Extended tanh-function method and its applications to nonlinear equations

4 December 000 Physics Letters A 77 (000) 1 18 www.elsevier.nl/locate/pla Extended tanh-function method and its applications to nonlinear equations Engui Fan Institute of Mathematics Fudan University Shanghai 00433 PR China Received 6 July 000; received in revised form 5 October 000; accepted 5 October 000 Communicated by C.R. Doering Abstract An extended tanh-function method is proposed for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The key idea of this method is to take full advantages of a Riccati equation involving a parameter and use its solutions to replace the tanh function in the tanh-function method. It is quite interesting that the sign of the parameter can be used to exactly judge the numbers and types of these travelling wave solutions. In addition by introducing appropriate transformations it is shown that the extended tanh-function method still is applicable to nonlinear PDEs whose balancing numbers may be any nonzero real numbers. Some illustrative equations are investigated by this means and new travelling wave solutions are found. 000 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear partial differential equation; Travelling wave solution; Riccati equation; Symbolic computation 1. Introduction In recent years directly searching for exact solutions of nonlinear PDEs has become more and more attractive partly due to the availability of computer symbolic systems like Maple or Mathematica which allow us to perform some complicated and tedious algebraic calculation on a computer as well as help us to find new exact solutions of PDEs [1 9]. One of most effectively straightforward methods to construct exact solution of PDEs is tanh-function method [3 7 10 1]. Recently Parkes and Duffy have further developed a powerful automated tanh-function method (ATFM) that automates the tanh-function method and in which the Mathematica package ATFM can deal with the tedious algebraic computation and output di- E-mail address: faneg@fudan.edu.cn (E. Fan). rectly the required solutions of nonlinear equations [3 5]. Other generalizations of tanh-function method have been carried out by Tian Gao and Gudkov [89 1314]. Let us simply describe the tanh-function method. Consider a given PDE say in two variables H(uu t u x u xx...)= 0. (1.1) The fact that the solutions of many nonlinear equations can be expressed as a finite series of tanh functions motivates us to seek for the solutions of Eq. (1.1) in the form m u(x t) = U(z)= a i w i (1.) i=0 where w(xt) = tanh(kz) z = x + ct m is a positive integer that can be determined by balancing the linear term of highest order with the nonlinear term in 0375-9601/00/$ see front matter 000 Elsevier Science B.V. All rights reserved. PII: S0375-9601(00)0075-8

