New explicit exact solutions for the Lienard equation and its applications

New explicit exact solutions for the Li ´ enard equation and it s applications Gui-Qiong Xu ∗ Department of Information Managemen t, Colleg e of Ma n agemen t, Shanghai Univ ersity , Shanghai 200444, PR China Abstract In this letter, new exact exp licit solutions are obtained for the Li´ enard equation, and th e ap - plications of the results to the generalized P o chhammer-Chree equation, the Kun du equation and the generalized long-short wa v e r esonance equations are presente d . P A CS : 02.30.Jr; 04.20.Jb Keyw ords : Li ´ enard equation; P erio dic w a ve; Solita r y w a ve; P o c hh ammer-Chree equation; Kun du equation 1. In tro duction Nonlinear partial differen tial equations (NLPDEs) describ e v arious nonlinear phenomena in natural an d applied sciences suc h as fluid dynamics, plasma physics, solid state physics, optical fib ers, acoustics, mec hanics, biology and mathematical finance. It is of significan t imp ortance to construct exact solutions of NLPDEs from b oth theoretical and practical p oin ts of view. Up to now, man y p o w erf ul metho ds for solving NLPDEs hav e b een p rop osed, s uc h as the inv erse scattering metho d[1], B¨ ac klund and Darb oux tr ansform[2]-[3], Hirota ’s bilinear method[4], truncated painlev´ e expansion metho d[5]-[10], homogeneous balance metho d[11], v ariational iteration metho d[12], h o- motop y p erturbation metho d[13], tanh-fu nction metho d[14], Jacobian elliptic fun ction expan s ion metho d[15]-[19], F an su b-equation method [20]-[22], auxiliary equation metho d[23]-[25 ], F-expansion metho d[26]-[28]and so on. The last fiv e metho ds mentioned ab o v e b elong to a class of method called subsidiary ordinary differen tial equ ation metho d(sub -O DE metho d f or short). Th e sub-ODE method whic h were often used the Riccati equation, Jacobian elliptic equation, p ro jectiv e Riccati equation, etc. In this letter, ∗ E-mail address: xugq@staff.shu.edu.cn (G.-Q. Xu) 1 w e choose the L i ´ enard equation a ′′ ( ξ ) + l a ( ξ ) + m a 3 ( ξ ) + n a 5 ( ξ ) = 0 , lm n 6 = 0 , (1) as th e sub s idiary ordinary differen tial equation. By m eans of some pr op er transformations, a n u m b er of NLPDEs with strong nonlin ear terms can b e reduced to Eq.(1), thus s eeking explicit exact solutions of th ese nonlinear equations can b e attributed to solv e (1). Therefore, to searc h for exact s olutions of the Li ´ enard equation (1) is a v ery imp ortant job and it h as attracted muc h atten tion. F or example, Behera and Khare[29] has shown that the exact solution of Eq.(1) can b e expressed in terms of the W eierstrass f unction. Dey et al.[30] inv estigated Eq.(1) and established the exact solution of a one-parameter family of generalized Li´ enard equation with p th order nonlin earit y b y mapping it to the field equatio n of the φ 6 -field theo ry . B y mea n s of differen t met ho ds, Kong[31], Zhang[32]-[33] and F en g[34 ]-[36] h a ve giv en some exp licit exact solitary w av e solutions of Eq.(1). In Refs.[32]-[36 ], Zhang and F eng deriv ed three kind s of so litary w a ve s olutions of Eq.(1) as follo ws: If l < 0 , m > 0 , n ≤ 0 or l < 0 , m ≤ 0 , n > 0, Eq .(1 ) p ossesses the solitary w a ve solution, a 1 ( ξ ) = ±       4 r 3 l 2 3 m 2 − 16 nl sec h 2 √ − l ξ 2 + − 1 + √ 3 m √ 3 m 2 − 16 nl ! sec h 2 √ − l ξ       1 2 . (2) If l < 0 , m > 0 and 3 m 2 − 16 nl = 0 , Eq.(1) admits exact solutions, a 2 ( ξ ) = ±  − 2 l m ( 1 + tanh ( √ − l ξ )  1 2 , a 3 ( ξ ) = ±  − 2 l m ( 1 − tanh ( √ − l ξ )  1 2 . (3) The v arious metho d s used in [29]-[36] are very us efu l and the applications of the solutions of the Li´ enard equ ation to some imp ortan t NLPDEs are quite p erfect. Ho we v er, it is natural to ask whether Eq.(1) can supp ort other new e x act solutions. The p resen t letter is motiv ated by the desire to impro ve the work m ad e in [31]-[36] b y in tro du cing more solutions of Eq.