On Some Classes of Breather Lattice Solutions to the sinh-gordon Equation

Transcription

1 On Soe Clsses of Brether Lttice Solutions to the sinh-gordon Eqution Zunto Fu,b nd Shiuo Liu School of Physics & Lbortory for Severe Stor nd Flood Disster, Peing University, Beijing, 0087, Chin b Stte Key Lbortory for Turbulence nd Coplex Systes, Peing University, Beijing, 0087, Chin Reprint requests to Z F; E-il: fuzt@pueducn Z Nturforsch 6, (007); received Mrch 6, 007 In this pper, dependent nd independent vrible trnsfortions re introduced to solve the sinh- Gordon eqution by using the nowledge of the elliptic eqution nd Jcobin elliptic functions It is shown tht different inds of solutions cn be obtined to the sinh-gordon eqution, including brether lttice solutions nd periodic wve solutions Key words: Jcobin Elliptic Function; sinh-gordon Eqution; Periodic Wve Solution; Brether Lttice Solution PACS nuber: 0365Ge Introduction The sinh-gordon (ShG) eqution 8] u xt = γ sinh u () is widely pplied in physics nd engineering, for exple in integrble quntu field theories ], noncouttive field theories ], fluid dynics 3] Due to the wide pplictions of the sinh-gordon eqution, ny chieveents hve been obtined in different spects 3 9] For instnce, the ShG eqution is nown to be copletely integrble ] becuse it possesses siilrity reductions to the third Pinlevé eqution 5] Due to the specil for of the sinh-gordon eqution, it is difficult to solve it directly, so we need soe trnsfortions In this pper, bsed on the introduced trnsfortions, we will show systeticl results of the brether lttice solutions 0 ] for the ShG eqution () by using the nowledge of the elliptic eqution nd Jcobin elliptic functions 3 5] The Brether Lttice Solutions to the ShG Eqution In order to obtin the brether-type solutions to the ShG eqution, we introduce the dependent vrible trnsfortion u =tnh w or w =tnh u, () which will let us rewrite the ShG eqution () s ( w )w xt +ww x w t γw( + w )=0 (3) Here one point ust be stressed tht not ll trnsfortions cn be pplied to solve the ShG eqution to derive the brether-type solutions In 9], the trnsfortions u =sinh w or w =sinh u () nd u =cosh w or w =cosh u (5) hve been introduced to derive the periodic wve solutions to the ShG eqution, nd ny different inds of the periodic solutions expressed in ters of the Jcobi elliptic functions hve been listed 9] However, it cn esily be checed tht these two trnsfortions cn not let us derive the brether-type solutions to the ShG eqution Siilrly, the trnsfortions u =tnh w or w =tnh u (6) nd u =tnh w or w =tnhu (7) cn not be pplied to obtin the brether-type solutions to the ShG eqution / 07 / $ 0600 c 007 Verlg der Zeitschrift für Nturforschung, Tübingen

