) 2 } { ( a n + a j ) 2 + ( b n + b j ) 2 }{ ( a n + a j ) 2 + ( b n − b j ) 2 } , (4 . 30 b ) 18 β ( − ) n = 2 M X j = n +1 tan − 1 b n − b j a n − a j + tan − 1 b n + b j a n − a j − tan − 1 b n + b j a n + a j − tan − 1 b n − b j a n + a j ! . (4 . 30 c ) As t → + ∞ , u and x tak e the same asymptotic forms as (4.2 9) with α ( − ) n and β ( − ) n replaced b y α (+) n and β (+) n , resp ectiv ely where 2 n − 2 X j =1 ln  p 2 n − 1 − p j p 2 n − 1 + p j  2 = α (+) n + i β (+) n . (4 . 31) The expressions of α (+) n and β (+) n corresp onding to (4.30b,c) f o llo w if one r e- places the sum P M j = n +1 b y P n − 1 j =1 in (4.30b,c). Observing the asymptotic b e- ha vior of the solution in the rest f r ame of reference, we see that it represen ts a sup erposition of M breathers, eac h has a form giv en by (4.24 a ). The effect of the interaction is the phase shift give n b y the sum o f the quantities α ( ± ) n and β ( ± ) n whic h is caused b y the pair wise collisions of M breathers and a term due to the co ordinate transformation. The form ula for the total phase shift will no t b e defined definitely since t he solution ta k es the form of w a ve pack ets. If w e consider the small-amplitude limit, how ev er, the oscillating part and the en v elop e ar e shown to b e separated completely so that one can obta in the formula lik e (4.17) fo r the N -lo op soliton solution. Actually , in the small-a mplitude limit a n → 0 ( n = 1 , 2 , ..., M ), expressions (4.29) and (4.30) are approxim at ed b y u ( y , t ) ∼ 4 a n b 2 n sin  χ n + β ( − ) n  cosh  θ n + α ( − ) n + ln b n a n  , (4 . 32 a ) x ( y , t ) ∼ y − 4 a n b 2 n tanh  θ n + α ( − ) n + ln b n a n  − 4 a n b 2 n + c, (4 . 32 b ) α ( − ) n ∼ − M X j = n +1 8( b 2 n + b 2 j ) a j a n ( b 2 n − b 2 j ) 2 , (4 . 33 a ) β ( − ) n ∼ M X j = n +1 8 a j b n b 2 n − b 2 j . (4 . 33 b ) 19 Let ¯ ∆ n b e the phase shift of the cente r p osition ( a p oin t corresp onding to the maxim um amplitude) of the env elop e for the n th breather. By p erforming the asymptotic analysis similar to that f o r the N -lo op soliton case, w e find that ¯ ∆ n = M X j = n +1 8( b 2 n + b 2 j ) a j ( b 2 n − b 2 j ) 2 − n − 1 X j =1 8( b 2 n + b 2 j ) a j ( b 2 n − b 2 j ) 2 , ( n = 1 , 2 , ..., M ) . (4 . 34) 4.3.3 2-breather solution The solution describing the inte ra ction o f t w o breat hers is parameterized b y (4.26) and (4.27) with M = 2. Since the par a metric solution has a length y expression, it is no t appropriate to write it do wn here. Instead, we sho w the time ev olution of the solution graphically and demonstrate its solitonic b eha vior. Fig. 9 depicts the pro file of the t w o- breather solution for three differen t times ( a ) t = − 40 , ( b ) t = − 5 , ( c ) t = 35. The v alues of the parameters are chos en as a 1 = 0 . 1 , b 1 = 0 . 5 , a 2 = 0 . 16 , b 2 = 0 . 8 , λ 1 = λ 2 = 0 , µ 1 = µ 2 = 0 . The v elo cit y of the larg e breather is 3 . 85 while that o f the s mall breather is 1 . 50 (see (4.27e)). Note in this example P 2 j =1 ( a j /b j ) = 0 . 4 so tha t the inequalit y (4.28) is satisfied. F or la rge negativ e time, the solution b ehav es lik e t w o indep enden t breathers, eac h has a form give n b y (4.24) and propagates to the left. As time go es, b oth breathers merge and then they separate eac h other with lea ving the original w a ve profiles. Appar ently , the presen t example exhibits a t ypical feature common t o the in teraction of tw o-solito n solutio ns. (a) 0 50 100 150 200 x - 2 - 1 0 1 2 u t =- 40 20 (b) - 100 - 50 0 50 100 x - 2 - 1 0 1 2 u t =- 5 (c) - 200 - 150 - 100 - 50 0 x - 2 - 1 0 1 2 u t = 35 Fig. 9 : The profile o f the 2-breather solution for (a) t = − 40, (b) t = − 5, and (c) t = 35. 4.4 Remark The 1- and 2- lo op soliton solutions as well as the 1-breather solution ha ve b een o bta ined by different metho ds [1 4, 15]. 5 P ERIODIC SOLUTIONS 5.1 1-phase solutions 21 The p erio dic solutions of the SP equation can b e constructed by using an exact metho d of solution described in Sec. 3 [16]. Here, w e deal with solutions whic h dep end on t he single v ariable η . Then, the sG equation (3.11) reduces to an ODE for φ φ ′′ = − sin φ, (5 . 1) where the prime app ended to φ denotes the differentiation with resp ect to η . There exist sev eral particular solutions of Eq. (5.1). Among them, we lo ok for solutions expressed b y Jacobi’s elliptic functions. Corresp ondingly , these solutions yield the 1- pha se solutio ns of the SP equation. W e in v estigate the prop erties of the solutions b y t he f ollo wing t w o examples. 5.1.1 Example 1 The first example of t he solution of Eq. (5.1) is giv en by Jacobi’s sn function. Explicitly φ = − 2 sin − 1 sn  η k , k  , (5 . 2) where the parameter k is the mo dulus of the elliptic function. Substituting (5.2) in to (3 .1 2) gives the para metric represen tation of u u = 2 k a dn  η k , k  , (5 . 3) where dn( u, k ) is Jacobi’s dn function. If we intro duce the relation cos φ = 1 − 2 sn 2  η k , k  whic h is deriv ed from (5.2) in to (3.1 4 ) , we obt a in x = y − 2 Z sn 2  η k , k  dy + c. (5 . 4) One can see that the inte gra tion constan t dep ends on t whose time ev olutio n is determined by the second equation of (3.13) with u give n b y (5.3) . Indeed, using the iden tit y k 2 sn 2  η k , k  + dn 2  η k , k  = 1 , (5 . 5) w e obtain an OD E for c , c ′ ( t ) = − 2 / ( k a ) 2 . This equation can b e in tegrated immediately to giv e c ( t ) = − 2 ( k a ) 2 t + d , (5 . 6) where d is an inte gr a tion constan t . Substituting (5.5) and (5.6) into (5.4 ) a nd rearranging terms, w e find a parametric represen ta tion of x x + 2 − k 2 ( ak ) 2 t − x 0 = 1 ak  − 1 k (2 − k 2 ) η + 2 E  η k , k   + d, (5 . 7) 22 where x 0 = − η 0 /a and E ( u, k ) is the elliptic inte gr a l of the second kind defined b y [17] E ( u, k ) = Z u 0 dn 2 v d v = Z τ 0 r 1 − k 2 t 2 1 − t 2 dt, ( t = sn v , τ = sn u ) . (5 . 