ANALYTICAL SOLUTIONS TO SINE-GORDON EQUATION WITH VARIABLE COEFFICIENT

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1 Romaia Reports i Physics, Vol. 66, No., P. 6 73, 4 ANALYICAL SOLUIONS O SINE-GORDON EQUAION WIH VARIABLE COEFFICIEN ZHENGPING YANG, WEI-PING ZHONG,,* Shude Polytechic, Departmet of Electroic ad Iformatio Egieerig, Guagdog Provice, Shude 583, Chia exas A&M Uiversity at Qatar, P.O. Box 3874 Doha, Qatar *Correspodig author: zhogwp6@6.com Received August 4, 3 Abstract: We study exact solito solutios of the Sie-Gordo (SG) equatio with variable coefficiet. Based o the similarity trasformatio ad Hirota s biliear method, we report both i-type ad ati-i-type oe-solito ad two-solito solutios of the SG equatio. I particular, we aalyze the emergig solito structures by a special selectio of two self-similar variables. Our results show that the shapes of both i-type ad ati-i-type solitos ca be effectively cotrolled by the specific form of those two self-similar variables. Key words: Sie-Gordo equatio, Hirota method, similarity trasformatio.. INRODUCION he Sie-Gordo (SG) equatio is oe of the most importat dyamical models i oliear sciece. It appears i may braches of moder sciece, icludig the study of propagatio of ultra-short optical pulses i resoat laser media [], propagatio of magetic fluxes i Josephso juctios [], trasmissio of ferromagetic waves [3], epitaxial growth of thi films [4], motio of dislocatios i crystals [5], DNA-solito dyamics [], ad so o. It is ow that the classical solito cocept emerges from the autoomous systems which have costat coefficiets. Geerally, these systems have o sigificat effects o the cotrol of solito shape. A more geeral situatio is oe i which a system receives some form of exteral time-depedet ad spacedepedet force, amely a o-autoomous system. Recetly, a o-autoomous system with variable coefficiets has attracted a lot of attetio because of its iterestig features ad potetial applicatios [6]. More geerally, o-autoomous systems with time- ad space depedet variable coefficiets have very iterestig properties but have bee the subject of relatively fewer studies.

2 Aalytical solutios to Sie-Gordo equatio 63 hus, i realistic physical systems, the SG model usually appears ot i its pure form, but with ihomogeeities, which spoil the most importat property of the model, e.g. the itegrability. For example, SG models with variable coefficiets appears i fluxo dyamics i Josephso juctios with impurity [7], i DNAsolito dyamics due to o-uiformity iduced by specific base sequeces i the promoter regio [8], i spi wave propagatio with variable iteractio stregth [9] etc. However i all such ihomogeeous SG models with variable coefficiets, due to o-itegrability the solutios ca be extracted oly umerically or at best perturbatively [7, 8, 9, ]. his paper is orgaized as follows. I Sec., we recapitulate the steps of similarity trasformatio to reduce the SG model with variable coefficiet ito the stadard SG model; the, by usig Hirota s biliear method we obtai its exact solutios. I Sec. 3, we costruct differet varieties of localized solito solutios. We also aalyze the uique features of i -type ad ati-i -type solitos by choosig the two arbitrary self-similar variables as certai special fuctios, amely, Chebyshev polyomials of the secod id. Fially, i Sec. 4, we summarize our results.. HE MODEL AND HE HIROA BILINEAR MEHOD We cosider the followig geeralized SG equatio with variable coefficiet, which describes the dyamics of the system [6], V = M xt x t (, ) si( V), () where V = V ( x, t) is a scalar-valued fuctio, t is time, ad x is the spatial coordiate. M ( xt, ) is a variable coefficiet, ad it is a space-time depedet variable. Whe M ( x, t ) =, Eq. () is simplified as the stadard SG equatio. Accordig to the self-similar method [, ], the solutio of Eq. () is assumed to be the followig self-similar trasformatio, V ( x, t) = u( X, ), () where X = X ( x) ad = ( t) are real fuctios of the spatial coordiate x ad X time t, respectively. If we choose M ( x, t) =, substitutig the t x trasformatio () ito Eq. (), we fid that Eq. () becomes u X = si ( u). (3)