2 556 Z Fu nd S Liu The sinh-gordon Eqution Fro the bove discussion we now tht the suitble trnsfortion is relly iportnt for us to derive different inds of solutions to the ShG eqution () Next, we will introduce the independent vrible trnsfortion r = x + bt + r 0, s = cx + dt + s 0, (8) where r 0 nd s 0 re two constnts Considering the trnsfortion (8), (3) cn be rewritten s ( w )bw rr +(d + bc)w rs + cdw ss ] +w(bw r + dw s )(w r + cw s ) γw( + w ) (9) =0 Copred to the trnsfortion given in 6, 7], trnsfortion (8) hs less constrints, of course, this will let us hve different types of solutions to the ShG eqution (3) Inspired by the trnsfortion given in 6] nd the results in, ], we choose the dependent vrible trnsfortion w = αu(r)v (s), (0) where α is constnt plitude to be deterined, U nd V stisfy the following elliptic equtions: Ur = nu + p U + q, Vs = βnv + p V + q, () where β, n, p, p, q nd q re deterined constnts for the specific choice of U nd V Here one point which ust be stressed is tht the introduction of β will let us hve ore choices to obtin different inds of solutions to the ShG eqution, nd the specific choice de in () is the result of vrible seprtion; siilr result cn be found in 6] for the sine-gordon eqution Substituting (0) nd () into (9) yields the lgebric equtions p p c = γ b, βnc + q α =0, q α c + n =0, d + bc =0, () (b) (c) (d) fro which we cn deterine α = βn q q, p p c = γ b, c = βn q α = q α n, (3) Fro (3) it is obvious tht the deterined constnts in () ust stisfy the constrints β q > 0, q q n < 0, βn < 0 () q This iplies tht not ll cobintions of Jcobi elliptic functions re solutions to the ShG eqution () under the bove-entioned trnsfortions Only the cobintion of couple of the Jcobi elliptic functions stisfies the constrint (); it cn be solution to the ShG eqution () Actully, there exist only 8 of these inds of cobintions; we will ddress the in detil Cse When U =dn(r, ) nd V = dn(s, ), where dn(r, ) nd dn(s, ) re the Jcobi elliptic function of the third ind, nd nd re their odulus 8 0], then fro (), we hve n =, p =, q = ( ), βn =, p =, q = ( ) (5) Substituting (5) into (3), the preters cn be deterined s c =, b = γ ( ) c ( )], ], α = ± (6) ( )( ) Then the solution to the ShG eqution w = ± ( )( ) ] dn(r, )dn(s, )], (7) which is ind of the brether lttice solution When 0 (or 0), the brether lttice solution (7) turns to be periodic wve solution w = ± ( ) ] dn(s, )] (8) Cse When U = sn(r, ) nd V = sn(s, ), where sn(s, ) is the Jcobi sine elliptic function 8 0], then n =, p = ( + ), q =,

3 Z Fu nd S Liu The sinh-gordon Eqution 557 βn =, p = ( + ), q = Fro (3) the preters cn be deterined s =, c = ( + ) c ( + )], b γ bc, α = ± Then the solution to the ShG eqution w = ± sn(r, )sn(s, )], d= (A) (9) (0) which is nother ind of the brether lttice solution Cse 3 When U = sn(r, ) nd V = cd(s, ) = where cn(s, ) is the Jcobi cosine elliptic function 8 0], then n =, p = ( + ), q =, βn =, p = ( + ), q = Fro (3) the preters cn be deterined s cn(s,) dn(s,), =, c (B) = ( + ) c ( + )], b γ bc, α = ± () Then the brether lttice solution to the ShG eqution w3 = ± sn(r, )cd(s, )] () d= Figure A shows the spce-tie evolution of the brether lttice solution of () nd (), where the preters re chosen s =, c =, = 8, γ =, s0 = r0 = 0, fro which the other preters cn be 6 deterined s b = 3 5, d = 5, =, α = Figures B nd C show the sptil profile t t = 0 nd t = 0, respectively Figure A shows the spce-tie evolution of the periodic solution of () nd (), where the preters re chosen s =, c =, =, γ =, s0 = r0 = 0, fro which the other preters cn be deterined s b = 89, d = 6 9, =, α = Figure B shows the sptil profile t t = 0 Figures nd describe the spce-tie evolution of the periodic solution of () nd (), for different vlues of nd ; their behviours re quite different Figure shows the evolution of the brether lttice solution, while Fig is just the norl periodic solution Cse When U = cd(r, ) nd V = cd(s, ), then n =, p = ( + ), q =, βn =, (C) Fig (A) Spce-tie evolution of the brether lttice solution of () nd (), where the preters re chosen s =, c =, = /8, γ =, s0 = r0 = 0, fro which the other preters cn be deterined s b = 3/5, d = 6/5, = /, α = / (B) Sptil profile t t = 0 (C) Sptil profile t t = 0