8 ) Th us, (5.3) and (5.7) g iv e the parametric solution of the 1 -phase solution. This solution b ecomes a m ulti- v alued function. In fact, applying the criterion (3.15) to (5.2 ), we see that u x exhibits singularity when sn( η /k , k ) = ± 1 / √ 2. T o in v estigate the prop erties of the solution, it is con v enien t t o tak e the amplitude and the mo dulus as indep enden t pa rameters. The amplitude A of the w a ve ma y b e defined by the relation A = ( u max − u min ) / 2 where u max and u min are the maxim um and minim um v alues of u , resp ectiv ely . In the presen t example, it follow s fro m (5.3) that A = 1 − √ 1 − k 2 k a , ( 5 . 9) whic h enables us to expres s the par ameter a in terms of A a nd k . The v elo cit y V and t he wa vele ngth Λ of the wa v e are then giv en resp ectiv ely by V = 2 − k 2 ( ak ) 2 = (2 − k 2 ) A 2 (1 − √ 1 − k 2 ) 2 , (5 . 10) Λ = 2 A 1 − √ 1 − k 2  (2 − k 2 ) K ( k ) − 2 E ( k )  , (5 . 11) where K ( k ) and E ( k ) are the complete elliptic integrals of t he first a nd second kinds, resp ectiv ely . In deriving (5.11), we hav e used (5.7 ) and the p erio dicit y relation E ( u + 2 K ( k ) , k ) = E ( u, k ) + 2 E ( k ) . (5 . 12) The wa ve num b er K (whic h should not b e confused with K ( k )) and the angula r frequency W are defined respectiv ely b y K = 2 π / Λ and W = V K . The limiting forms of thes e w a ve para meters for k → 0 and k → 1 are give n resp ectiv ely by K ∼ 8 Ak 2 , V ∼ 8 A 2 k 4 , W ∼ 64 A 2 k 6 , ( k → 0) , (5 . 13 a ) K ∼ 2 π A 1 ln 8 1 − k , V ∼ A 2 , W ∼ 2 π A ln 8 1 − k , ( k → 1) . (5 . 13 b ) 23 It follow s from (5.13 ) that W ∼ A 2 K ( K → 0) , (5 . 14 a ) W ∼ A 5 K 3 / 8 ( K → ∞ ) . (5 . 14 b ) 0 10 20 30 40 K 0 1000 2000 3000 4000 5000 6000 W Fig. 10 : The disp ersion relation W = W ( K , A ) with A = 1 . 0 for Example 1 as a function of K . -3 -2 -1 0 1 2 3 x 0 0.5 1 1.5 2 2.5 3 u Fig. 11 : A t ypical profile of the p erio dic solution for Example 1. The disp ersion relation W = W ( K , A ) with A = 1 . 0 is plotted in Fig. 10 as a function o f K . Figure 11 illustrates the t ypical profile of the p erio dic 24 solution represen ted b y (5.3) and (5.7) where the parameters are chosen as A = 1 . 0 , k = 0 . 95 , x 0 = d = η 0 = 0 and the time t is set to zero. In this example, one can see that the p erio d Λ is 1.853. The figure represen ts a p erio dic lo o p trav eling to the left at a constan t ve lo city V = 2 . 320. When the w av elength of the perio dic w av e b ecomes ve ry long, it degen- erates in to a single lo op soliton, as w e shall now demonstrate. As seen from (5.13), t he long-w av e limit Λ → ∞ (or K → 0) is at t a ined when k tends to 1. Using the relations dn( u, 1) = sec h u and E ( u, 1) = tanh u , the parametric solution represen t ed by (5.3) and (5 .7) reduces respective ly to u = 2 A sec h η , (5 . 15 a ) x + A 2 t + x 0 = − Aη + 2 A tanh η + d, (5 . 15 b ) with η = y / A − At + η 0 . W e see from this expression that the limiting solution is essen tially the same as that of the 1- lo op soliton solution giv en b y (4.10a) and (4.11). See Fig. 1. 5.1.2 Example 2 The second example of the solution of Eq. (5.1) is giv en by Jacobi’s dn function φ = − 2 cos − 1 dn( η , k ) . (5 . 16) The parametric represen tation of the solution can b e written in the form u = 2 k a cn( η , k ) , (5 . 17 a ) x − 1 a 2 (1 − 2 k 2 ) t − x 0 = 1 a {− η + 2 E ( η , k ) } + d, (5 . 17 b ) where cn( η , k ) is Jacobi’s cn function. This solution is c haracterized by the w a ve par ameters A = 2 k a , (5 . 18) V = 1 a 2 (1 − 2 k 2 ) = A 2 4 k 2 (1 − 2 k 2 ) , (5 . 19) Λ = 2 A k | − K ( k ) + 2 E ( k ) | . (5 . 20) Figure 12 sho ws the disp ersion relatio n W = W ( K , A ) with A = 1 . 0. The disp ersion curve ha s tw o branches dep ending on the v alue of k . The upp er branch plotted by the solid line corresp onds to the disp ersion relation 25 for 0 ≤ k ≤ k c whereas the lo we r one (brok en line) represen ts the disp ersion relation for k c < k ≤ 1 where k c (= 0 . 9089) is a solution of the transcenden tal equation K ( k ) = 2 E ( k ) . Note that the w a v elength Λ becomes zero when k = k c . 0 5 10 15 20 25 30 K -4 -2 0 2 4 W Fig. 12 : The disp ersion relation W = W ( K , A ) with A = 1 . 0 for Example 2 as a function of K . The limiting forms o f the w av e parameters for b oth k → 0 a nd k → 1 are giv en resp ectiv ely b y K ∼ 2 k A , V ∼ A 2 4 k 2 , W ∼ A 2 k , ( k → 0) . (5 . 21 a ) K ∼ 2 π A 1 ln 8 1 − k , V ∼ − A 2 4 , W ∼ − π A 2 ln 8 1 − k , ( k → 1) . ( 5 . 21 b ) W e see from (5.2 1) that W ∼ 1 /K ( K → 0) for the upp er branc h and K ∼ − A 2 K/ 4 ( K → 0) for the low er branc h. As K → ∞ , b oth branc hes approac h a straigh t line W = − V c K with V c = (2 k 2 c − 1) A 2 / (4 k 2 c ) ≃ 0 . 19 74 A 2 . It is easy to see that the parametric solution (5.17 ) b ecomes a single-v alued function when k lies in the range 0 < k < 1 / √ 2. Note that the upp er limit of this inequality coincides with the v alue of the mo dulus k for whic h the v elo cit y giv en by (5.19 ) b ecomes zero. Figure 13 illustrat es the profile of the nonsingular p erio dic solution at t = 0 with the parameters A = 1 . 0 , k = 0 . 65 , x 0 = d = η 0 = 0. In this example, Λ = 3 . 027. It r epresen ts a p erio dic w av etrain trav eling to the righ t at a constan t v elo cit y V = 0 . 091 7. 26 -4 -2 0 2 4 x -1 -0.5 0 0.5 1 u Fig. 13 : A t ypical profile of the nonsingular p erio dic solution for Example 2. If the parameter k lies in the range 1 / √ 2 < k < 1, the solution exhibits singularities. Figure 14 illustrates the profile of the nonsingular p erio dic solu- tion at t = 0 with the parameters A = 1 . 0 , k = 0 . 8 , x 0 = d = η 0 = 0. In this example, Λ = 1 . 394. It represen ts a p erio dic w av etrain trav eling to the left at a constan t velocity V = 0 . 1094. -2 -1 0 1 2 x -1 -0.5 0 0.5 1 u Fig. 14 : A t ypical profile of the singular p erio dic solution for Example 2. In conclusion, it will b e w orthwhile to consider the small amplitude limit o f the solution. As suggested b y the asymptotic relations (5.21 a), the appropriate 27 limiting pro cedure is tak en b y the limit k → 0 while k eeping the v a lues of K , V and W finite. It turns out that the magnitude of the amplitude A is of order k . If w e substitute t he relations cn( η , 0) = cos η and E ( η , 0) = η in to (5.17), the limiting form of the solution can b e written as u = A cos  ax − t a − b  , (5 . 