Symbolic Computation of Exact Solutions of Two Nonlinear Lattice Equations

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1 3rd Itertiol Coferece o Mchiery Mterils d Iformtio Techology Applictios (ICMMITA 5) Symbolic Compttio of Exct Soltios of Two Nolier Lttice Eqtios Sheg Zhg d Yigyig Zhob School of Mthemtics d Physics Bohi iversity Jizho 3 PR Chi szhgchi@6.com b37776@qq.com Keywords: Nolier lttice eqtio; Discrete G /G-expsio method; Exct soltio Abstrct. I this pper modified discrete G /G-expsio method is sed to costrct exct soltios of Tod lttice eqtio d Ablowitz-Ldik lttice eqtios. With the id of compter symbolic compttio we obtied i iform wy hyperbolic fctio soltios trigoometric fctio soltios d rtiol soltios of these two olier lttice eqtios. Whe the prmeters re tke s specil vles some kow soltios re recovered. It is show tht the modified method with symbolic compttio provides more effective mthemticl tool for solvig olier lttice eqtios i sciece d egierig. Itrodctio Solvig olier lttice eqtios plys importt role i my fields of sciece d egierig. I the pst severl decdes my effective methods for costrctig exct soltios of olier prtil differetil eqtios (PDEs) hve bee proposed sch s those i [-5]. slly it is hrd to geerlize oe method for olier PDEs to solve olier lttice eqtios. I 8 Wg Li d Zhg [6] proposed ew method clled the G /G-expsio method to fid trvellig wve soltios of olier PDEs. Some reserchers sch s Wg et l. [7] d Ebdi d Bisws [8-] hve doe sigifict work sig the method to costrct hyperbolic fctio soltios trigoometric fctio soltios d rtiol soltios of some importt eqtios. This method ws geerlized by Zhg Tog d Wg [] for olier PDEs with vrible coefficiets. More recetly Zhg et l. [] fod the itertive reltios betwee the lttice idices by crefl lysis d devised effective discrete lgorithm for sig the G /G-expsio method [6] to costrct hyperbolic fctio soltios d trigoometric fctio soltios of olier differetil- differece eqtios (DDEs). Lter Zhg et l. [3] employed embedded prmeter to modify the lgorithm i [] for ot oly hyperbolic fctio soltios trigoometric fctio bt lso rtiol soltios of olier DDEs. I order to show the vlidity d dvtges of the improved method we shll se the modified discrete method [3] to solve the Tod lttice eqtio d the Ablowitz-Ldik lttice eqtios i []. Exct soltios of Tod lttice eqtio Let s first cosider the fmos Tod lttice eqtio [36]: d (t ) d (t ) + [ + (t ) + (t ) (t )] =. dt dt We se the wve trsformtio (t ) = (ξ ) ξ = d + ct + ζ the Eq. () becomes c (c + )( + + ) =. () We sppose Eq. () hs the soltio i the form 5. The thors - Pblished by Atltis Press () 668

2 G ( ξ ) = + G( ξ ) d+ ελ µ G ( ξ ( ) + + ε ) d+ ελ ( µ ) d+ ελ ( µ ) G( ξ ) λ + + G ( ξ ) d ελ ( µ ) λ ) G( ξ ) = d+ ελ ( µ ) d+ ελ µ G ( ξ ( ) + ε ) d+ ελ ( µ ) d+ ελ ( µ ) G( ξ ) λ + G ( ξ ) d ελ ( µ ) λ + + ) G( ξ ) = where ( ) G ξ stisfies d+ ελ ( µ ) (3) () (5) G ξ G ξ + λ + µ G( ξ ) =. (6) d ( ) d ( ) dξ dξ Sbstittig Eqs. (3)-(5) log with Eq. (6) ito Eq. () d sig Mthemtic we obti set of lgebric eqtios for d d c. Solvig the set of lgebric eqtios we hve three cses λ µ λ µ sih( d) sih( d) = = c = d = d λ µ λ µ ε = d = µ λ µ λ si( d) si( d) = = c = d = d µ λ µ λ ε = d = (7) (8) λ = d = c = d d = d ε = d = µ =. (9) Whe λ µ > we obti hyperbolic fctio soltio of Eq. (): λ µ λ µ µ λ λ µ Csih( ξ) + Ccosh( ξ) λsih( d) = sih( d ) + () λ µ λ µ λ µ Ccosh( ξ) + Csih( ξ) λ µ sih( d) where ξ = d Settig λ µ ζ = c d C = from soltio () we obti λ µ = λ µ λ sih( d) λ µ d = k sih( k)th( k sih( k) t + c) () which is the kow kik-type solitry wve soltio i []. If set obti λ + µ = λ µ λsih( d) λ µ d = k ζ = c d C = from soltio () we 669