4 558 Z Fu nd S Liu The sinh-gordon Eqution (A) preters cn be deterined s c =, b = γ ( ) c ( )],, α = ± ] (5) Then the brether lttice solution to the ShG eqution (B) Fig (A) Spce-tie evolution of the periodic solution of equtions () nd (), where the preters re chosen s =/, c =, =/, γ =, s 0 = r 0 =0, fro which the other preters cn be deterined s b =8/9, d = 6/9, =, α =/ (B) Sptil profile t t =0 p = ( + ), q = Fro (3) the preters cn be deterined s c =, b = γ ( + ) c ( + )],, α = ± (3) Then the brether lttice solution to the ShG eqution w = ± cd(r, )cd(s, )] () Cse 5 When U = dn(r, ) nd V = nd(s, ) = dn(s,),thenn =, p =, q = ( ), βn =, p =, q = Fro (3) the ] w 5 = ± dn(r, )nd(s, )] (6) Cse 6 When U =nd(r, ) nd V = nd(s, ), then n =, p =, q =, βn =, p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( ) c ( )],, α = ±( )( )] (7) Then the brether lttice solution to the ShG eqution w 6 = ±( )( )] nd(r, )nd(s, )] (8) Aprt fro the nonsingulr solutions given bove, there re still soe solutions which stisfy the constrint () but y blow up for specific vlues of independent vribles Although they y be rther unphysicl, for resons of theticl trctbility, they cn be solutions to the ShG eqution () in the theticl for, too We list the in the following prts Cse 7 When U =dc(r, ) nd V =dc(s, ), then n =, p = ( + ), q =, βn =, p = ( + ), q = Fro (3) the preters cn be deterined s c =, b = γ ( + ) c ( + )], (9), α = ±

5 Z Fu nd S Liu The sinh-gordon Eqution 559 Then the brether lttice solution to the ShG eqution w 7 = ± dc(r, )dc(s, )] (30) Cse 8 When U =ns(r, ) = sn(r,) nd V = ns(s, ), thenn =, p = ( + ), q =, βn =, p = ( + ), q = Fro (3) the preters cn be deterined s c =, b = γ ( + ) c ( + )], (3), α = ± Then the brether lttice solution to the ShG eqution w 8 = ± ns(r, )ns(s, )] (3) Cse 9 When U =ns(r, ) nd V =dc(s, ) = dn(s,) cn(s,),thenn =, p = ( + ), q =, βn =, p = ( + ), q = Fro (3) the preters cn be deterined s c =, b = γ ( + ) c ( + )], (33), α = ± Then the brether lttice solution to the ShG eqution w 9 = ± ns(r, )dc(s, )] (3) Cse 0 When U =sc(r, ) = sn(r,) cn(r,) nd V = sc(s, ), thenn = ( ), p =, q =, βn = ( ), p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( ) c ( )], (35), α = ±( )( )] Then the brether lttice solution to the ShG eqution w 0 = ±( )( )] sc(r, )sc(s, )] (36) Cse When U =sc(r, ) nd V =cs(s, ) = cn(s,) sn(s,),thenn = ( ), p =, q =, βn =, p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( ) c ( )], (37) ], α = ± Then the brether lttice solution to the ShG eqution ] w = ± sc(r, )cs(s, )] (38) Cse When U =cs(r, ) nd V =cs(s, ), then n =, p =, q =, βn =, p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( ) c ( )], (39) ], α = ± ( )( ) Then the brether lttice solution to the ShG eqution w = ± ( )( ) ] cs(r, )cs(s, )] (0) Cse 3 When U =sn(r, ) nd V =ns(s, ), then n =, p = ( + ), q =, βn =, p = ( + ), q = Fro (3) the preters