22) where b = a ( x 0 + d ) is a phase constan t. The disp ersion relatio n of this linear w a ve is give n b y W = 1 /K a s is consisten t with the asymptotics (5.21a). W e also remark that (5.22) satisfies the linearized SP equation u xt = u . 5.2 2-phase solutions 5.2.1 Separation of v ariables The general N -phase solution is now a v ailable for the sG equation. See [18], for instance. How ev er, it will b e difficult t o p erform the in tegral in (3.14) ev en f o r the 2 - phase solution. An alternat iv e approac h for constructing the general N -phase solutions will b e discussed in Sec. 6. Here, w e address the follo wing sp ecific form in tro duced b y La mb [19 , 20] φ = 4 tan − 1  f ( ξ ) g ( η )  . (5 . 23) W e substitute (5 .23) in to the sG equation ( 3 .11) and see that the v ariables ξ and η can b e separated if f and g satisfy the following nonlinear OD Es f ′ 2 = − κf 4 + µf 2 + ν, (5 . 24 a ) g ′ 2 = κg 4 + ( µ − 1) g 2 − ν , (5 . 24 b ) where κ, µ and ν are a r bitrary constan ts. F or sp ecial c hoice of these param- eters, one can obtain solutions for f and g whic h are expressed in terms of elliptic functions. If w e substitute (5.23) in to (3.12), we immediately obta in the parametric represen tat io n of u u = 4 a f ′ g + f g ′ f 2 + g 2 . (5 . 25) On the other hand, it follows from (5.23) b y an elemen ta ry calculation using form ulas of trigo no metric functions that cos φ = 1 − 8 f 2 g 2 ( f 2 + g 2 ) 2 . (5 . 26) 28 The righ t- hand side of (5.26) can b e mo dified in suc h a w ay that the inte gra l in ( 3 .14) can b e p erformed analytically . T o this end, w e in tro duce the function Y = Y ( ξ , η ) Y = c 1 ( f 2 ) ′ + c 2 ( g 2 ) ′ f 2 + g 2 , (5 . 27) where c 1 and c 2 are constan ts to b e determined later. Note that Y dep ends on the v ariables y and t through the relation (3 .1 0). Now, w e differen tiate Y b y y and use ( 5.24) to simplify the resultan t expression. After some calculatio ns, w e obtain Y y = a ( f 2 + g 2 ) 2 h − 2 κ ( c 1 f 6 + 3 c 1 f 4 g 2 − 3 c 2 f 2 g 4 − c 2 g 6 ) − 4 c 2 f 2 g 2 +2( c 1 + c 2 )  − 2 f g f ′ g ′ + 2 µf 2 g 2 − ν ( f 2 − g 2 )  i . (5 . 28) W e set c 1 + c 2 = 0 and c 1 = − 2 /a to reduce (5.28) in the form Y y = 4 κ ( f 2 + g 2 ) − 8 f 2 g 2 ( f 2 + g 2 ) 2 . (5 . 29) Comparing (5.26) and (5.29), w e find that cos φ = 1 + Y y − 4 κ ( f 2 + g 2 ) . (5 . 30) Finally , w e substitute (5.30 ) in to (3.14) and tak e account of (5.27) with c 1 = − c 2 = − 2 /a . Then, the in tegratio n with resp ect to y can b e p erformed trivially to giv e the expression of x in terms of f and g x = y − 4 a f f ′ − g g ′ f 2 + g 2 − 4 κ Z ( f 2 + g 2 ) dy + c. (5 . 31) The time dep endence of c can b e determined b y the second equation of ( 3 .13) with u and x giv en resp ective ly by (5.25) and (5.3 1 ). It turns out that c ′ ( t ) = 0 so that c = d (=const.). The expressions (5.25) and (5.31) pro vide the parametric represe ntation of the the t w o-phase p erio dic solutions of the SP equation. W e can obt a in sev era l 2 - phase p erio dic solutions dep ending on the c hoice of the functions f and g . Here, w e exemplify three solutions whic h r educe, in the long-wa ve limit, to breather solution (Example 1), 2- lo op soliton solution (Example 2) and lo op-antiloo p soliton solution (Example 3). 5.2.2 Example 1 29 The first example of f and g assumes the form f ( ξ ) = A cn( β ξ , k f ) , g ( η ) = 1 cn(Ω η , k g ) , (5 . 32) where A , β and Ω are p ositive parameters and k f and k g are mo duli of the elliptic function. If w e substitute (5.32) in to (5.2 4 ), w e can determine the parameters κ , µ and ν as w ell as k f , k g and Ω in terms of A and β . In particular, k 2 f = A 2 1 + A 2  1 + 1 β 2 (1 + A 2 )  , (5 . 3 3 a ) k 2 g = A 2 1 + A 2  1 − 1 Ω 2 (1 + A 2 )  , (5 . 33 b ) Ω 2 = β 2 + 1 − A 2 1 + A 2 , (5 . 33 c ) κ = β 2 k 2 f A 2 , µ = β 2 (2 k 2 f − 1) , ν = β 2 A 2 (1 − k 2 f ) . (5 . 33 d ) Note from (5.33) and the inequality 0 ≤ k f ≤ 1 that the parameter β m ust b e restricted b y t he condition A √ 1 + A 2 ≤ β , (5 . 34) with arbitrary p ositiv e A . No w, the par a metric represen tation of u follows from (3 .1 2), (5.23 ) and (5.32). It reads u = 4 A a − β sn( β ξ , k f )dn( β ξ , k f )cn(Ω η , k g ) + Ω cn( β ξ , k f )sn(Ω η , k g )dn(Ω η , k g ) A 2 cn 2 ( β ξ , k f )cn 2 (Ω η , k g ) + 1 . (5 . 35) The expression of x is dervied b y substituting (5.32) in to (5.31) as x = y + 4 β a cn( β ξ , k f )cn(Ω η , k g ) A 2 cn 2 ( β ξ , k f )cn 2 (Ω η , k g ) + 1 n A 2 sn( β ξ , k f )dn( β ξ , k f )cn(Ω η , k g ) − β k 2 f Ω k ′ g 2 cn( β ξ , k f )sn(Ω η , k g )dn(Ω η , k g ) o − 4 β a " E ( β ξ , k f ) − k ′ f 2 β ξ − β k 2 f A 2 Ω k ′ g 2 n E (Ω η , k g ) − k ′ g 2 Ω η o # + d. (5 . 36) 30 Here w e ha v e used the follo wing in tegral formulas for the Jacobi cn function in p erforming the in tegral in (5.5 1) Z cn 2 ( u, k ) du = 1 k 2 n E ( u, k ) − k ′ 2 u o , (5 . 37 a ) Z 1 cn 2 ( u, k ) du = 1 k ′ 2  sn( u, k )dn( u, k ) cn( u, k ) − E ( u, k ) + k ′ 2 u  , (5 . 37 b ) where k ′ = √ 1 − k 2 . Note that R ( f 2 + g 2 ) dy = a − 1 R f 2 ( ξ ) dξ + a − 1 R g 2 ( η ) dη b y virtue of (3.10 ). Let us now describ e some prop erties of the parametric solution giv en b y (5.35) and (5.36) . In general, u is a m ultiply p erio dic function of x for fixed t . Under certain condition, ho w ev er, it becomes a simply p erio dic function. T o see this, w e define the tw o parameters L ξ and L η b y L ξ ≡ 4 K ( k f ) /β and L η ≡ 4 K ( k g ) / Ω. In view of the p erio dicity of Ja cobi’s elliptic functions lik e sn( u + 4 K ( k ) , k ) = sn( u , k ), L ξ ( L η ) is the p erio d of u with resp ect to ξ ( η ). In accordance with the v a lue of A , there arise tw o p ossible cases for the p erio d of u . When 0 < A ≤ A c ( A c ≃ 2 . 