3 64 Zhegpig Yag, Wei-Pig Zhog 3 I geeral, Eq. () is difficult to treat aalytically. herefore, as log as we get solito solutios of the stadard SG equatio (3), by the self-similar trasformatio (), we will obtai the solito solutio of the SG equatio () with variable coefficiets. Next, we aim at obtaiig aalytical solutio to Eq. (3), represeted i the Hirota biliear form. I order to do this, we mae use of the followig trasformatio: u( X, ) = 4 arcta g, (4) f where g = g( X, ) ad f f ( X, ) = are two fuctios of X ad, to be determied. Substitutig Eq. (4) ito Eq. (3), ad equatig the coefficiets of f g ad gf, the followig biliear equatios are obtaied, ( DXD )( g f ) X =, (5a) D D f f g g =, (5b) where D X is Hirota s biliear operator [3, 4, 5], defied as DXD ( g f ) = g( X, ) f ( X, ) X X. X = X = I order to obtai a solutio of Eq. (5), we expad the two fuctios, f X, ito power series of a parameter ε as follows: g ( X ) ad 3 3 g( X, ) = g +ε g +ε g +ε g3 + ad (, ) Isertig g ( X, ) ad f (, ) f X = +ε f +ε f +ε f +. 3 X ito Eqs. 5(a,b) ad equatig the coefficiets of each power of ε, the followig formulas are give as: ε : ε : ε : ( DXD )( g ) =, (6a) D D g g =, (6b) X ( DXD )( g f g ) [ ] + =, (6c) D D f + f g g g g =, (6d) X

4 4 Aalytical solutios to Sie-Gordo equatio 65 3 ε : DXD [ f f g g g g] + =, (6e) D D f + f f + f g g g g g g =, (6f) X ( DXD )( g f3 g f g f g3 ) =, (6g) D D f + f f + f f + f g g g g g g g g =. (6h) X ax + b +θ Next we will solve Eqs. (6), step by step. If we tae g = e ad f = as a special solutio of (6a), from Eqs. (6b), (6c), ad (6d) we have g = ad f =, ( ). From Eq. (6a) we fid that b= a. hus we obtai a solutio solutio of the form: u ( X, ) = 4 arcta λe ax + a, (7) where λ= e θ. his is the oe-solito solutio of the Sie-Gordo equatio (), for arbitrary costats λ ad a. he solutio with a > is usually called the i oe-solutio solutio, ad that with a < is the ati-i oe-solutio solutio. he, the two-solito solutio of (6a) ca be obtaied by usig ax + b +θ g ax = e, + b +θ g = e, ad f =. We fid that g = ad f = ( ). From Eqs. (6a) (6h) we have b = a, b = a ad f a + a a + a X+ +θ +θ a a aa e =, where a a. a + a hus, we obtai the two-solito solutio i the form: u a e e, = 4 arcta a a + e a + a ( X ) ax + +θ ax+ +θ a a+ a a+ a X+ +θ +θ aa. (8) For a > a > the above expressio is a i two-solito solutio, while for a > a > is a ati-i two-solito solutio. I a similar fashio, we ca obtai the th-order solito solutios of Eq. (3).

5 66 Zhegpig Yag, Wei-Pig Zhog 5 3. ANALYSIS AND DISCUSSION I this sectio, we will costruct aalytical localized solito solutios to Eq. () by choosig two arbitrary self-similar variables X ( x ) ad ( t ), ad we will study the spatial ad temporal shapes of the solutios. aig ito accout the temporal ad spatial symmetry, ad the fact that X ( x ) ad ( t ) are arbitrary fuctios of x ad t, we will select X ( x ) ad ( t ) to be some special fuctios. Next we will tae X ( x) = U ( x) ad ( t) = U ( t), where U is the Chebyshev polyomial of the secod id, which is defied as: ( ) ( + ) x ( + ) + ( x ) d = U x.!! dx It satisfies the followig differetial equatio [6], d d U x U x x 3x ( ) U x dx dx + + =, (9) where is a o-egative iteger. For U t, we have a similar expressio. he correspodig variable coefficiets are give by (, ) M x t ( + ) U ( x) xu ( x) ( + ) U ( t) tu ( t) ( x )( t ) =. () able lists some variable coefficiet values for several Chebyshev polyomials of low order. As see from the table, the fuctio M ( xt, ) is ot a sigular oe. able Some M ( xt, ) values for several Chebyshev polyomials of low order X ( x) = U ( x) ( t) = U ( t) M ( xt, ) x t 4 + 4x + 4t 64tx 3 4x + 8x3 4t + 8t3 6( 6t )( 6x ) 4 x + 6x4 t + 6t4 64tx( 3 8t )( 3 8x ) 5 x( 3 6x + 6x 4 ) t( 3 6t + 6t 4 ) 4( 3 48t + 8t 4 )( 3 48x + 8x 4 ) 6 + 4x 8x4 + 64x6 + 4t 8t4 + 64t6 56xt( 3 t + 4t 4 )( 3 x + 4x 4 )

6 6 Aalytical solutios to Sie-Gordo equatio 67 Whe choosig the parameters θ = ad a =, from Eqs. () ad (7) we obtai the i oe-solutio family of solutios of Eq. () U ( x) + U ( t) ( x t) = V, 4arcta e, ad the correspodig profiles of V, x t for four differet positive eve values of are show i Fig.. As see from these figures, the solitos have cocave shapes. he solitos emerge with both cocave circular shapes ad cocave quadrilateral shapes. Whe x ad t, V ( x, t ) = π, thus, V ( x, t ) is a localized solutio. For this solito family, there exist pea values (Fig. ). Fig. Profiles of i -type oe-solito solutios with the parameters θ = ad a = for =, 4,6,8 (color olie).

