Separation Transformation and ew Exact Solutions for the (1-Dimensional Triple Sine-Gordon Equation Yifang Liu a Jiuping Chen b andweifenghu c and Li-Li Zhu d a School of Economics Central University of Finance and Economics Beijing 10001 China b Guanghua School of Management Peing University Beijing 10071 China c School of Mathematics and Physics Shandong Institute of Light Industry Jinan 505 China d School of Information Science and Technology Taishan College Taian 7101 China Reprint requests to Y. L.; E-mail: yfl191@16.com Z. aturforsch. 66a 19 (011; received October 19 009 / revised March 19 010 The separation transformation method is extended to the (1 -dimensional triple Sine-Gordon equation and a special type of implicitly exact solution for this equation is obtained. The exact solution contains an arbitrary function which may lead to abundant localized structures of the highdimensional nonlinear wave equations. The separation transformation method in this paper can also be applied to other inds of high-dimensional nonlinear wave equations. Key words: Separation Transformation; Triple Sine-Gordon Equation; Implicitly Exact Solution. PACS numbers: 0.0.I 0.75.Hh 05.45.Yv 1. Introduction It is well nown that the Sine-Gordon-type equations [1 10] including one-dimensional single Sine- Gordon (SG equation u xx u tt = asin(u double Sine-Gordon equation u xx u tt = asin(ubsin(u (1 and triple Sine-Gordon equation u xx u tt = asin(ubsin(ucsin(u ( are widely applied in physics and engineering. These equations are special cases of nonlinear wave equations. Especially the single Sine-Gordon equation is Lax integrable and has nontrivial prolongation structures[1 11]. The (11-dimensional Sine-Gordon equation [1 ] was originally considered in the nineteenth century in the course of studying surfaces of constant negative curvature. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions [1 14]. The double SG equation arises in many physical systems lie the spin dynamics in the B phase of superfluid He [4] some features of the propagation of resonant ultrashort optical pulses through degenerate media [5] quasi-one-dimensional charge density wave condensate theories of the organic linear conductors lie TTF-TCQ [6] and nonlinear excitations in a compressible chain of XY dipoles under conditions of piezoelectric coupling macromolecules [7]. In the case of resonant fivefold degenerate media propagation and creation of ultrashort optical pulses the anisotropic magnetic liquids HeA and HeB at temperatures below the transition to the A phase at.6 mk and the propagation of spin waves one usually uses the single SG and the double SG model. However in some other cases one has to use the higher [ 10] which describes the propagation of strictly resonant sharp line optical pulses through unexcited absorbing media with Q( symmetry and the extrapolation to Q(J. In these cases the selection rules for the optical transitions are J = 0 MJ = 0 where J is an angular momentum quantum number. In the experiments carried out by Bullough and Caudrey [10] J is the hyperfine structure quantum number F = and the transition is F 1 = 5-fold degenerate. The (1 -dimensional Sine-Gordon-type equation [15 16] appears in many physically relevant systems and its properties have been studied from both mathematical and physical viewpoints. Recently Wang et al. [17] and Yan [1] introduced an effective separation transformation to study the various (1 - dimensional nonlinear systems and some wonderful results are obtained. In this paper we will apply this 09 074 / 11 / 0100 0019 $ 06.00 c 011 Verlag der Zeitschrift für aturforschung Tübingen http://znaturforsch.com