1797), one can sho w with use of (5.33) that the inequalit y L η < L ξ holds for a r bitrary p ositiv e v alues of β , and at fixed A , b oth L ξ and L η are monotonically decreasing f unctions of β and v anish as β → ∞ . If the ratio of b oth p erio ds b ecomes a ratio nal n umber, i.e., L ξ /L η = m η /m ξ with ( m ξ , m η ) = 1 and m ξ < m η , then u has a perio d L with respect to y giv en by L = 1 a m ξ L ξ = 1 a m η L η . (5 . 38) With use of the relatio n (5.38 ), the p erio d Λ with resp ect to x is determined from (5.36). Indeed, using the p erio dicity of the elliptic in tegral o f t he second kind E ( u + 2 mK ( k ) , k ) = E ( u , k ) + 2 mE ( k ) ( m : in teger ) as w ell as (5.33) and (5.38), the spatial p erio d is found to b e as Λ = L  1 − 4 β 2  E ( k f ) K ( k f ) − k 2 f A 2 (1 − k 2 g ) E ( k g ) K ( k g ) + 1 β 2 (1 + A 2 )  . (5 . 39) When A c < A , on the other hand, the equation L ξ = L η has a unique solution for β and the corresp onding expression of the p erio d is also giv en by (5.39). W e remark that the solution presen ted here b ecomes a single-v alued func- tion when the parameter lies in the rang 0 < A < √ 2 − 1 . This fo llo ws fro m (4.8) and (5.32) with the aid of the inequalit y | cn( β ξ , k f )cn(Ω η , k g ) | ≤ 1. 31 Figure 15 depicts a profile of u at t = 0. The para meters chos en here are A = 0 . 2 , m ξ = 1 , m η = 2 , a = 1 . 0 , d = ξ 0 = η 0 = 0. Solving (5.38) f or β , one obtains β = 0 . 5832 so that Ω = 1 . 124 , k f = 0 . 3837 , k g = 0 . 0958 , L = 11 . 21. Substituting these v a lues into (5.3 9 ), the p erio d Λ is found to b e 10 . 37. -10 -5 0 5 10 15 x -0.75 -0.5 -0.25 0 0.25 0.5 0.75 u Fig. 15 : A t ypical profile of the p erio dic solution for Example 1. W e no w consider the limiting profile of the p erio dic solution when the p erio d Λ tends t o infinit y . T o b e more sp ecific, w e tak e the limits k f → 1 and k g → 0. It then turns out from (5.33) that β → A/ √ 1 + A 2 and Ω → 1 / √ 1 + A 2 . The limiting v alue of β corresponds to the low er limit of the inequalit y (5.34). Using the relations sn( u, 0) = sin u, cn( u, 0) = cos u, dn( u, 0) = 1 , (5 . 40 a ) sn( u, 1) = tanh u, cn( u, 1) = sec h u, dn( u, 1 ) = sec h u, (5 . 40 b ) E ( u, 1) = tanh u, E ( u , 0) = u, (5 . 40 c ) (5.35) and (5.36) reduces resp ectiv ely to u ∼ 4 A Ω a − A sinh β ξ cos Ω η + cosh β ξ sin Ω η cosh 2 β ξ + A 2 cos 2 Ω η . ( 5 . 41 a ) x ∼ y − 2Ω a sinh 2 β ξ + A sin 2Ω η cosh 2 β ξ + A 2 cos 2 Ω η + d. (5 . 41 b ) 32 This parametric solution is ess entially the same a s the breather solution al- ready giv en by (4.24 ) . See Fig. 7. 5.2.3 Example 2 The second example is giv en b y the following f and g f ( ξ ) = A sn( β ξ , k f ) cn( β ξ , k f ) , g ( η ) = 1 dn(Ω η , k g ) , (5 . 42) where k 2 f = 1 − A 2 + A 2 β 2 (1 − A 2 ) , (5 . 43 a ) k 2 g = 1 − 1 A 2 + 1 Ω 2 (1 − A 2 ) , (5 . 43 b ) Ω = β A, (5 . 4 3 c ) κ = − β 2 (1 − k 2 f ) A 2 , µ = β 2 (2 − k 2 f ) , ν = β 2 A 2 . (5 . 43 d ) The inequalities 0 ≤ k f ≤ 1 and 0 ≤ k g ≤ 1 imp ose the condition for β 1 √ 1 − A 2 ≤ β ≤ 1 1 − A 2 , (5 . 44) with A in the r a nge 0 < A < 1. The exressions of u and x follows from (5.25), (5.31) and (5.42) with use of the formulas Z sn 2 ( u, k ) cn 2 ( u, k ) du = 1 k ′ 2  dn( u, k )sn( u, k ) cn( u, k ) − Z dn 2 ( u, k ) du  , (5 . 45 a ) Z 1 dn 2 ( u, k ) du = 1 k ′ 2  − k 2 sn( u, k )cn( u, k ) dn( u, k ) + Z dn 2 ( u, k ) du  . (5 . 45 b ) The resulting parametric solution reads in the fo rm u = 4 A a β dn( β ξ , k f )dn(Ω η , k g ) + k 2 g Ω sn( β ξ , k f )cn( β ξ , k f )sn(Ω η , k g )cn(Ω η , k g ) A 2 sn 2 ( β ξ , k f )dn 2 (Ω η , k g ) + cn 2 ( β ξ , k f ) . (5 . 46) x = y − 4 β a 1 A 2 sn 2 ( β ξ , k f )dn 2 (Ω η , k g ) + cn 2 ( β ξ , k f ) × × h ( A 2 dn 2 (Ω η , k g ) − 1)sn ( β ξ , k f )cn( β ξ , k f )dn( β ξ , k f ) 33 + k 2 g A 3 sn 2 ( β ξ , k f )sn(Ω η , k g )cn(Ω η , k g )dn(Ω η , k g ) i + 4 β a ( − E ( β ξ , k f ) + AE (Ω η , k g )) + d. (5 . 47) Although the solution giv en ab o v e is a multiply p erio dic function, it has a single p erio d if the condition L = 1 2 a m ξ L ξ = 1 2 a m η L η , (5 . 4 8 ) is satisfied. Using the relation (1 − k 2 f ) /k ′ g 2 = A 4 whic h fo llo ws from (5.43), the spatial p erio d is fo und to b e as Λ = L  1 − 4 β 2  E ( k f ) K ( k f ) − A 2 E ( k g ) K ( k g )  . (5 . 49) Unlik e Example 1, the solution alwa ys exhibit singularities as confirmed easily from (4.8) and (5.42). Figure 16 plots a profile of u at t = 5. The parameters are ch osen as A = 0 . 2 , m ξ = 2 , m η = 1 , a = 1 . 0 , d = ξ 0 = η 0 = 0. In t his example, β = 1 . 02 7 , Ω = 0 . 2053 , k f = 0 . 9998 , k g = 0 . 8421 , L = 20 . 35 , Λ = 5 . 9 38. -4 -2 0 2 4 6 8 x 0 0.5 1 1.5 2 2.5 3 u Fig. 16 : A t ypical profile of the p erio dic solution for Example 2. In considering the limiting pro files, there arise tw o cases a ccording to the inequalit y (5.44). The upp er limit of the inequ ality for β is attained when 34 k g → 0. In this limit, one has the limiting for ms Ω ∼ A 1 − A 2 , k 2 f ∼ 1 − A 4 , f ∼ A sn( β ξ , k f ) cn( β ξ , k f ) , g ∼ 1 . (5 . 50) The expressions (5.46) and (5.47) then reduce resp ectiv ely to u ∼ 4 a β A dn( β ξ , k f ) A 2 sn 2 ( β ξ , k f ) + cn 2 ( β ξ , k f ) = 4 a β A (1 + A 2 ) dn( β ξ , k f ) dn 2 ( β ξ , k f ) + A 2 , (5 . 51) x ∼ y + 4 a (1 + A 2 )sn( β ξ , k f )cn( β ξ , k f )dn( β ξ , k f ) dn 2 ( β ξ , k f ) + A 2 + 4 β a ( − E ( β ξ , k f ) + A Ω η ) + d, (5 . 52) Using (3.10), (5.52) is mo dified in the form x + V t − x 0 = 1 a (1 + 4 β 2 A 2 ) ξ + 4 a (1 + A 2 )sn( β ξ , k f )cn( β ξ , k f )dn( β ξ , k f ) dn 2 ( β ξ , k f ) + A 2 − 4 β a E ( β ξ , k f ) + d, (5 . 53 a ) with V = 1 + 6 A 2 + A 4 [ a (1 − A 2 )] 2 , x 0 = 1 a  (1 + 4 β 2 A 2 ) ξ 0 − 4 β 2 A 2 η 0  . (5 . 53 b ) 0 0.5 1 1.5 2 2.5 3 3.5 x 0 0.5 1 1.5 2 2.5 3 u Fig. 17 : Pe rio dic lo op ar ising from the limit k g → 0 of the p erio dic solution depicted in Fig. 16. 35 Since the parametric solution (5.51) and (5 .5 3) dep ends only o n the single v ariable ξ , it b ecomes a 1-phase p erio dic function. Indeed, as sho wn in Fig. 17, it represen ts a p erio dic train of lo ops propag a ting to the left at a constan t v elo cit y V . See also Fig . 