6 560 Z Fu nd S Liu The sinh-gordon Eqution cn be deterined s c =, b = γ ( + ) c ( + )],, α = ± () Then the brether lttice solution to the ShG eqution w 3 = ± sn(r, )ns(s, )] () Cse When U =cd(r, ) = cn(r,) dn(r,) nd V = ns(s, ), thenn =, p = ( + ), q =, βn =, p = ( + ), q = Fro (3) the preters cn be deterined s c =, b = γ ( + ) c ( + )],, α = ± (3) Then the brether lttice solution to the ShG eqution w = ± cd(r, )ns(s, )] () Cse 5 When U =sn(r, ) nd V =dc(s, ) = dn(s,) cn(s,),thenn =, p = ( + ), q =, βn =, p = ( + ), q = Fro (3) the preters cn be deterined s c =, b = γ ( + ) c ( + )],, α = ± (5) Then the brether lttice solution to the ShG eqution w 5 = ± sn(r, )dc(s, )] (6) Cse 6 When U =cd(r, ) nd V =dc(s, ), then n =, p = ( + ), q =, βn =, p = ( + ), q = Fro (3) the preters cn be deterined s c =, b = γ ( + ) c ( + )],, α = ± (7) Then the brether lttice solution to the ShG eqution w 6 = ± cd(r, )dc(s, )] (8) Cse 7 When U =cn(r, ) nd V =nc(s, ) = cn(s,),thenn =, p =, q =, βn = ( ), p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( ) c ( )], (9), α = ± ( ] ) ( ) Then the brether lttice solution to the ShG eqution w 7 = ± ( ] ) cn(r, )nc(s, )] (50) ( ) Cse 8 When U =sd(r, ) = sn(r,) dn(r,) nd V = ds(s, ) = dn(s,) sn(s,),thenn = ( ), p =, q =, βn =, p =, q = ( ) Fro (3) the preters cn be deterined s c =, b = γ ( ) c ( )], (5), α = ± ( ] ) ( )

7 Z Fu nd S Liu The sinh-gordon Eqution 56 Then the brether lttice solution to the ShG eqution ( ] ) w 8 = ± sd(r, )ds(s, )] (5) ( ) Cse 9 When U =cn(r, ) nd V =ds(s, ), then n =, p =, q =, βn =, p =, q = ( ) Fro (3) the preters cn be deterined s c =, b = γ ( ) c ( )],, α = ± ( )( ) ] (53) Then the brether lttice solution to the ShG eqution w 9 = ± ( )( ) ] cn(r, )ds(s, )] (5) Cse 0 When U =nc(r, ) nd V =sd(s, ), then n = ( ), p =, q =, βn = ( ), p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( ) c ( )], (55), α = ± ( )( ] ) Then the brether lttice solution to the ShG eqution ( )( ] ) w 0 = ± nc(r, )sd(s, )] (56) Cse When U =sn(r, ) nd V =cs(s, ), then n =, p = ( + ), q =, βn =, p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( + )+c ( )],, α = ± ] (57) Then the brether lttice solution to the ShG eqution ] w = ± sn(r, )cs(s, )] (58) When nd 0, the brether lttice solution (58) turns to the solution w = ±tnh (r)cot(s)], (59) with = c, b = γ, d = ± γ (60) Cse When U =cd(r, ) nd V =cs(s, ), then n =, p = ( + ), q =, βn =, p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( + )+c ( )], (6), α = ± ] Then the brether lttice solution to the ShG eqution ] w = ± cd(r, )cs(s, )] (6) Cse 3 When U =ns(r, ) nd V =cs(s, ), then n =, p = ( + ), q =, βn =, p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( + )+c ( )],, α = ± ( ) ] (63)

