New exact solutions for the combined sinh-cosh-gordon equation

Transcription

1 Sociedad Colobiaa de Mateáticas XV Cogreso Nacioal de Mateáticas 2005 Aputes Lecturas Mateáticas Volue Especial (2006), págias New exact solutios for the cobied sih-cosh-gordo equatio César A. Góez S. Uiversidad Nacioal de Colobia, Bogotá Álvaro Salas Uiversidad de Caldas, Maizales Uiversidad Nacioal de Colobia, Maizales Abstract. We preset the geeral projective Riccati equatios ethod to obtai exact solutios for the cobied sih-cosh-gordo equatio. The Pailevé property v = e u will be used to back up the ethod to derive travellig wave solutios of distict physical structures. I additio we showed the behavior of the solutios with the graph of soe of the. The ethod ca also be applied to other oliear partial differetial equatio (NLPDE s) or systes i atheatical physics. Key words ad phrases. Noliear differetial equatio, travellig wave solutio, Matheatica, projective Riccati equatio ethod AMS Matheatics Subject Classificatio. 35C05. Resue. Presetaos el étodo proyectivo de ecuacioes de Riccati geeral, para obteer solucioes exactas para la ecuació sih-cosh- Gordo cobiada. La propiedad de Paivelé v = e u se usará para alcazar el étodo, y derivar solucioes por odas viajeras de distitas estructuras físicas. Adeas ostrareos el coportaieto de las solucioes co el gráfico de alguas de ellas. El étodo puede adeás ser aplicado a otras ecuacioes difereciales parciales o lieales (NLPDEs) o sisteas e física ateatica. 1. Itroductio I the study of oliear wave pheoea, the travellig wave solutios of partial differetial equatio (PDEs) have physical relevace. The kowledge of closed for solutios of oliear PDEs ad ODEs facilitates the testig of uerical solvers, ad aids i the stability aalysis of solutios. It is well-kow

2 88 César A. Góez S. & Álvaro Salas that searchig the exact solutios for oliear partial differetial equatios is the great iportace for ay researches. A variety of powerful ethods such that tah ethod, geeralized tah ethod, geeral projective Riccati equatio ethod, Bäcklud trasforatio, Hirota biliear fors, ad ay other ethods have bee developed i this directio. Practically, there is ot a uified ethod that ca be used to hadle all types of oliear probles. I this paper, we will use the geeral projective Riccati equatio ethod, to costruct exact solutios for the cobied sih-cosh-gordo equatio. 2. The geeral projective Riccati equatios ethod For a give oliear equatio that does ot explicitly ivolve idepedet variables P (u, u x,u t,u xx,u xt,u tt,...)=0, (2.1) whe we look for its travellig wave solutios, the first step is to itroduce the wave trasforatio, which have by defiitio the for u(x, t) =v(ξ), ξ = x + λt, (2.2) where λ is a costat ad chage (1.1) to a ordiary differetial equatio (ODE) for the fuctio v(ξ) P (v, v,v,...)=0. (2.3) The ext crucial step is to itroduce ew variables σ(ξ), τ(ξ) which are solutios of the syste { σ (ξ) =eσ(ξ)τ(ξ) τ (ξ) =eτ 2 (2.4) (ξ) µσ(ξ)+r. It is easy to see that the first itegral of this syste is give by τ 2 = e[r 2µσ(ξ)+ µ2 + ρ σ 2 (ξ)], (2.5) r where ρ = ±1. Fro this itegral we obtai the followig particular solutios: 1. Case I: If r = µ = 0 the τ 1 (ξ) = 1 eξ, σ 1(ξ) = C ξ. (2.6) 2. Case II: If e =1adρ = 1 τ 2 = σ 2 = r ta( rξ) µ sec( rξ)+1 r sec( rξ) µ sec( rξ)+1 (r >0) (r >0). (2.7)