1 whic h illustrates a ty pical profile of a 1 - lo op soliton solution. T he maxim um and minimum v alues of u are ev alua t ed from (5.48). They read as u max = 2(1 + A 2 ) / [ a (1 − A 2 )] (at dn( β ξ , k f ) = A ) a nd u min = 4 A/ [ a (1 − A 2 )] (at dn( β ξ , k f ) = A 2 , 1). The spatia l p erio d Λ is giv en b y (5.49) with L = K ( k f ) / ( aβ ). In this example, u max = 2 . 167 , u min = 0 . 833 , β = 1 . 042 , Ω = 0 . 2 083 , k f = 0 . 999 2 , L = 4 . 442 , Λ = 1 . 0 10 , V = 1 . 347. The low er limit of β in (5.44) is realized when k f → 1 , k g → 1, whic h leads to the asymptotics Ω ∼ A √ 1 − A 2 , f ∼ A sinh β ξ , g ∼ cosh Ω η . (5 . 54) Then, the parametric solution b ecomes u ∼ 4 β A a cosh β ξ cosh Ω η + A sinh β ξ sinh Ω η A 2 sinh 2 β ξ + cosh 2 Ω η , (5 . 55 a ) x ∼ y − 2 β a A 2 sinh 2 β ξ − A sinh 2Ω η A 2 sinh 2 β ξ + cosh 2 Ω η + d. (5 . 55 b ) This expression coincides with the parametric fo r m of the 2- lo o p soliton solu- tion giv en by (4 .19). See Fig. 3. 5.2.4 Example 3 The third example of f and g tak es the form f ( ξ ) = A dn( β ξ , k f ) , g ( η ) = cn(Ω η , k g ) sn(Ω η , k g ) , (5 . 56) where k 2 f = 1 − 1 A 2 + 1 β 2 ( A 2 − 1 ) , ( 5 . 57 a ) k 2 g = 1 − A 2 + A 2 Ω 2 ( A 2 − 1 ) , (5 . 57 b ) Ω = β A , (5 . 57 c ) κ = β 2 A 2 , µ = β 2 (2 − k 2 f ) , ν = β 2 A 2 ( k 2 f − 1) . (5 . 57 d ) 36 The inequalities 0 ≤ k f ≤ 1 and 0 ≤ k g ≤ 1 require that the parameter β alw ay s m ust lie in the r a nge A √ A 2 − 1 ≤ β ≤ A 2 A 2 − 1 , (5 . 58) with A > 1 . Using (5.25), (5.31 ) and (5.56), the parametric represen tatio n of the solution can b e found to b e a s u = − 4 A a Ω dn( β ξ , k f )dn(Ω η , k g ) + β k 2 f sn( β ξ , k f )cn( β ξ , k f )sn(Ω η , k g )cn(Ω η , k g ) A 2 dn 2 ( β ξ , k f )sn 2 (Ω η , k g ) + cn 2 (Ω η , k g ) , (5 . 59 a ) x = y − 4 β a 1 A 2 dn 2 ( β ξ , k f )sn 2 (Ω η , k g ) + cn 2 (Ω η , k g ) × × h 1 A (1 − A 2 dn 2 ( β ξ , k f ))sn (Ω η , k g )cn (Ω η , k g )dn(Ω η , k g ) − k 2 f A 2 sn( β ξ , k f )cn( β ξ , k f )dn( β ξ , k f )sn 2 (Ω η , k g ) i − 4 β a  E ( β ξ , k f ) − 1 A E (Ω η , k g )  + d. (5 . 59 b ) As demonstrated readily by using (4.8) and (5.56), this solution alw ay s b e- comes a m ulti-v alued function. A spatial p erio d of the ab o v e solution can b e found if there exist in- tegers m ξ and m η satisfying the relation (5.4 8). Since in the presen t case K ( k f ) /β < K ( k g ) / Ω, one mus t imp ose t he condition m η < m ξ , ( m ξ , m η ) = 1. The expression of the spatial p erio d is no w giv en b y Λ = L  1 − 4 β 2  E ( k f ) K ( k f ) − 1 A 2 E ( k g ) K ( k g )  . (5 . 60) Figure 18 plots the profile of u at t = 5. The para meters are c hosen as A = 5 , m ξ = 2 , m η = 1 , a = 1 . 0 , d = 12 , ξ 0 = η 0 = 0 . In this example, β = 1 . 02 7 , Ω = 0 . 2053 , k f = 0 . 9998 , k g = 0 . 8421 , L = 20 . 35 , Λ = 5 . 9 38. Last, we consider tw o limiting cases. When k g → 0, β attains the upp er limit of (5.58) and other parameters b ehav e like Ω ∼ A A 2 − 1 , k f ∼ √ A 4 − 1 A 2 , f ∼ A dn( β ξ , k f ) , g ∼ cot Ω η . (5 . 61) 37 The solution (5.59) reduces to u ∼ − 4 A a Ω dn( β ξ , k f ) + β k 2 f sn( β ξ , k f )cn( β ξ , k f )sinΩ η cosΩ η A 2 dn 2 ( β ξ , k f )sin 2 Ω η + cos 2 Ω η , (5 . 62 a ) x ∼ y − 4 β a 1 A 2 dn 2 ( β ξ , k f )sin 2 Ω η + cos 2 Ω η h 1 A (1 − A 2 dn 2 ( β ξ , k f ))sin Ω η cos Ω η − k 2 f A 2 sn( β ξ , k f )cn( β ξ , k f )dn( β ξ , k f )sin 2 Ω η i − 4 β a  E ( β ξ , k f ) − 1 A Ω η  + d. (5 . 62 b ) In the case o f m ξ = 2 and m η = 1, the relation (5.48) determines A uniquely as A = 6 . 5 53 so that β = 1 . 024 , Ω = 0 . 1562 , k f = 0 . 9 9 97 , L = 20 . 11 , Λ = 5 . 669. The solution (5.62) exhibits a profile similar to that depicted in Fig. 18. 0 2 4 6 8 10 x -4 -3 -2 -1 0 1 u Fig. 18 : A t ypical profile of the p erio dic solution for Example 3. The lo w er limit of β in (5.58) is established when k f → 1 and k g → 1. Consequen tly , o ne ha s Ω ∼ 1 √ A 2 − 1 , f ∼ A sec h β ξ , g ∼ cosec h Ω η . (5 . 63 ) The solution then b ecomes u ∼ − 4 β a cosh β ξ cosh Ω η + A sinh β ξ sinh Ω η cosh 2 β ξ + A 2 sinh 2 Ω η , (5 . 64 a ) 38 x ∼ y − 2 β a sinh 2 β ξ − A sinh 2Ω η cosh 2 β ξ + A 2 sinh 2 Ω η + d. (5 . 64 b ) It represen ts the interaction b et w een a lo op soliton and an an tilo op soliton. See Fig. 3. 5.3 Remarks 1. An elemen tary metho d for obtaining 1-pha se solutions is a v aila ble whic h reduces the SP equation to a tractable ODE by a ssu ming solution o f trav eling t yp e [21]. 2. The solutions (5.32), (5 .42) and (5.56) hav e b een deriv ed in the con text of a finite-length sG system [22]. See also [23 , 24] for analog ous works . 6 A L TERNA TIV E METHOD OF SOLUTION 6.1 Bilinear transformation metho d The bilinear t r ansformation metho d enables us to construct particular solu- tions of nonlinear ev olutio n equations [25- 28]. Although t his metho d has b een emplo y ed to o bt a in soliton solutions of the sG equation (see Sec. 4), it is appli- cable to p erio dic solutions as w ell. Indeed, Nak am ura deve lop ed a systematic pro cedure f or constructing p erio dic solutions o f v arious t yp es of soliton equa- tions [2 9, 30]. Here, w e shall use his metho d to obta in p erio dic solutions of the sG equation. As already demonstrated in Sec. 4 for constructing soliton solu- tions, the τ - functions pla y an essen tial role in the bilinear formalism. In the p erio dic pro blem, w e in t ro duce t he same dependen t v ariable transforma t io n as (4.1) φ = 2i ln f ′ f . ( 6 . 1) Then, w e can transform the sG equation (3.9) to the the following system of bilinear equations for the τ -f unctions f and f ′ f f y t − f y f t − 1 4 ( f 2 − f ′ 2 ) = λf 2 , (6 . 2 a ) f ′ f ′ y t − f ′ y f ′ t − 1 4 ( f ′ 2 − f 2 ) = λf ′ 2 , (6 . 2 b ) where λ is a complex parameter to b e determined later and the v ariable τ has b een replaced b y the v ariable t b y virtue o f (3.3a ). The parametric represen- tation of u follo ws immediately from (3.8) and (6.1) u ( y , t ) = 2i  ln f ′ f  t . (6 . 3) 39 W e then use (6.1) and (6.2) to deriv e the relation cos φ = 1 + 4 λ − 2(ln f ′ f ) y t . (6 . 4) In tro ducing (6.4) into (3.14 ) and in tegrating with resp ect to y yield the para- metric represen tatio n of the co ordinate x x ( y , t ) = (1 + 4 λ ) y − 2(ln f ′ f ) t + c. (6 . 