E. Fan / Physics Letters A 77 (000) 1 18 13 Eq. (1.1) [11115] and kca 0...a m are parameters to be determined. Substituting (1.) into (1.1) will yield a set of algebraic equations for kca 0...a m because all coefficients of w i have to vanish. From these relations kca 0...a m can be obtained. There are two questions involving this method. First the balancing number m in (1.) is required to be a positive integer since it represents the number of terms ofaseries.butwefindthatm may not be a positive integer for some nonlinear PDEs. In this case it was shown that these equations still can be dealt with by a tanh-function method [3]. Second we can see that only tanh-type solitary wave solutions of Eq. (1.1) are obtained by tanh-function method. If tanh(kz) is replaced by another function (for example tan(kz)) maybe a tan-type travelling wave solution is obtained. But it involves similarly repetitious calculation many times [1617]. So it is natural to ask whether there is a method to directly and uniformly construct multiple travelling wave solutions of Eq. (1.1)? The purpose of this paper is to present an extended tanh-function method and give this question an affirmative answer.. Method and its applications The key idea of our method is to take full advantage of a Riccati equation and use its solutions to replace tanh(kz) in (1.). Other processes are similar to tanhfunction method [3 710 1]. The desired Riccati equation reads w = b + w (.1) where := d/dz andb is a parameter to be determined. By using (.1) repeatedly we can express all derivatives of w in term of series in w. It is found that the Riccati equation (.1) admits several types of solutions { w = b tanh bz as b<0 (.) b coth bz as b<0 w = 1 as b = 0 (.3) z { w = b tan bz as b>0 (.4) b cot bz as b>0. The tanh function in (1.) is only a special function in (.) (.4) so we conjecture that Eq. (1.1) may admit other types of travelling wave solutions in (.) (.4) in addition to a tanh-type one. Moreover we hope to construct them in a unified way. For this purpose we shall use the Riccati equation (.1) once again to generate an associated algebraic system but not use one of functions in (.) (.4). Another advantage of the Riccati equation (.1) is that the sign of b can be used to exactly judge the type of travelling wave solution for Eq. (1.1). For example if b<0 we are sure that Eq. (1.1) admits tanh- and coth-type travelling wave solutions. Especially Eq. (1.1) will possess five types of travelling wave solutions if b is an arbitrary constant. In this way we can successfully recover the previously known solitary wave solutions that had been found by the tanh-function method and other more sophisticated methods. More importantly for some equations with no extra effect we also find other new and more general types of solutions at the same time. The algorithm presented here is also a mechanization method in which generating an algebraic system from Eq. (1.1) and solving it are two key procedures and laborious to do by hand. But they can be implemented on computer with help of Mathematica. The outputs of solving the algebraic system from computer comprise a list of the form {bca 0...}. In general if b or any of the parameter is left unspecified then it is to be regarded as arbitrary for the solution of Eq. (1.1). Especially we are concerned with the sign of bwhich shows the number and types of travelling wave solutions for a PDE. Further we deal with the case in which the balancing number m is not a positive integer. Following the idea [3] we can introduce a transformation u = v 1/m and change Eq. (1.1) into another equation for v whose balancing number will be a positive integer. Then it can be dealt with by the above method. In the following we illustrate this method by considering some important equations. For simplification cothand cot-type travelling wave solutions are omitted in this paper since they always appear in pairs with tanhand tan-type solutions. Example 1. Consider the KdV Burgers Kuramoto equation [31819] u t + uu x + pu xx + qu xxx + ru xxxx = 0 (.5)