8 56 Z Fu nd S Liu The sinh-gordon Eqution Then the brether lttice solution to the ShG eqution w 3 = ± ( ) ] ns(r, )cs(s, )] (6) Cse When U =dc(r, ) nd V =cs(s, ), then n =, p = ( + ), q =, βn =, p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( + )+c ( )], (65) ], α = ± ( ) Then the brether lttice solution to the ShG eqution w = ± ( ) ] dc(r, )cs(s, )] (66) Cse 5 When U =dc(r, ) nd V =sc(s, ), then n =, p = ( + ), q =, βn = ( ), p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( + )+c ( )], (67) ], α = ± Then the brether lttice solution to the ShG eqution ] w 5 = ± dc(r, )sc(s, )] (68) Cse 6 When U =ns(r, ) nd V =sc(s, ), then n =, p = ( + ), q =, βn = ( ), p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( + )+c ( )],, α = ± ] (69) Then the brether lttice solution to the ShG eqution ] w 6 = ± ns(r, )sc(s, )] (70) Cse 7 When U =sn(r, ) nd V =sc(s, ), then n =, p = ( + ), q =, βn = ( ), p =, q = Fro (3) the preters cn be deterined s c =, b = γ ( + )+c ( )], (7), α = ± ( )] Then the brether lttice solution to the ShG eqution w 7 = ± ( )] sn(r, )sc(s, )] (7) Cse 8 When U =cd(r, ) nd V =sc(s, ), then n =, p = ( + ), q =, βn = ( ), p =, q = Fro (3) the preters cn be deterined s c =, b = γ (+ )+c ( )],, α = ± ( )] (73) Then the brether lttice solution to the ShG eqution w 8 = ± ( )] cd(r, )sc(s, )] (7) 3 Conclusion In this pper, the independent vrible trnsfortion nd the dependent vrible trnsfortion were

9 Z Fu nd S Liu The sinh-gordon Eqution 563 introduced to solve the sinh-gordon eqution by using the nowledge of the elliptic eqution nd Jcobin elliptic functions It ws shown tht suitble trnsfortions re required in order to obtin the brether-type solutions to the sinh-gordon eqution Soe solutions with quite different structures hve not been reported in the literture, including brether lttice solutions The i of this pper ws to obtin ore inds of brether lttice solutions We did not touch on the stbility of those solutions which re non-singulr in the whole doin Although we did not give the stbility nlysis to our solutions, fro the results given by Kevreidis et l, ], we cn sy tht not ll solutions given in our nuscript re unstble Even though the solutions re unstble, they cn be stbilized by c driving nd dping; this hs been reported by Kevreidis et l, ] Due to wide pplictions of the ShG eqution, the nlyticl solutions given in this pper will be helpful in relted reserch Acnowledgeent Mny thns to nonyous referees for vluble suggestions nd to the Ntionl Nturl Science Foundtion of Chin for support (No nd No ) ] P Mosconi, G Mussrdo, nd V Rid, Nuc Phys B 6, 57 (00) ] I Cbrer-Crnero nd M Moriconi, Nuc Phys B 673, 37 (003) 3] K W Chow, Wve Motion 35, 7 (00) ] MJAblowitz,DJKup,ACNewell,ndHSehur, J Mth Phys 5, 85 (97) 5] P A Clrson, J B McLeod, P J Olver, nd A Rni,SIAMJMthAnl7, 798 (986) 6] A Khre, Phys Lett A 88, 69 (00) 7] Z J Qio, Physic A 3, (997) 8] G Cub nd R Punov, Phys Lett B 38, 55 (996) 9] Z T Fu, S K Liu, nd S D Liu, Coun Theor Phys 5, 55 (006) 0] P G Kevreidis, A Sxen, nd A R Bishop, Phys Rev E 6, 0663 (00) ] P G Kevreidis, A Khre, nd A Sxen, Phys Rev E 68, 0770 (003) ] P G Kevreidis, A Khre, A Sxen, nd G Herring, J Phys A 37, 0959 (00) 3] Z T Fu, S D Liu, nd S K Liu, Coun Theor Phys 39, 53 (003) ] Z T Fu, S K Liu, S D Liu, nd Q Zho, Phys Lett A 90, 7 (00) 5] S K Liu, Z T Fu, S D Liu, nd Q Zho, Phys Lett A 89, 69 (00) 6] G L Lb, Jr, Eleents of Soliton Theory, John Wiley nd Sons, New Yor 980 7] P G Drzin nd R S Johnson, Solitons: n Introduction, Cbridge University Press, New Yor 989 8] S K Liu nd S D Liu, Nonliner Equtions in Physics, Peing University Press, Beijing 000 9] Z X Wng nd D R Guo, Specil Functions, World Scietific Publishing, Singpore 989 0] P F Byrd nd M D Friedn, Hndboo of Elliptic Integrls for Engineers nd Physicists, Springer Verlg, Berlin 95