5) Comparing (6.5) with (4.6), one sees that a new parameter λ comes in the p erio dic solution whic h w ould disapp ear in t he long-w av e (or soliton) limit. Th us, if w e can solv e the bilinear equations (6.2), then we can o btain solutions of the SP equation throug h the parametric represen tation (6.3 ) and (6.5). It should b e remarke d that unlik e the soliton solutions, the constan t c in (6.5) dep ends on t . This constan t can b e determined b y using (4.7). 6.2 Metho d of solution In accordance with Nak am ura’s pro cedure, w e construct perio dic solu- tions of the bilinear equations (6.2) . T o t his end, w e first intro duce the N - dimensional theta function θ ( z | τ ) = ∞ X n 1 ,n 2 ,...,n N = −∞ exp 2 π i N X j =1 n j z j + π i N X j,k =1 n j τ j k n k ! , (6 . 6) where z = ( z 1 , z 2 , ..., z N ) is an N -dimensional v ector and τ = ( τ j k ) 1 ≤ j,k ≤ N is an N × N symmetric matrix. First, we seek solution of t he bilinear equation (6.2a) in terms of the theta functions as f = θ  z + d 4    τ  , (6 . 7 a ) f ′ = θ  z − d 4    τ  , (6 . 7 b ) where d = (1 , 1 , ..., 1 ) is an N - dimensional v ector whose en tries are all unit y and z j ( j = 1 , 2 , ..., N ) are phase v a r iables defined b y z j = k j y + ω j t + z j 0 , ( j = 1 , 2 , ..., N ) . (6 . 7 c ) Here, k j , ω j and z j 0 are complex parameters. Substituting (6.7) into (6.2a), w e find that the bilinear equation can b e tr a nsformed to the form ∞ X m 1 ,m 2 ,...,m N = −∞ F ( m 1 , m 2 , ..., m N )exp 2 π i N X j =1 m j z j ! = 0 , (6 . 8 a ) 40 where F ( m 1 , m 2 , ..., m N ) = ∞ X n 1 ,n 2 ,...,n N = −∞ " − 2 π 2 ( N X j =1 (2 n j − m j ) k j ) ( N X l =1 (2 n l − m l ) ω l ) − 1 4 n 1 + 4 λ − ( − 1) P N j =1 m j o # × × exp " π i ( N X j,k =1 n j τ j k n k + N X j,k =1 ( m j − n j ) τ j k ( m k − n k ) ) + π 2 i N X j =1 m j # . (6 . 8 b ) By shifting the s th summation index n s as n s + 1 in (6.8b), w e see that F ( m 1 , m 2 , ..., m N ) = − F ( m 1 , ..., m s − 1 , m s − 2 , m s +1 , ..., m N ) × × exp " 2 π i N X l =1 τ sl m l − τ ss !# . (6 . 9) Th us, if the relatio ns F ( m 1 , m 2 , ..., m N ) = 0 , (6 . 1 0 ) hold for all p ossible combinations of m 1 = 0 , 1 , m 2 = 0 , 1 , ..., m N = 0 , 1, then all F ′ s become zero for arbitrary integer v alues of m 1 , m 2 , ..., m N , implying that Eq. (6.8) holds iden t ically . Consequen tly , the τ -functions (6.7 ) satisfy the bilinear equation (6.2 a ). A similar analysis show s that the bilinear equation (6.2b) reduces to Eq. (6.8a) where t he function F has the same form as (6.8b) except that the factor π 2 i P N j =1 m j in the expo nential function is replaced simply by − π 2 i P N j =1 m j . It turns out tha t the relations (6.10) assure that the τ -functions (6.7) satisfy the bilinear equation (6.2b) as w ell. Th us, if w e can determine par a meters such that relations (6.10 ) are satisfied, then w e obtain p erio dic solutions of the sG equation. W e can regard (6.10) as a system of 2 N nonlinear equations for the unkno wn parameters ω j ( j = 1 , 2 , ..., N ) , τ j k (1 ≤ j < k ≤ N ) and λ with giv en v a lues of k j and τ j j ( j = 1 , 2 , ., N ). The total n umber of unkno wns is N ( N − 1) / 2 + N + 1 = N ( N + 1) / 2 + 1. F o r N = 1 , 2, t he tota l n um b er of equations is equal t o the total n um b er o f unkno wn para meters. Hence, w e ha v e 1- a nd 2- phase solutions in terms of the theta functions. F or N ≥ 3, on the other hand, the tot al num b er of equations alw ay s exceeds that of unkno wns. 41 In this case, Eqs. (6.10) b ecome an ov erdetermined system and w e need a separate consideration as for the existence of the solution. 6.3 1-phase solutions Here, we deriv e a 1-phase solution of the sG equation by means of the metho d described ab o v e. F or N = 1 , the relations (6.10) b ecome F ( m ) = ∞ X n = −∞  − 2 π 2 (2 n − m ) 2 k ω − 1 4 { 1 + 4 λ − ( − 1) m }  × × exp h π i { ( n − m ) 2 + n 2 } τ + π 2 i m i = 0 , ( m = 0 , 1 ) , (6 . 1 1 ) where w e ha v e put k = k 1 , ω = ω 1 , τ = τ 11 , m = m 1 , n = n 1 for simplicit y . Explicitly , these read ∞ X n = −∞ (8 π 2 n 2 k ω + λ )e 2 π i n 2 τ = 0 , (6 . 12 a ) ∞ X n = −∞ { 2 π 2 (2 n − 1) 2 k ω + 1 2 (1 + 2 λ ) } e π i { ( n − 1) 2 + n 2 } τ = 0 . (6 . 12 b ) W e can rewrite (6.12) in terms of the following 1- dimensional theta f unctions θ 1 ( z | τ ) = − i ∞ X n = −∞ ( − 1) n exp " π i(2 n + 1) z + π i  n + 1 2  2 τ # , (6 . 13 a ) θ 2 ( z | τ ) = ∞ X n = −∞ exp " π i(2 n + 1) z + π i  n + 1 2  2 τ # , (6 . 13 b ) θ 3 ( z | τ ) = ∞ X n = −∞ exp  2 π i nz + π i n 2 τ  , (6 . 13 c ) θ 4 ( z | τ ) = ∞ X n = −∞ ( − 1) n exp  2 π i nz + π i n 2 τ  . (6 . 13 d ) Before pro ceeding, it is con v enien t to in tro duce a new parameter q by q = e π i τ , (6 . 14) 42 and write the ab o ve four theta functions a s θ j ( z | τ ) = θ j ( z , q ) , ( j = 1 , 2 , 3 , 4) . (6 . 15) Th us, the relation θ j ( z | nτ ) = θ j ( z , q n ) holds for an y in teger n . This notation will b e used in the follow ing. No w, using (6.1 5 ), Eqs. (6.12) can b e recast into the following system of linear algebraic equation for the tw o unknow ns k ω and λ 2 θ ′′ 3 (0 , q 2 ) k ω − θ 3 (0 , q 2 ) λ = 0 , (6 . 16 a ) 2 θ ′′ 2 (0 , q 2 ) k ω − 1 2 θ 2 (0 , q 2 )(1 + 2 λ ) = 0 , (6 . 16 b ) where θ ′′ j (0 , q 2 ) = d 2 θ j ( z , q 2 ) /dz 2 | z =0 , ( j = 2 , 3). So lving this system, w e obtain k ω = − 1 4 θ 2 (0 , q 2 ) θ 3 (0 , q 2 ) θ 2 (0 , q 2 ) θ ′′ 3 (0 , q 2 ) − θ ′′ 2 (0 , q 2 ) θ 3 (0 , q 2 ) , (6 . 17 a ) λ = − 1 2 θ 2 (0 , q 2 ) θ ′′ 3 (0 , q 2 ) θ 2 (0 , q 2 ) θ ′′ 3 (0 , q 2 ) − θ ′′ 2 (0 , q 2 ) θ 3 (0 , q 2 ) . (6 . 17 b ) If w e use the iden tity θ ′′ 3 (0 , q 2 ) θ 3 (0 , q 2 ) − θ ′′ 2 (0 , q 2 ) θ 2 (0 , q 2 ) = π 2 θ 4 4 (0 , q 2 ) , (6 . 18) w e can recast (6.1 7 ) to the compact expressions k ω = − 1 4 π 2 θ 4 4 (0 , q 2 ) , (6 . 19 a ) λ = − θ ′′ 3 (0 , q 2 ) 2 π 2 θ 3 (0 , q 2 ) θ 4 4 (0 , q 2 ) . (6 . 19 b ) No w, t he τ -functions f and f ′ can b e expressed in t he f orm f = θ  z + 1 4    τ  = ∞ X n = −∞ exp  2 π i n  z + 1 4  + π i n 2 τ  , (6 . 