14 E. Fan / Physics Letters A 77 (000) 1 18 where p q and r are real constants. This equation occupies a prominent position in describing physical processes in motion of turbulence and other unstable systems [318 1]. To look for the travelling wave solution of Eq. (.5) we make transformation u(x t) = U(z) z = x + ct and change Eq. (.5) into the form cu + UU + pu + qu + ru (4) = 0. (.6) Balancing U (4) with UU yields m = 3. Therefore we may choose U = a 0 + a 1 w + a w + a 3 w 3. (.7) Substituting (.7) into Eq. (.6) and using Mathematica yields a set of algebraic equations for a 0 a 1 a a 3 b and c bca 1 + b qa 1 + ba 0 a 1 + b pa + 16b 3 ra + 6b 3 qa 3 = 0 bpa 1 + 16b ra 1 + ba1 + bca + 16b qa + ba 0 a + 6b pa 3 + 10b 3 ra 3 = 0 ca 1 + 8bqa 1 + a 0 a 1 + 8bpa + 136b ra + 3ba 1 a + 3bca 3 + 60b qa 3 + 3ba 0 a 3 = 0 pa 1 + 40bra 1 + a1 + ca + 40bqa + a 0 a + ba + 18bpa 3 + 576b ra 3 + 4ba 1 a 3 = 0 6qa 1 + 6pa + 40bra + 3a 1 a + 3ca 3 + 114bqa 3 + 3a 0 a 3 + 5ba a 3 = 0 4ra 1 + 4qa + a + 1pa 3 + 816bra 3 + 4a 1 a 3 + 3ba3 = 0 10ra + 60qa 3 + 5a a 3 = 0 360ra 3 + 3a3 = 0 for which with the aid of Mathematicawefind a 0 = c a = 0 q = 0 b= 0 a 0 = c a = 0 q = 0 a 1 = 60p 19 a 3 = 10r a 1 = 90p 19 a 3 = 10r b= p 76r (.8) (.9) a 0 = c a 1 = 70p 19 a = 0 a 3 = 10r q = 0 b= 11p 76r a 0 = c 11p r a 1 = 30p a = 60 r a 3 = 10r q =±4 r b = p 4r a 0 = c ± 9p r a 1 = 30p a = 60 r a 3 = 10r q =±4 r b = p 4r a 0 = c ± 45p 47 47r r a = 180 47 a 3 = 10r r q =±1 a 1 = 90p 47 47 b= p 188r a 0 = c ± 60p 73 73r r a = 40 73 a 3 = 10r r q =±16 73 b= p 9r a 1 = 150p 73 (.10) (.11) (.1) (.13) (.14) where c is arbitrary constant. In the following according to the sign of b we discuss the multiple travelling wave solutions that Eq. (.5) possesses. Since b = 0 in (.8) according to (.3) we obtain a rational-type travelling wave solution u 1 = c + 60p 19z + 10r z= x + ct z3 (q = 0) As pr < 0 we see that b<0 in (.9) and b>0in (.10). According to (.) and (.4) the corresponding travelling wave solutions respectively are u = c 15rk 3 tanh(z) [ 3 tanh (z) ] p 19r z= 1 k(x + ct) (q = 0)

E. Fan / Physics Letters A 77 (000) 1 18 15 [ ] 9 u 3 = c 15rk 3 tan(z) 11 + tan (z) Example. For the two-dimensional KdV Burgers equation [114] 11p 19r z= 1 k(x + ct) (q = 0). (u t + uu x + pu xxx qu xx ) x + ru yy = 0 (.15) we consider its travelling solution u(x t) = U(z) z = As pr > 0 we see that b<0 in (.10) (.1) x + dy + ct then Eq. (.15) reduces to (.13) and (.14); b>0 in (.9) and (.11). For these cases by using (.) and (.4) we have (cu + UU + pu qu ) + rd U = 0. (.16) [ ] 9 u 4 = c 15rk 3 tanh(z) 11 tanh (z) Balancing U (3) with UU gives m = so we may choose 11p 19r z= 1 k(x + ct) (q = 0) U = a 0 + a 1 w + a w. (.17) u 5 = c ± 9rk 3 15rk 3 Substituting (.17) into (.16) and using Mathematica tanh(z) [ 1 ± tanh(z) tanh (z) ] engenders the following set of