0 Y. Liu et al. Exact Solutions of (1-Dimensional Triple Sine-Gordon Equation separation transformation approach to the (1 - dimensional in the form u tt x j x j = u 1 u asin( ( u bsin csin(u. ( The rest of this paper is organized as follows. In Section some special type of implicitly exact solutions for ( are obtained by means of the separation transformation approach. Conclusions are presented in Section.. Separation Transformation and ew Exact Solutions for Equation ( In this part we extend the separation transformation method to the (1-dimensional (. The following proposition reduces ( to a nonlinear ordinary differential equation (ODE and a system of partial differential equations (PDEs with a nonlinear PDE and a linear PDE. Proposition. Function u(x 1 x x ;t =U[w (x 1 x x ;t] is the solution of ( if the function U[w(x 1 x x ;t] solves the following system of nonlinear ODE: U (w= U w = 1 asin ( U bsin ( U (4 csin(u and the function w = w(x 1 x x ;t satisfies ( w x i ( w = t w x i w = 0 (5 t where is a nonzero constant. The proof of this proposition is similar to that of References [17 1]. From this proposition we find that ( is now separated into two groups of differential equations namely (4 and (5. Once we have the exact solutions of (4 and (5 the exact solutions of ( are obtained immediately from the transformation u(x 1 x x ;t=u[w(x 1 x x ;t]. We first solve the system of partial differential equations (5. Case 1. When = 1 (5 becomes ( w ( w w = x 1 t x w 1 t = 0 with w(x 1 t=c 1 (x 1 t (x 1 tc (6 4c 1 where c 1 and c are integral constants. Case. When (5 has the following solution w(x 1 x x ;t=h εδ = l j δ ( λ j l j ε = ( λ j x j εt c l j x j δt (7 λ j ( where h = h(x 1 x ;t is an arbitrary function λ j x j (j = 1 ε δ c are all constants. In what follows we solve the ODE (4 by using special transformations [19 5] and direct integral technique. Multiplying both sides of ODE (4 with U we have U U = U [ ( 1 U asin ( ] U bsin csin(u. (9 Then integrating (9 and letting the integral constant be C 1 we arrive at (U = [ ( U ccos(u acos ( ] (10 U bcos C 1. In order to reduce (10 to a solvable equation we introduce a basic transformation of field U as U = arccos(v A (11 with v being a function of w and A being a constant. The substitution of (11 into (10 yields (v = ( 40A 9 v5 9 4 v 4 9 ( 16A 9 4 0A v 9
Y. Liu et al. Exact Solutions of (1-Dimensional Triple Sine-Gordon Equation 1 4A A ( 0A 9 1 9 C 1 v ( 4 9 4A 4A 16A 9 40A4 9 9 C 1A 9 4A 4A 9 A A5 9 4A4 9K 1 9 C 1A which can be further rewritten as v α 4 = 4d 4( d β 5d 4 β 0 β a = 9(4d 4d 0d β 15d 0 β β b = 9(d 0β d (d 5d 0 β 16β c = 9(d 0β d β with (v = α 0 α 1 vα v α v α 4 v 4 α 5 v 5 (1 α 0 = 9 4A 4A 9 A A5 9 4A4 9K 1 9 C 1A α 1 = 4 9 4A 4A 16A 9 40A4 9 9 C 1A α = 4A A 0A 9 1 9 C 1 α = 16A 9 4 0A 9 α 4 = 40A 9 4 9 α 5 = 9. According to the previous results on the φ 5 model [6] with some special parameters we can express the solutions of the five degree ODE (1 as v(w=β 0 β 1 f (wβ f (w. (1 Substituting (1 into (1 we want to transform (1 into f w = d 0 d 1 f d f d f d 4 f 4 (14 so the parameters must be satisfied α 1 = 4 [ 5d β 4 β0 4 d 4 β0 d β 6β 0 d 0 d 4 β d 0β 4 4β 0d 0 d β ] α = 4 [ 10d4 β β0 1d 4β0 d β 6β 0 d 0 d 4 β β 0 d β d 0 d β ] α = 4(10d 4 β 0 d 4β 0 d β d 0 d 4 β d β β C 1 = 9(4d 1d 0d β 1d 0 β β β 1 = 0 α 5 = 4d 4 β d 1 = d = 0 d 4 = 1 4 β d 0 1 d β A = β 0 1 with β 0 β d 0 d being arbitrary constants. From these relations we find f = f (w satisfies f w = d 0 d f ( 14 β d 0 1 d β f 4 (15 and v = v(w satisfies v = β 0 β f. (16 Solving (15 by direct integral we have its implicit exact solution as ( (d0 β d arctan (d 0 β d f d 0 (d 0 β d w (d 0 β d d 0 (d 0 β d ( β d arctanh f β C 1 = 0. d 0 β d (17 Combining (16 with (11 the implicit exact solution for the (1 -dimensional ( is obtained as u(x 1 x x ;t=arccos(β f 1 (1 where f = f (w satisfies (17 and w = w(x 1 x x ;t are determined by (6 (. In what follows we choose special parameters d 0 d β to analyze the physical properties of our implicit exact solution.