20 a ) f ′ = θ  z − 1 4    τ  = ∞ X n = −∞ exp  2 π i n  z − 1 4  + π i n 2 τ  , (6 . 2 0 b ) 43 with z = k y + ω t + z 0 . (6 . 20 c ) In order to obta in a real p erio dic solution, we in tro duce the new real quan tities with tilde b y k = − i 2 π ˜ k , ω = − i 2 π ˜ ω , z 0 = − i 2 π ˜ z 0 , ( 6 . 21 a ) z = − i 2 π ˜ z = − i 2 π ( ˜ k y + ˜ ω t + ˜ z 0 ) , (6 . 21 b ) and put τ = i b, ( b > 0) , (6 . 21 c ) to assure the conv ergence of the series (6.20). Then, f and f ′ are rewritten as f = ∞ X n = −∞ exp h n  ˜ z + π 2 i  − π n 2 b i , (6 . 22 a ) f ′ = ∞ X n = −∞ exp h n  ˜ z − π 2 i  − π n 2 b i , (6 . 22 b ) with ˜ z = ˜ k y + ˜ ω t + ˜ z 0 . (6 . 22 c ) In terms of the new parameters ˜ ω and ˜ k , the disp ersion relation (6.19a) b e- comes ˜ ω = 1 θ 4 4 (0 , q 2 ) ˜ k , ( q = e − π b ) . (6 . 23) Ob viously , f ′ = f ∗ and λ is real. Hence, the parametric solution giv en by (6.3) a nd (6.5) yields a real 1-phase p erio dic solution of the SP equation. F or computing u from (6.3 ), we rewrite f in terms of the theta functions θ 1 and θ 4 as f = θ 4  ˜ z π i , q 4  + i  i θ 1  ˜ z π i , q 4  . (6 . 24 a ) Note that θ 4 ( ˜ z /π i , q 4 ) and i θ 1 ( ˜ z /π i , q 4 ) are real functions of ˜ z . The τ -f unction f ′ is giv en b y the complex conjugate of f . It reads f ′ = θ 4  ˜ z π i , q 4  − i  i θ 1  ˜ z π i , q 4  . (6 . 24 b ) 44 It follow s from (6.24a ) and the definition of the sn function in terms of the theta functions that Im f Re f = i θ 1  ˜ z π i , q 4  θ 4  ˜ z π i , q 4  = i θ 2 (0 , q 4 ) θ 3 (0 , q 4 ) sn( v , κ ) , (6 . 25 a ) where v = − i θ 2 3 (0 , q 4 ) ˜ z , κ = θ 2 2 (0 , q 4 ) θ 2 3 (0 , q 4 ) . (6 . 25 b ) F urthermore, using the formula sn(i v , κ ) = i sn( v , κ ′ ) cn( v , κ ′ ) , κ ′ = √ 1 − κ 2 = θ 2 4 (0 , q 4 ) θ 2 3 (0 , q 4 ) , (6 . 26) (6.25) b ecomes Im f Re f = θ 2 (0 , q 4 ) θ 3 (0 , q 4 ) sn( θ 2 3 (0 , q 4 ) ˜ z , κ ′ ) cn( θ 2 3 (0 , q 4 ) ˜ z , κ ′ ) . (6 . 27) The relation f ′ = f ∗ mak es it p ossible to write (6.3) as u = 4 [tan − 1 (Im f / Re f )] t . Substitution o f ( 6 .27) in to this expression yields u giv en b y (5.51) with t he iden tification a mong the parameters A = θ 2 (0 , q 4 ) θ 3 (0 , q 4 ) , a = θ 2 4 (0 , q 2 ) ˜ k , β = θ 2 3 (0 , q 4 ) θ 2 4 (0 , q 2 ) , κ ′ = √ 1 − A 4 . (6 . 28) T o deriv e the expression of x from (6.5), on the other hand, one needs the time dep endence of c whic h can b e determined fr om ( 6 .24) and (4.7). After some calculations using the iden tities of the theta functions, w e find c ( t ) = − 4 ˜ ω 2 π 2  θ ′′ 4 (0 , q 4 ) θ 4 (0 , q 4 ) + π 2 θ 2 2 (0 , q 4 ) θ 2 3 (0 , q 4 )  t + d , (6 . 29) where d is an a r bitr ary real constant. It can b e demonstrated by substituting (6.24) a nd (6.29) in to (6.5) that the express ion (6.5) coincides with (5.52 ). In- deed, a straightforw a r d calculation using (6.5), (6.20) , (6.28) and some form u- las fo r the theta f unctions and Jacobi elliptic functions leads to the express ion of x . W e write it in the form x = y + 4 a (1 + A 2 ) sn( β ξ , k f )cn( β ξ , k f )dn( β ξ , k f ) dn 2 ( β ξ , k f ) + A 2 + 4 β a  − E ( β ξ , k f ) + θ ′′ 4 (0 , q 4 ) π 2 θ 2 3 (0 , q 4 ) θ 4 (0 , q 4 ) ˜ z  + 4 λy + c ( t ) . (6 . 30) 45 Note f r om (3.10a ), (6.22 c), (6.23) and ( 6 .28) tha t ˜ z = β ξ /θ 3 (0 , q 4 ) and f rom (3.10) that the last tw o terms on the right-hand side of (6.30) can b e expressed in terms of ξ and η . With these facts in mind, we t hen use the formulas θ ′′ 3 (0 , q 2 ) θ 2 3 (0 , q 2 ) = 2 θ 4 (0 , q 4 ) θ ′′ 4 (0 , q 4 ) θ 3 (0 , q 2 ) θ 4 (0 , q 2 ) − π 2 2 θ 4 2 (0 , q 2 ) , (6 . 31 a ) θ 2 (0 , q 4 ) θ 3 (0 , q 4 ) = 1 2 θ 2 2 (0 , q 2 ) , (6 . 31 b ) and see that (6.30) coincides p erfectly with t he expression of x giv en b y (5 .52). Last, w e consider t he soliton limit. T o t his end, w e first shift the phase constan t ˜ z 0 as ˜ z 0 → ˜ z 0 + π b and tak e the limit b → ∞ (or q → 0). In this limit, the theta functions θ 3 and θ 4 ha v e the p o w er series expansions θ 3 (0 , q ) = 1 + 2 q + O ( q 4 ) , (6 . 32 a ) θ 4 (0 , q ) = 1 − 2 q + O ( q 4 ) . (6 . 32 b ) Then, the asymptotic fo rm of λ from (6.17b) a nd that of ˜ ω from (6 .2 3) b ecome λ ∼ 0 , ˜ ω ∼ 1 ˜ k . (6 . 33) The τ -function fr o m (6.22a ) b ehav es lik e f ∼ 1 + ie ˜ z , ˜ z = ˜ k y + 1 ˜ k t + ˜ z 0 . (6 . 34) This coincides with the τ -function (4 .9 ) for the 1-lo op solito n solution. Another type of real 1-phase solutions can b e construc ted b y a similar pro cedure to that described ab ov e. W e list tw o of them for reference. If w e replace z by z + 1 4 and put τ = 1 2 + i b in (6.20), the τ -functions f and f ′ turn out to b e f = ∞ X n = −∞ exp  2 π i nz − π i 2 n 2 − π i n 2 τ  , (6 . 35 a ) f ′ = ∞ X n = −∞ exp  2 π i nz + π i 2 n 2 − π i n 2 τ  , (6 . 35 b ) where w e ha v e used the formula e π i n ( n +1) = 1. In view of the formulas θ 3 ( z | τ + 1) = θ 4 ( z , τ ) , θ 4 ( z | τ + 1) = θ 3 ( z , τ ), the relations (6.19) then b ecome k ω = − 1 4 π 2 θ 4 3 (0 , q 2 ) , (6 . 36 a ) 46 λ = − θ ′′ 4 (0 , q 2 ) 2 π 2 θ 4 (0 , q 2 ) θ 4 3 (0 , q 2 ) . (6 . 36 b ) Using (6.35a), w e find Im f Re f = − A cn( w , κ ) dn( w , κ ) , (6 . 37 a ) where w = 2 π 2 θ 2 3 (0 , q 4 )( k y + ω t + z 0 ) , A = θ 2 (0 , q 4 ) θ 3 (0 , q 4 ) , κ = A 2 , q = e − π b . (6 . 37 b ) By replacing z b y i z in (6.35), w e obatin Im f Re f = − A dn( w , κ ′ ) , (6 . 38) with κ ′ = √ 1 − κ 2 , where the expressions (6 .36) remain the same forms. The expressions (6.37) and (6.38) giv e rise to t he 1-phase solutions of the sG equa- tion. 6.4 2-phase solutions In order to construct 2-phase solutions of t he sG equation, w e first solve the system of equations (6.10). Giv en v alues of k 1 , k 2 , τ 11 and τ 22 , this system can b e solv ed in principle for the unkno wn parameters ω 1 , ω 2 , τ 12 and λ. Because of the transcenden tal nature of the system of equations, ho w ev er, it is v ery difficult to obtain analytical solution. Under a few sp ecial situations in whic h the 2-dimensional theta function is expressed b y a finite sum of 1 - dimensional theta functions, the sys tem can b e solv ed algebraically . Indeed, there exist sev eral examples to realize the situatio n men tioned ab ov e [31]. Among them, w e consider the pa r ticularly imp ortant case τ 11 = τ 22 . (6 . 