algebraic equations for a 0 a 1 a b and c: p r z= 1 k(x + ct) ( ) q =±4 pr b qa 1 + b a1 + b ca + 16b 3 pa u 6 = c ± 45rk 3 + 15rk 3 + b d ra + b a 0 a = 0 tanh(z) [ 3 3tanh(z) + tanh (z) ] bca 1 + 16b pa 1 + bd ra 1 + ba 0 a 1 47r z= 1 16b qa + 6b a 1 a = 0 k(x + ct) 8bqa 1 + 4ba1 + 8bca + 136b pa + 8bd ra ( ) pr + 8ba 0 a + 6b a q =±1 = 0 47 ca 1 + 40bpa 1 + d ra 1 + a 0 a 1 40bqa u 7 = c ± 60rk 3 + 15rk 3 tanh(z) [ 5 4tanh(z) + tanh (z) ] + 18ba 1 a = 0 6qa 1 + 3a1 + 6ca + 40bpa + 6d ra p 73r z= 1 + 6a 0 a + 16ba k(x + ct) = 0 ( 4pa ) 1 4qa + 1a 1 a = 0 pr q =±16 10pa + 10a = 0. 73 u 8 = c 15rk 3 tan(z) [ 3 + tan (z) ] Solving them by means of Mathematica gives 19r z= 1 a 0 = c + q k(x + ct) (q = 0) 5p d r a 1 = 1q 5 a = 1p b = 0 (.18) u 9 = c 11rk 3 15rk 3 tan(z) [ 1 ± tan(z) + tan (z) ] a 0 = c + 3q 5p d r a 1 = 1q 5 r z= 1 k(x + ct) ( ) q =±4 pr. a = 1p b = q (.19) 100p Among these solutions u u 4 u 5 u 6 and u 7 were ever found by Parkes and Duffy [3]. For special case q = 0 we can find that u 1 u u 3 u 4 and u 8 satisfy the Kuramoto Sivashinsky equation [180] u t + uu x + pu xx + ru xxxx = 0. where c and d are arbitrary constants. Since b 0 so from (.18) and (.19) it follows that u 1 = c + q 5p d r 1q 5z 1p z z = x + dy + ct

16 E. Fan / Physics Letters A 77 (000) 1 18 u = 3q [ ] 6q 1 + tanh(z) + 5p 5p ( c + d r ) z = q (x + dy + ct). (.0) 10p Obviously these solutions contain two arbitrary speeds c and d. Making transformation d (5p/q)d c + d r c then the solution (.0) is exactly the same with that in Ref. [1]. Example 3. Consider the Variant Boussinesq equation [] u t + v x + uu x + pu xxt = 0 v t + (uv) x + qu xxx = 0. (.1) (.) Let u(x t) = U(z) v(xt) = V(z) z = x + ctthen Eqs. (.1) and (.) become cu + V + UU + pu = 0 cv + (UV ) + qu = 0. (.3) (.4) Balancing U with UU andu with (UV ) leads to the following ansatz: U = a 0 + a 1 w + a w V = b 0 + b 1 w + b w. Substituting (.5) into (.3) and (.4) gives bca 1 + b cpa 1 + ba 0 a 1 + bb 1 = 0 ba1 + bca + 16b cpa + ba 0 a + bb = 0 ca 1 + 8bcpa 1 + a 0 a 1 + 3ba 1 a + b 1 = 0 a1 + ca + 40bcpa + a 0 a + ba + b = 0 6cpa + 3a 1 a = 0 4cpa + a = 0 b qa 1 + ba 1 b 0 + bcb 1 + ba 0 b 1 = 0 16b qa + ba b 0 + ba 1 b 1 + bcb + ba 0 b = 0 8bqa 1 + a 1 b 0 + cb 1 + a 0 b 1 + 3ba b 1 + 3ba 1 b = 0 40bqa + a b 0 + a 1 b 1 + cb + a 0 b + 4ba b = 0 6qa 1 + 3a b 1 + 3a 1 b = 0 4qa + 4a b = 0. (.5) With the help of Mathematicawefind a 0 = a + 8pba + 7pq 1pa a 1 = b 1 = 0 b 0 = 4(9q bqa ) a b = 6q c = a 1p (.6) where b and a being arbitrary constants. From (.) (.4) and (.6) we obtain three types of travelling wave solutions: [ ( b u 1 = a 0 a b tanh x a )] 1p t (b < 0) [ ( b v 1 = b 0 + 6qbtanh x a )] 1p t (b < 0) a u = a 0 + ( x a 1p t) (b = 0) v = b 0 6q ( x a 1p t) (b = 0) u 3 = a 0 + 4b tan [ b ( x a 1p v 3 = b 0 6qbtan [ b ( x a 1p where a 0 and b 0 are given by (.6). ) ] t (b > 0) ) ] t (b > 0) In the following we consider two equations whose balancing numbers are not positive integers and show how they are dealt with by the extended tanh-function method. Example 4. Consider the generalized Burgers Fisher equation [33] u t + pu r u x u xx qu ( 1 u r) = 0 (.7) where p q and r are some parameters. Let u(x t) = U(z) z = x + ct then Eq. (.7) reduces to cu + pu r U U qu ( 1 U r) = 0. (.8) Obviously balancing U with U r U gives m = 1/r which is not an integer as r 1. But we need the balancing number to be a positive integer so