Y. Liu et al. Exact Solutions of (1-Dimensional Triple Sine-Gordon Equation (i If the constants d 0 d β in (17 are taen as d 0 = 1 d = β = 4 then the coefficients a b c of the (1 -dimensional ( are a = 7 b = 7 c = 9 (19 and its implicit exact solution reads u(x 1 x x ;t=arccos(4 f 1 (0 with f = f (w satisfying w 1 arctan( f 6 arctanh( f 90 = 0. (1 (ii If the constant d 0 d β in (17 are taen as d 0 = 1 d = 7 β = 4 then the coefficients a b c of the (1 -dimensional ( are a = 15 b = 177 16 79 c = ( and its implicit exact solution is just (0 with f = f (w satisfying w 11 arctan( f 11 arctanh( f 1665 = 0. ( (iii If the constant d 0 d β in (17 are taen as d 0 = 1 d = 1 β = 4 then the coefficients a b c of the (1 -dimensional ( are a = 567 1 b = 16 c = 9 (4 and its implicit exact solution is just (0 with f = f (w satisfying w 1 arctan( f arctanh( f 69 = 0. (5 (iv If the constant d 0 d β in (17 are taen as d 0 = 1 d = 7 β = 4 then the coefficients a b c of the (1 -dimensional ( are a = 15 b = 177 16 79 c = (6 and its implicit exact solution is just (0 with f = f (w satisfying w 11 arctan( f 11 arctanh( f 1665 = 0. (7 When = 1 ( becomes the (11-dimensional u tt u xx = 1 ( u asin ( u bsin csin(u ( by replacing x 1 with x. In this case we have w = c 1 (x t 4c 1 (x tc. Therefore for the coefficients a b c in (19 ( (4 (6 the exact solutions of the (11-dimensional ( is (0 with f described by (1 ( (5 (7 respectively. When = we arrive at the (1-dimensional u tt u xx u yy = 1 ( u asin ( u bsin csin(u (9 by replacing x 1 x with x y respectively. In this case we have w = h(λ 1 x λ y εt c (l 1 x l y δt (0 l 1 l δ where h( is an arbitrary function and εδ = λ 1 l 1 λ l ε = λ1 λ. Therefore for the coefficients a b c in (19 ( (4 (6 the exact solutions of the (1-dimensional (9 is (0 with w described by (0 and f described by (1 ( (5 (7 respectively. When = we arrive at the (1-dimensional u tt u xx u yy u zz = 1 asin ( u bsin ( u csin(u (1 by replacing x 1 x x with x y z respectively. In this case we have w = h(λ 1 x λ y λ z εt c (l 1 x l y l z δt ( l1 l l δ where h( is an arbitrary function and εδ = λ 1 l 1 λ l λ l ε = λ1 λ λ. Therefore for the coefficients a b c in (0 ( (5 (7 the exact solutions of the (1-dimensional ( is (1 with w described by ( and f described by ( (4 (6 ( respectively.