3 sih( k)coth( k sih( k) t + c) () which is the kow siglr trvellig wve soltio i []. Whe λ µ < we obti hyperbolic fctio soltio of Eq. (): µ λ µ λ µ λ µ λ Csi( ξ) + Ccos( ξ) λsi( d) = si( d ) + (3) µ λ µ λ µ λ Ccos( ξ) + Csi( ξ) µ λ si( d) where ξ = d Settig µ λ ζ = c d C = from soltio (3) we obti λ + µ = µ λ λsi( d) λ µ d = k si( k)t( k si( k) t + c) () which is the kow periodic wve soltio i []. If set obti λ + µ = µ λ λs i( d) µ λ d = k ζ = c d C = from soltio (3) we si( k)cot( k si( k) t + c) (5) which is the kow periodic wve soltio i []. Whe λ µ = we obti rtiol soltio of Eq. (): dc d λ (6) = C + Cξ where ξ = d dt + Settig dλ d = k ζ = c d C = from soltio (6) we obti k (7) k kt + c which is the kow rtiol soltio i []. Exct soltios of Ablowitz-Ldik lttice eqtios We ext cosider the Ablowitz-Ldik lttice eqtios []: d () t [ + () t v()][ t + () t + ()] t + ()= t (8) dt d v () t + [ + () t v()][ t v+ () t + v ()] t v()=. t (9) dt We se the wve trsformtio = ( ξ ) v = V( ξ ) ξ = d + ct + ζ the Eqs. (8) d (9) become c ( + V )( + ) + = () + cv + ( + V )( V + V ) V =. () + 67

4 Accordig to the homogeeos blce procedre we sppose tht Eqs. () d () hve the followig forml soltios: G ( ξ ) = + G( ξ ) + G ( ξ d ελ ( µ ) + + ε ) d+ ελ ( µ ) d+ ελ ( µ ) G( ξ ) λ + + G ( ξ ) d ελ ( µ ) λ ) G( ξ ) = d+ ελ ( µ ) () (3) V d+ ελ µ G ( ξ ( ) + ε ) d+ ελ ( µ ) d+ ελ ( µ ) G( ξ ) λ + G ( ξ ) d ελ ( µ ) λ + + ) G( ξ ) = () d+ ελ ( µ ) G ( ξ ) = β + β β G( ξ ) (5) V d+ ελ µ G ( ξ ( ) + + ε ) d+ ελ ( µ ) d+ ελ ( µ ) G( ξ ) λ + β + β G ( ξ ) d ελ ( µ ) λ ) G( ξ ) = d+ ελ ( µ ) (6) V + G ( ξ d ελ ( µ ) + ε ) d+ ελ ( µ ) d+ ελ ( µ ) G( ξ ) λ β + β G ( ξ ) d ελ ( µ ) λ + + ) G( ξ ) = (7) d+ ελ ( µ ) where G( ξ ) stisfies Eq. (6). Sbstittig Eqs. ()-(7) log with Eq. (6) ito Eqs. () d () d sig Mthemtic we obti set of lgebric eqtios for β β d d c. Solvig the set of lgebric eqtios we hve three cses λ µ ( λ λ µ )sih ( d) ( λ µ ) = = ( λ λ µ ) β = λ µ λ µ sih ( d) sih ( d) β = c = d = d ε = d = ( λ µ µ µ λ ( λ µ λ )si ( d) ( µ λ ) = = ( λ i µ λ ) β = (8) (9) (3) 67