E. Fan / Physics Letters A 77 (000) 1 18 17 as to apply formulas (1.) and (.1). We make a transformation U = V 1/r and change Eq. (.8) into the form qr V (1 V)+ (1 r)v rv(cv + pv V V ) = 0. (.9) Now balancing V with VV in Eq. (.9) we find m = 1. So we can assume that U = a 0 + a 1 w. Substituting (.30) into (.9) yields qr a0 + qr a0 3 + bcra 0a 1 + bpra0 a 1 b a1 + b ra1 = 0 bra 0 a 1 qr a 0 a 1 + 3qr a0 a 1 + bcra1 + bpra 0 a1 = 0 cra 0 a 1 + pra0 a 1 ba1 qr a1 + 3qr a 0 a1 + bpra1 3 = 0 ra 0 a 1 + cra1 + pra 0a1 + qr a1 3 = 0 a1 ra 1 + pra3 1 = 0 which have a solution by means of Mathematica a 0 = 1 a 1 = 1 + r pr b = p r 4(1 + r) + q(1 + r) c= p. p(1 + r) Since b<0 it follows that { 1 u = 1 [ tanh 1 (1 + r) (x p + q(1 + r) )] } 1/r t. p(1 + r) (.30) Example 5. Consider the nonlinear heat conduction equation [4] u t ( u ) (.31) xx = pu qu where pq 0 are constants. Let u(x t) = U(z) z = x + ct then Eq. (.31) reduces to cu ( U ) pu + qu = 0. (.3) Balancing between U and (U ) yields m = 1 which is not a positive integer. Let U = V 1 then Eq. (.3) becomes pv 3 6V + V (q cv ) + VV = 0. (.33) Now balancing terms V V and VV gives the desired balancing number m = 1. In this case we can assume that V = a 0 + a 1 w. Substituting (.34) into (.33) yields qa0 pa3 0 bca 0 a 1 6b a 1 = 0 4ba 0 a 1 + qa 0 a 1 3pa0 a 1 bca 0 a1 = 0 ca0 a 1 8ba1 + qa 1 3pa 0a1 bca3 1 = 0 4a 0 a 1 ca 0 a1 pa3 1 = 0 a1 ca3 1 = 0 which have a solution a 0 = q p a 1 =± q p c = b= q q 16. (.34) Since we require a 1 and c to be real numbers which imply b<0 we have { [ q u 1 = p q ± q tanh (x ± p )] } 1 q. 4 3. Summary and discussion We have proposed an extended tanh-function methodandusedittosolvesome(1 + 1)- and( + 1)- dimensional nonlinear PDEs. In the fact this method is readily applicable to a large variety of nonlinear PDEs. In contrast to tanh-function method some merits are available for our method. First it can be used to construct multiple travelling wave solutions of nonlinear equations in a unified way by which we can successfully recover the previously known solitary wave solutions that had been found by the tanh-function method. More importantly for some equations with no extra effect we also pick up other new and more general types of solutions at the same time. The travelling wave solutions derived in this paper include soliton solutions periodical solutions rational solutions and singular solutions. The physical

18 E. Fan / Physics Letters A 77 (000) 1 18 relevance of soliton solutions and periodical solutions is clear to us. The rational solutions are a disjoint union of manifolds and the particle system describing the motion of pole of rational solutions of KdV and Boussinesq equation was analyzed in [5 8]. We also can see that some solutions develop singularity at a finite point i.e. for any fixed t = t 0 there exist x 0 at which these solutions blow up. There is much current interest in the formation of so called hot spots or blow-up of solutions [199 3]. It appears that these singular solutions will model this physical phenomena. Second it is quite