3 New exact solutios for the cobied sih-cosh-gordo equatio Case III: If e = 1 adρ = 1 r tah( rξ) τ 3 = µsech( rξ)+1 σ 3 = rsech( rξ) µsech( rξ)+1 4. Case IV: If e = 1 adρ =1 τ 4 = σ 4 = r coth( rξ) µcsch( rξ)+1 r csch( rξ) µ csch( rξ)+1 5. Case V: If e =1adρ =1 τ 5 = r coth( rξ) µcsch( rξ)+1 σ 5 = r csch( rξ) µ csch( rξ)+1 We seek a solutio of (1.1) i the for u(x, t) =v(ξ) =a 0 + (r >0) (r >0). (r >0) (r >0). (r <0) (r <0). (2.8) (2.9) (2.10) M σ i 1 (ξ)(a i σ(ξ)+b i τ(ξ)), (2.11) i=1 where σ(ξ), τ(ξ) satisfy the syste (2.4). The iteger M ca be deteried by balacig the highest derivative ter with oliear ters i (2.3), before the a i ad b i ca be coputed. Substitutig (2.11), alog with (2.4) ad (2.5) ito (2.3) ad collectig all ters with the sae power i σ i (ξ)τ j (ξ), we get a polyoial i the variables σ(ξ) ad τ(ξ). Equalig the coefficiets of this polyoial to zero, we obtai a syste of algebraic equatios, fro which the costats µ, r, λ, a i,b i (i =1, 2,...,M) are obtaied explicitly. Usig the solutios of the syste (2.14) alog with (2.11), we obtai the explicit solutios for (2.1) i the origial variables. 3. The cobied sih-cosh-gordo equatio This is the equatio u tt ku xx + α sih(u)+β cosh(u) =0, (3.1) where subscripts idicate partial derivatives, u is a real scalar fuctio of the two idepedet variables x ad t, while α ad β are all odel paraeters ad they are arbitrary, ozero costats. This equatio has bee discussed i [8] by ea the variable separated ODE ad the tah ethods. I this paper, we

4 90 César A. Góez S. & Álvaro Salas obtai ew exact solutio for ay values of k, α ad β. First itroduce the trasforatios sih u = V V 1, cosh u = V 1 + V 1, V = e u, (3.2) 2 2 after which we obtai the equatio 2V (V tt kv xx )+2(kVx 2 Vt 2 )+(β + α)v 3 +(β α)v =0. (3.3) The substitutio V = v(ξ) =v(x + λt) i (2.15) gives us the equatio ()v 3 (α β)v +2(λ 2 k)vv 2(λ 2 k)(v ) 2 =0. (3.4) Accordig to the ethod described above, we seek solutios of (2.13) i the for u(x, t) =v(ξ) =a 0 + a 1 σ(ξ)+b 1 τ(ξ), (3.5) where σ(ξ) adτ(ξ) satisfy the syste (1.5). Substitutig (2.17), alog with (1.5) ad (1.6) ito (2.16) ad collectig all ters with the sae power i σ i (ξ)τ j (ξ) we get a polyoial i the variables σ(ξ) adτ(ξ). Equalig the coefficiets of these polyoial to zero ad after siplificatios (usig e = ±1, r 0) we get the followig algebraic syste: 4ea 1 b 1 (k λ 2 )(µ 2 + ρ) =0, 2e(k λ 2 )(µ 2 + ρ)( ra eb 2 1(µ 2 + ρ)) = 0, a 3 0() a 0 (α β +3erb 2 1()) = 0, 2µb 2 1(r(k λ 2 )+3ea 0 ()) + a 1 ( +2era 0 (k λ 2 )+3a 2 0(α + β) 3erb 2 1()) = 0, a 0 (6erµa 1 (k λ 2 ) 3ra 2 1()+3eb 2 1()(µ 2 + ρ)) + 2r( b 2 1(k λ 2 )(3µ 2 +2ρ)+3eµa 1 b 2 1()) = 0, 4ea 0 a 1 (k λ 2 )(µ 2 +ρ) 2erµa 2 1(k λ 2 )+ra 3 1(α+β)+6µb 2 1(k λ 2 )(µ 2 + ρ) 3ea 1 b 2 1()(µ 2 + ρ) =0, b 1 (α β 3a 2 0()+erb 2 1()) = 0, 2b 1 (a 0 ( eµ(k λ 2 )+3a 1 ()) + e(ra 1 (k λ 2 )+µb 2 1())) = 0, b 1 ( 4ea 0 (k λ 2 )(µ 2 + ρ) +4erµa 1 (k λ 2 ) 3ra 2 1() +eb 2 1(α + β)(µ 2 + ρ)) = 0. Solvig the previous syste respect to the ukow variables r, a 0,a 1,b 1 we obtai the solutios b 1 =0,a 0 = ± α β α2 β,a 2 1 = 2µe(k λ2 ),r= where µ 2 + ρ =0,ρ = ±1, e = ±1. α2 β 2 e(k λ 2 ), Therefore, accordig (2.17) ad usig (1.6) to (1.10), ad after siplificatios we obtai the followig classificatio of soe exact solutios for the equatio (2.13): (i all cases u(x, t) =v(ξ) =a 0 +a 1 σ(ξ), b 1 =0, = α 2 β 2 0, =(λ 2 k) 0adξ = x + λt): For e =1adρ =1:

5 New exact solutios for the cobied sih-cosh-gordo equatio 91 N r µ a 0 a 1 u 1 2 ı ı α β α β 2ı 2 ( ) (α β) csc( ξ)+1 ( ) )(<0adα>β) csc( ξ) 1 ( ) (β α) csc( ξ)+1 ( ) )(>0adα<β) csc( ξ) 1 For e = 1 adρ =1: N r µ a 0 a 1 u 3 4 ı ı α β α β 2ı 2ı ( ) (α β) csc( ξ) 1 ( ) )(<0adα>β) csc( ξ)+1 ( ) (β α) csc( ξ) 1 ( ) )(>0adα<β) csc( ξ)+1 For e =1adρ = 1: N r µ a 0 a 1 u 5 1 α β 2 (α β) cot 2 ( 1 2 ξ) )(<0adα>β) α β 2 (β α) cot 2 ( 1 2 ξ) 6 1 )(>0adα<β) 7 1 α β 2 (β α) coth 2 ( 1 2 ξ) )(>0adα<β) 8 α β 2 (β α) ta 2 ( 1 2 ξ) 1 )(>0adα<β) 9 α β 2 (α β) tah 2 ( 1 2 ξ) 1 )(<0adα>β) 10 1 α β 2 (β α) tah 2 ( 1 2 ξ) )(>0adα<β) (α β) coth 2 ( 1 2 ξ) )(<0adα>β) 12 1 α β 2 (α β) ta 2 ( 1 2 ξ) )(<0adα>β)

6 92 César A. Góez S. & Álvaro Salas The surface i Figure 1 correspods to solutio (1) with ξ = x + λt, k =2, λ =1,α =2adβ =1,forx = 14 to x =14adt = 1 tot =1. The surface i Figure 2 correspods to solutio (8) with ξ = x + λt, k =1, λ =2,α = 192 ad β =1,forx = 115 to x = 115 ad t = 1 tot =1. The surface i Figure 3 correspods to solutio (7) with ξ = x + λt, k =1, λ =2,α = 192 ad β =1,forx = 0,5 tox =0,5 adt = 1 tot =1. Figure 1 Figure 2 Figure 3 4. Coclusios The projective Riccati equatio ethod is a powerful ethod to search exact solutios for NLPDE s. The projective ethod is ore coplicated tha other ethods, i the sese that deads ore coputer resources sice the algebraic syste ay require a lot of tie to be solved. I soe cases, this syste is so coplicated that o coputer algorith ay solve it, specially if the value of M is greater tha four. I this paper, this ethod has bee applied to the cobied sih-cosh-gordo equatio with M = 1. Ackowledgets: The authors wat to express their gratitude to professor A. Siitsy for his helpful suggestios ad recoedatios about this paper. Refereces [1] R. Cote & M. Musette, Lik betwe solitary waves ad projective Riccati equatios,j. Phys. A Math. 25 (1992), [2] E. Ic & M. Ergüt, New Exact Tavellig Wave Solutios for Copoud KdV-Burgers Equatio i Matheatical Physics, Applied Matheatics E-Notes 2 (2002),

7 New exact solutios for the cobied sih-cosh-gordo equatio 93 [3] J. Mei, H. Zhag & D. Jiag, New exact solutios for a Reactio-Diffusio equatio ad a Quasi-Caassa-Hol Equatio, Applies Matheatics E-Notes, 4 (2004), [4] Z. Ya, The Riccati equatio with variable coefficiets expasio algorith to fid ore exact solutios of oliear differetial equatio, MMRC, AMSS, Acadeia Siica (Beijig) 22 (2003), [5] Z. Ya, A iproved algebra ethod ad its applicatios i oliear wave equatios, MMRC, AMSS, Acadeia Siica (Beijig) 22 (2003), [6] D. Baldwi, U. Goktas, W. Herea, L. Hog, R. S. Martio & J. C.Miller, Sybolic coputatio of exact solutios expressible i hyperbolic ad elliptic fuctios for oliear PDFs, J. Sybolic Copt 37 (6) (2004), Preprit versio: li.si/ (arxiv.org) [7] A. Salas & C. Goez, El software Matheatica e la búsqueda de solucioes exactas de ecuacioes difereciales o lieales e derivadas parciales ediate la ecuació de Riccati.EMeorias Prier Seiario Iteracioal de Tecologías e Educació Mateática. 1. Uiversidad Pedagógica Nacioal (2005), [8] A. M. Wazwaz, The tah ethod: exact solutios of the Sie-Gordo ad the Sih- Gordo equatios, Applied Matheatics ad Coputatio 49 (2005), (Recibido e Marzo de Aceptado para publicació agosto de 2006) Departeto de Mateáticas Uiversidad Nacioal de Colobia Bogotá, Colobia e-ail:[email protected] Departeto de Mateáticas Uiversidad de Caldas Maizales, Colobia e-ail:[email protected]