5 µ λ µ λ si ( d) isi ( d) β = c = d = d ε = d = ( µ λ ) µ λ (3) λd = = λ β = d λ β = = = ε = d = µ = (3) c d d Whe λ µ > we obti hyperbolic fctio soltios of Eqs. (8) d (9): λ µ λ µ Csih( ξ ) cosh( ) + C ξ = λ µ λ µ λ µ λ µ Ccosh( ξ) + Csih( ξ) (33) v λ µ λ µ λ µ λ µ sih ( d) Csih( ξ) + Ccosh( ξ) sih ( d) = λ µ λ µ λ µ λ µ Ccosh( ξ) + Csih( ξ) λ µ sih ( d) where = λ ξ d Settig µ = λ µ (3) we hve (3) d C = from soltios (33) d = + th( d sih ( d) t+ ζ) (35) sih ( d) sih ( d) v = th( d sih ( d) t ζ) + (36) which re eqivlet to the kik-type solitry wve soltios i []. λ If set µ = d C = the soltios (33) d (3) become = + coth( d sih ( d) t+ ζ) (37) sih ( d) sih ( d) v = coth( d sih ( d) t ζ) + (38) which re eqivlet to the siglr trvellig wve soltios i []. Whe λ µ < we obti hyperbolic fctio soltios of Eqs. (8) d (9): v µ λ µ λ Csi( ξ ) cos( ) + C ξ i = µ λ µ λ µ λ µ λ Ccos( ξ) + Csi( ξ) µ λ µ λ µ λ λ µ si ( d) Csi( ξ) + Ccos( ξ) isi ( d) = µ λ µ λ µ λ µ λ Ccos( ξ) + Csi( ξ) µ λ isi ( d) where ξ = d µ λ (39) () 67

6 Whe λ µ = we obti rtiol soltios of Eqs. (8) d (9): C = C + Cξ v where ξ = d + Cd = ( C + C ξ ) () Ackowledgemets This work ws spported by the PhD Strt-p Fds of Bohi iversity (bsqd35) d Lioig Provice of Chi (37) the Lioig BiQiW Tlets Progrm (3955). Refereces [] S.Y. Lo d X.Y. Tg: Method of Nolier Mthemticl Physics (Sciece Press Beijig 6) [] C.S. Grder J.M. Greee M.D. Krskl d R.M. Mir: Phys. Rev. Lett. Vol. 9 (965) p. 95 [3] M.J. Ablowitz d P.A. Clrkso: Solito Nolier Evoltio Eqtios d Iverse Sctterig (Cmbridge iversity Press Cmbridge 99) [] M.R. Mirs: Bäckld Trsformtio (Spriger-Verlg Berli 978) [5] R. Hirot: Phys. Rev. Lett. Vol. 7 (97) p. 9 [6] M.L. Wg: Phys. Lett. A Vol. 3 (996) p. 79 [7] E.G. F: Phys. Lett. A Vol. 3 () p. 3 [8] S. Zhg d T.C. Xi: Comm. Theor. Phys. Vol. 5 (6) p. 985 [9] S. Zhg Y.Y. Zho IAENG It. J. Appl. Mth. Vol. () p. 77 []S. Zhg B. X d H.Q. Zhg It. J. Compt. Mth. Vol. 9 () p. 6 []S. Zhg d D. Wg: Therm. Sci. Vol. 8 () p. 555 []S. Zhg J.L. Tog d W. Wg: Compt. Mth. Appl. Vol. 58 (9) p. 9 [3]S. Zhg d B. Ci: Nolier Dy. Vol. 78 () p. 593 []S. Zhg B. X d A.X. Peg: Appl. Mech. Mter. Vol. 39 (3) p. 57 [5]S. Zhg Y.Y. Zho d B. Ci: Adv. Mter. Res. Vol () p. 76 [6]M.L. Wg X.Z. Li d J.L. Zhg: Phys. Lett. A 37 (8) p. 7 [7]M.L. Wg J.L. Zhg d X.Z. Li: Appl. Mth. Compt. Vol. 6 (8) p. 3 [8]G. Ebdi d A. Bisws: J. Frkli Ist. Vol. 37 () p. 39 [9]G. Ebdi d A. Bisws: Comm. Nolier Sci. Nmer. Siml. Vol. 6 () p. 377 []G. Ebdi d A. Bisws: Mth. Compt. Model. Vol. 53 () p. 69 []S. Zhg J.L. Tog d W. Wg: Phys. Lett. A Vol. 37 (8) p. 5 []S. Zhg L. Dog J.M. B d Y.N. S Phys. Lett. A Vol. 373 (9) p. 95 [3]S. Zhg L. Dog J.M. B d Y.N. S: Prm-J. Phys. Vol. 7 () p. 7 []Z. Wg: Compt. Phys. Comm. Vol. 8 (9) p. 673

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