interesting that we can use the sign of b in a Riccati equation to judge the numbers and types of these travelling wave solutions that a equation possesses. Third this method is also a computerizable method which allow us to perform complicated and tedious algebraic calculation on computer. By introducing a appropriate transformation it is shown that this method is also applicable to equations whose balancing numbers are any non-zero real number which further enlarges application of this method to nonlinear PDEs. Due to the complexity of nonlinear PDEs we also aware of the fact that not all fundamental equations can be treated with our method. For example the nonlinear Schrödinger and sine Gordon equations are notable exceptions although their solutions contain hyperbolic functions as well. We are investigating how our method is further improved to treat complicated and other kinds of nonlinear PDEs. Acknowledgements I am grateful to Professor Gu Chaohao Professor Hu Hesheng and Professor Zhou Zixiang for their enthusiastic guidance and help. I also would like to express my sincere thanks to referees for their many helpful advice and suggestions. This work has been supported by the Postdoctoral Science Foundation of China Chinese Basic Research Plan Nonlinear Science and Mathematics Mechanization and a Platform for Automated Reasoning. References [1] W. Hereman M. Takaoka J. Phys. A 3 (1990) 4805. [] W. Hereman Comp. Phys. Commun. 65 (1991) 143. [3] E.J. Parkes B.R. Duffy Comp. Phys. Commun. 98 (1996) 88. [4] B.R. Duffy E.J. Parkes Phys. Lett. A 14 (1996) 71. [5] E.J. Parkes B.R. Duffy Phys. Lett. A 9 (1997) 17. [6] Z.B. Li Exact solitary wave solutions of nonlinear evolution equations in: X.S. Gao D.M. Wang (Eds.) Mathematics Mechanization and Application Academic Press 000. [7] Z.B. Li M.L. Wang J. Phys. A 6 (1993) 607. [8] B. Tian Y.T. Gao Mod. Phys. Lett. A 10 (1995) 937. [9] Y.T. Gao B. Tian Comp. Math. Appl. 33 (1997) 115. [10] S. Lou G. Huang H. Ruan J. Phys. A 4 (1991) L584. [11] W. Malfliet Am. J. Phys. 60 (199) 650. [1] E.J. Parkes J. Phys. A 7 (1994) L497. [13] V.V. Gudkov Comp. Math. Math. Phys. 36 (1996) 335. [14] V.V. Gudkov J. Math. Phys. 38 (1997) 4794. [15] M.L. Wang Z.B. Li Phys. Lett. A 16 (1996) 67. [16] J.F. Zhang Int. J. Theor. Phys. 35 (1996) 1793. [17] E.G. Fan H.Q. Zhang Phys. Lett. A 46 (1998) 403. [18] N.A. Kudryashov Phys. Lett. A 147 (1990) 87. [19] N.A. Kudryashov D. Zargaryan J. Phys. A 9 (1996) 8067. [0] G.I. Sivashinsky Physica D 4 (198) 7. [1] T. Kawahara Phys. Rev. Lett. 51 (1983) 381. [] R.L. Sachs Physica D 30 (1988) 1. [3] J. Satsuma in: M. Abolowitz B. Fuchssteiner M. Kruskal (Eds.) Topics in Soliton Theory and Exactly Solvable Nonlinear Equations World Scientific Singapore 1987. [4] H. Whilemsson Phys. Rev. A 36 (1990) 965. [5] M. Airault H. Mckean J. Moser Commun. Pure Appl. Math. 30 (1977) 95. [6] M. Adler J. Moser Commun. Math. Phys. 19 (1978) 1. [7] A. Nakamura R. Hirota J. Phys. Soc. Jpn. 54 (1985) 491. [8] R.L. Sachs Physica D 30 (1988) 1. [9] C.J. Coleman J. Aust. Math. Soc. Ser. B 33 (199) 1. [30] N.F. Smyth J. Aust. Math. Soc. Ser. B 33 (199) 403. [31] P.A. Clarkson E.L. Mansfield Physica D 70 (1993) 50. [3] E.G. Fan H.Q. Zhang Phys. Lett. A 45 (1998) 389.