39 ) It turns out that the 2-dimensional theta function (6.6 ) has the represen tatio n [31] θ ( z | τ ) = 1 2  θ 3  z + + 1 2    τ +  θ 3  z − + 1 2    τ −  + θ 3 ( z + | τ + ) θ 3 ( z − | τ − )  , (6 . 40 a ) where z ± = 1 2 ( z 1 ± z 2 ) , (6 . 40 b ) 47 τ ± = 1 2 ( τ 11 ± τ 12 ) , (Im τ ± > 0) . (6 . 40 c ) It follow s from (6.7) and (6 .40) that f = θ 3  z + − 1 4    τ +  θ 3  z − + 1 2    τ −  + θ 3  z + + 1 4    τ +  θ 3  z −   τ −  , (6 . 41 a ) f ′ = θ 3  z + + 1 4    τ +  θ 3  z − + 1 2    τ −  + θ 3  z + − 1 4    τ +  θ 3  z −   τ −  . (6 . 41 b ) In v oking t he definition of the dn function, the solution φ of t he sG equation can b e written in the form tan φ 4 = 1 i f − f ′ f + f ′ = p k ′ + − dn( u + + δ + , k + ) p k ′ + + dn( u + + δ + , k + ) p k ′ − − dn( u − , k − ) p k ′ − + dn( u − , k − ) , (6 . 42 a ) where u ± = π θ 2 3 (0 | τ ± ) z ± , (6 . 42 b ) k ± = q 1 − k ′ ± 2 , k ′ ± = θ 2 4 (0 | τ ± ) θ 2 3 (0 | τ ± ) , (6 . 42 c ) δ + = π 4 θ 2 3 (0 | τ + ) . (6 . 42 d ) Th us, the ab o v e solution has a form similar to (5 .2 3) in whic h the v ariables u + and u − are separated completely . The next step is to solv e the system of equations (6.10) with N = 2. W e observ e under the conditio n (6.39) that n 2 1 τ 11 + 2 n 1 n 2 τ 12 + n 2 2 τ 22 = ( n 1 + n 2 ) 2 τ + + ( n 1 − n 2 ) 2 τ − . (6 . 43) Using (6.43) , Eqs. (6.10) can b e solv ed analytically . The calculation inv olv ed is straigh tforward but somew hat tedious. Hence, w e outline o nly the main steps. It follows from the equations F (1 , 0) = 0 and F (0 , 1) = 0 that k 1 ω 1 = k 2 ω 2 . (6 . 44) Substituting ω 2 from (6.44 ) in to the equations F (0 , 0) = 0 and F (1 , 1 ) = 0, one obta ins a homogeneous system of linear eq uations for ω 1 and λ . The solv abilit y of this sytem yields the relation [ θ ′′ 2 (0 , q 8 + ) θ 3 (0 , q 8 + ) − θ 2 (0 , q 8 + ) θ ′′ 3 (0 , q 8 + )][ θ 2 2 (0 , q 8 − ) − θ 2 3 (0 , q 8 − )] 48 = α [ θ ′′ 2 (0 , q 8 − ) θ 3 (0 , q 8 − ) − θ 2 (0 , q 8 − ) θ ′′ 3 (0 , q 8 − )][ θ 2 2 (0 , q 8 + ) − θ 2 3 (0 , q 8 + )] , (6 . 45 a ) where q ± = e π i τ ± = e π i( τ 11 ± τ 12 ) / 2 , (6 . 45 b ) α =  k 1 − k 2 k 1 + k 2  2 . (6 . 45 c ) Through a sequence of tra nsformations using v a r ious form ulas of the theta functions, one can simplify (6.45) to the form θ 4 2 (0 , q 2 + ) = αθ 4 2 (0 , q 2 − ) . (6 . 46) This is a tra nsce ndental equation whic h determines q + for giv en q − and α . Once q + is obtained, the parameter τ 12 is found from the relation e π i τ 12 = q + q − , (6 . 47) whic h follows from (6.45b). The parameters ω 1 and λ are then determined from the equations F ( 0 , 0) = 0 and F (1 , 0) = 0 as w ell as the relation ( 6 .44), the explicit forms of whic h are not written dow n here. Th us, we hav e completed the construction o f a 2-phase solution of the bilinear equation (6.2a ). The corresp onding solution of the SP equation can b e obtained f rom (6.1) and (6.5). As in the case of 1 - phase solutions discusse d in Sec. 6.3, v a rious 2 - phase solutions of the SP equation arise by sp ecifying the parameters k 1 , k 2 , and τ 11 (= τ 22 ). The explicit solutions are not presen ted here and will rep orted elsewhere . 6.5 Remarks 1. The starting p oint of o ur discussion is the condition (6 .3 9) whic h mak es it p ossible to p erform all the calculations a lgebraically . The r esulting solution of φ has a separable form with resp ect to t he indep enden t v ariables u + and u − (see (6.42 )). This expression should b e compared with the separable solution in tro duced in (5.23). Th us, we can exp ect t hat t he metho d dev elop ed here w ould repro duce all solutions constructed by a differen t metho d describ ed in Sec. 5.2. 2. Unless the condition (6.39) is imp osed, w e w ould b e able to obtain a bro ader class o f 2-phase solutions. The structure of solutions is w orth studying in a future w ork. 3. The general N -phase solution of the sG equation has b een constructed b y means of the metho d of algebraic geometry [18]. It ha s a form giv en b y 49 (6.1) in terms of t he N -dimensional theta functions. How eve r, whether the corresp onding τ -functions satisfy a system of bilinear equations (6.2) or not is an op en problem to b e resolv ed in a differen t con text. On the other hand, our metho d first lo oks for the solution of (6.2 ) so that the expression of x follow s immediately from (6.5). As a lready men tioned, ho w ev er, the construction of the N -phase solutio ns with N ≥ 3 is a difficult task in the con text of t he bilinear formalism. Nev ertheless, it is undoubtedly a challenging pro blem. 7 C ONCLUSION W e ha v e presen ted soliton and p erio dic solutions of the SP equation b y means o f a new metho d of solution. The most difficult tec hnical problem in constructing solutions w as ho w to integrate the PDE whic h gov erns the in v erse mapping to the original co ordinate s ystem (see (3.13) and (3.1 4 )). In the case of soliton solutions, the explicit form of the co o rdinate x w as obtained in terms of the τ -functions f and f ′ as sho wn by (4.5 ). In the case of p erio dic solutions, on the other hand, t he sp ecial ansatz leads t o the explicit form of x in terms of Jacobi’s elliptic functions. Specifically , fo r 1- phase solutions, assuming the dep endence of single v ariable, the sG equation b ecomes a tractable OD E (5 .1) so that a n umber of sp ecial s olut io ns are av ailable. F or 2-phase solutions, under the a nsatz (5.23), w e w ere a ble to reduce the represen tat io n of x to single integrals whic h are easily integrated (see (5.31)). It should b e remarke d, ho w ev er, that the resulting solutions of the SP equation are sp ecial class of real 2-phase solutions. An a lternativ e metho d employing the bilinear transformation metho d describ ed in Sec. 6 enables us to construct a broader class of solutions tha n the solutions obtained in Sec. 5 . 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