7 68 Zhegpig Yag, Wei-Pig Zhog 7 Whe a <, we get covex solito clusters for differet positive eve values of. We tae a = ad θ =, ad from Eqs. () ad (7) we obtai the atii oe-solutio solutio of Eq. (): U ( x) U ( t) ( x t) = a V, 4 arcta e. Figure depicts the (x,t) distributio of the covex solito clusters for =, 4, 6,8 ; ote that the solitos arise with both covex circular shapes ad covex quadrilateral shapes. Further, we fid that the solito clusters have ( ) a pea values (Fig. ). We ca see that V ( x t ) localized solutio., = ; thus, it is also a Fig. Profiles of ati-i -type oe-solito solutios. he setup ad parameters are as i Fig., except for a = (color olie).

8 8 Aalytical solutios to Sie-Gordo equatio 69 Next, we discuss the shape of the two-solito solito solutio (8) possessig a special set of parameters a, a, θ, ad θ. For the i two-solito solutio, we choose the parameters a =, a =., θ =, ad θ = ; from Eqs. () ad (8) we obtai the expressio of a the i two-solito solutio, U( x) + U( t).u( x) U e e ( t) V ( x, t) = 4arcta. ().49383e.9U( x) 9U( t + ) his is a iterestig example, because we fid that there exist two differet solito shapes for the same set of parameters ad for positive eve ad odd values of. Figure 3 depicts the shapes of typical covex ellipse solitos for eve. Whe =, there is a cocave ellipse alog x -axis directio, while for = 4, 6, 8, there exist two, three, ad four cocave ellipses, respectively. I geeral, for such solito clusters there exist cocave ellipses alog x -axis directio. Fig. 3 Ki -type two-solito patters for,4,6,8 =. he setup ad parameters are give i the text (color olie).

9 7 Zhegpig Yag, Wei-Pig Zhog 9 he wave structures i these grooves are similar to those show i Fig. ; however they are somehow differet. First, i -type oe-solitos are imprisoed i cocave circular or quadrilateral structures, while i -type twosolitos display cocave ellipses. Secod, the amplitude of i -type oe-solito is positive, while that of i -type two-solito ca be either positive or egative. Whe is chose as a odd iteger, we obtai aother class of i -type two-solito solutio. Figure 4 displays some illustrative examples for = 3, 5, 7, 9, respectively. We fid that, for eve, V ( x ) = π ad V is also a localized solutio. V x = π; but for odd, x = π. herefore, the i -type two-solito Fig. 4 Same as i Fig. 3, except for = 3,5,7,9 (color olie).

10 Aalytical solutios to Sie-Gordo equatio 7 Fially, we will preset the aalytic expressio of the ati-i -type twosolito solutio. Oe assumes that a =, a =., ad θ =θ = ; the the explicit aalytical ati-i -type two-solito solutio ca be expressed as U( x) + U( t).u( x) + U e e ( t) a V ( x, t) = 4arcta. ().6694e.U( x) + U( t + ) From expressio (), it is obvious that the solito shape is govered by the U t metioed above. I what follows, we will costruct parity of U x ad differet types of ati-i -type two-solito patters depedig for either eve or odd. We assume that is a positive eve iteger. I Fig. 5 we preset the profiles of the ati-i -type two-solito solutios. Oe ca observe that a V x, t =. herefore, these solutios are localized solitos. Fig. 5 Profiles of ati-i -type two-solito solutios with the parameters a = ad a =., for eve (color olie).

11 7 Zhegpig Yag, Wei-Pig Zhog a Aother example is displayed i Fig. 6 for odd. We fid that V = or a V =± π whe x ad t. Fig. 6 Same as i Fig. 5, except for = 3,5,7,9 (color olie). 4. CONCLUSIONS By usig the similarity trasformatio ad Hirota s biliear method, we obtai both i -type ad ati-i -type oe-solito ad two-solito solutios of the geeralized Sie-Gordo equatio with variable coefficiet. Based o these aalytical solutios, we fid that for the oe-solito solutio, ( ) = π or V ( x t ) V x, t whereas for the two-solito solutio,, =,

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