Y. Liu et al. Exact Solutions of (1-Dimensional Triple Sine-Gordon Equation Remar. We have extended the separation transformation approach to the (1 -dimensional triple Sine-Gordon equation and derived the implicitly exact solutions of this equation. It has been shown that when > 1 there is an arbitrary function h = h ( λ jx j εt c in the implicitly exact solution of the (1 -dimensional triple Sine-Gordon equation which maybe lead to abundant localized structures [7 ]. We conclude that although most highdimensional nonlinear equations are non-integrable they may have rich localized nonlinear structures.. Conclusion In conclusion a special type of implicitly exact solution with an arbitrary function for the (1 - dimensional triple Sine-Gordon equation is proposed by extending the separation transformation method. The special exact solution may be useful to explain certain localized nonlinear phenomena in some scientific fields. Especially the result in this paper can be applied to describe the propagation of strictly resonant sharp line optical pulses in the near future. Acnowledgements This wor is supported by the ational atural Science Foundation of China (Grant o. 7100115 the Ministry of Education of the PRC for young people who are devoted to the researches of humanities and social sciences (Grant o. 09YJC9075 and is partially supported by the Third-stage of 11 Projects Foundation of Central University of Finance and Economics-Subject Construction Foundation of Central University of Finance. [1] V. E. Zaharov L. A. Tahtajan and L. D. Faddeev Dolady A SSSR 19 14 (197. [] R. K. Dodd and R. K. Bullough Proc. Roy. Soc. London Ser. A. 5 41 (1977. [] G. Liebbrandt Phys. Rev. Lett. 41 45 (197. [4] K. Mai and P. Kumar Phys. Rev. B 14 11 (1976 and 14 90 (1976. [5] R. K. Dodd R. K. Bullough and S. Ducworth J. Phys. A L64 (1975. [6] M. R. Rice Solitons and condensed matter physics. A. R. Bishop and T. Schneider (ed. Springer Berlin Heidelberg ew Yor 197 p. 46. [7] M. Remoissenet J. Phys. C 14 L5 (191. [] J. S. Yang and S. Y. Lou Chin. Phys. Lett. 1 60 (004. [9] R. K. Bullough P. J. Caudrey and H. M. Gibbs Solitons R. K. Bullough and P. J. Caudrey (ed. Springer Berlin 190 p. 107. [10] R. K. Bullough and P. J. Caudrey onlinear Evolution Equations Solvable by the Spectral Transform F. Calogero (ed. Pitman London 197 p. 10. [11] D. S. Wang J. Math. Phys. 50 151 (009; D. S. Wang Appl. Math. Lett. 665 (010; D. S. Wang D. Zhang and J. Yang J. Math. Phys. 51 0510 (010. [1]. J. Zabusy and M. D. Krusal Phys. Rev. Lett. 15 40 (1965. [1] M. J. Ablowitz and H. Segur Soliton and the inverse scattering transformation SIAM Philadelphia PA 191. [14] M. Wadati Y. H. Ichiawa and T. Shimizu Prog. Theor. Phys. 64 1959 (190. [15] H. C. Hu S. Y. Lou and K. W. Chow Chaos Solitons and Fractals 1 11 (007. [16] H. C. Hu and S. Y. Lou Phys. Scr. 75 4 (007. [17] D. S. Wang Z. Yan and H. Li Comput. Math. Appl. 56 1569 (00. [1] Z. Yan Phys. Lett. A 61 (007; Z. Yan Phys. Scr. 75 0 (007. [19] D. S. Wang and H. Q. Zhang Z. aturforsch. 60a 1 (005. [0] D. S. Wang and H. Q. Zhang Chaos Solitons and Fractals 5 601 (005. [1] D. S. Wang and H. Li Appl. Math. Comput. 1 76 (007. [] D. S. Wang and H. Li J. Math. Anal. Appl. 4 7 (00. [] D. S. Wang Comput. Math. Appl. 5 146 (009; D.S. Wang and Z. F. Zhang J. Phys. A: Math. Theor. 4 0509 (009. [4] Z. Yan Phys. Lett. A 1 19 (004. [5] Z. Yan Chaos Solitons and Fractals 16 759 (00. [6] J. S. Yang and S. Y. Lou Z. aturforsch. 54a 195 (1999. [7] D. S. Wang X. H. Hu J. Hu and W. M. Liu Phys. Rev. A 1 05604 (010. [] D. S. Wang X. H. Hu J. Hu and W. M. Liu Phys. Rev. A 061 (010.