(1) including all the solutions giv en in [31]-[36 ] b ut also other formal solutions. The rest of this letter is organized as follo ws. In Section 2, we find some new exact solutions for the Li ´ enard equation (1). In Section 3, w e use these sp ecial solutions to solv e the generalized P o chhammer-Chree equation, the Kundu equation and th e generalized long-short wa v e resonance equations. An d w e conclude the letter in the last section. 2 2. New exact solutions of the Li ´ enard equation Generally sp eaking, it is difficult to gi v e the general solution of Eq.(1). In what follo w s, we will consider some sp ecial cases. Based on Refs.[31]-[36 ], w e can ha ve the fol lo wing solutions of Eq.(1), a 1 ± ( ξ ) = ± " − 4 l m + ǫ p m 2 − 16 nl / 3 cosh(2 √ − l ξ ) # 1 2 , m 2 − 16 nl / 3 > 0 , l < 0 , (4a) a 2 ± ( ξ ) = ± " − 4 l m + ǫ p 16 nl/ 3 − m 2 sinh(2 √ − l ξ ) # 1 2 , m 2 − 16 nl / 3 < 0 , l < 0 , (4b) a 3 ± ( ξ ) = ±  − 2 l m  1 + ǫ tanh( √ − l ξ )   1 2 , m 2 − 16 nl / 3 = 0 , m > 0 , l < 0 , n < 0 , (4c) a 4 ± ( ξ ) = ±  − 2 l m  1 + ǫ coth( √ − l ξ )   1 2 , m 2 − 16 nl / 3 = 0 , m > 0 , l < 0 , n < 0 , (4d) a 5 ± ( ξ ) = ± " − 4 l m + ǫ p m 2 − 16 nl / 3 cos(2 √ l ξ ) # 1 2 , m 2 − 16 nl / 3 > 0 , l > 0 , (4e) where ǫ = ± 1. It is easily seen that a 3 ± ( ξ ) repr o duces t wo solutions giv en in E q.(3). T here is a tin y sym b olic error in the solution a 1 ( ξ )( the co efficien t of sec h 2 √ − l ξ in the numerator of fraction (2) should b e − 4 q 3 l 2 3 m 2 − 16 nl ). It is easily pro v ed th at the correct solution a 1 ( ξ ) and th e solution a 1 ± ( ξ ) with ǫ = 1 are actually the s ame and only different in the form. And the other solutions a 2 ± ( ξ ), a 4 ± ( ξ ) and a 5 ± ( ξ ) are firstly r ep orted h ere. T o our b est knowledge , the p erio dic wa ve solutions expressed in terms of J acobian elliptic function to Eq.(1) ha v e not b een c onsidered in existed literature. No w we assume JacobiSN( ξ , r ) = sn( ξ ), JacobiCN( ξ , r ) = cn( ξ ) and JacobiDN( ξ , r ) = dn( ξ ), and r is the mo du lus of Jacobian elliptic functions(0 ≤ r ≤ 1) . With the aid of sy mb olic computation soft w are suc h as MAPLE, after direct computations, we find thr ee kinds of elliptic p erio dic wa v e solutions of Eq.(1) when the parameter co efficien ts l, m, n satisfy certain conditions, a 6 ± ( ξ ) = ± " − 3 m 8 n 1 + ǫ sn √ 3 m 4 r √ − n ξ ! ! # 1 2 , l = 3 m 2 (5 r 2 − 1) 64 n r 2 , m > 0 , n < 0 , (5a) a 7 ± ( ξ ) = ± " − 3 m 8 n 1 + ǫ cn √ 3 m 4 r √ n ξ ! ! # 1 2 , l = 3 m 2 (4 r 2 + 1) 64 n r 2 , m < 0 , n > 0 , (5b) a 8 ± ( ξ ) = ± " − 3 m 8 n 1 + ǫ dn √ 3 m 4 √ n ξ ! ! # 1 2 , l = 3 m 2 ( r 2 + 4) 64 n , m < 0 , n > 0 . (5c) T o our k n o wledge, the solutions a 6 ± ( ξ ), a 7 ± ( ξ ) and a 8 ± ( ξ ) are firstly presented h ere. 3 It is we ll kno wn that there are many other Jacobian elliptic fu nctions wh ic h can b e generated b y sn( ξ ), cn ( ξ ) and dn( ξ ). F or the sak e of simplicit y , the solutions in terms of ns( ξ ), nd( ξ ), n c( ξ ), sc( ξ ), cs ( ξ ), sd( ξ ), d s( ξ ), cd( ξ ), dc( ξ ) are n ot considered here. 3. Applications Example 1. The generalized Pochhammer-Chree (PC) equation can b e written as u tt − u ttxx − ( a 1 u + a 3 u 3 + a 5 u 5 ) xx = 0 , (6) whic h describ es the propagation of longitudinal deform ation w a ves in an elastic ro d [37]. Zhang[32] and F eng[35] ha ve giv en some explicit solitary wa v e solutions of Eq.(6) by means of the metho d of solving algebraic equations. L i and Zhang[38] studied the bifu rcation problem of trav elling wa ve solutions for Eq.(6) by using the bifurcation theory of planar dyn amical s y s tems. In ord er to s olv e Eq .(6 ), its solutions ma y b e supp osed as: u ( x, t ) = u ( ξ ) , ξ = x − v t, (7) where v is a real constan t. Substituting ansatz (7) in to Eq.(6) yields, v 2 u ′′ ( ξ ) − v 2 u (4) ( ξ ) − ( a 1 u + a 3 u 3 + a 5 u 5 ) ξ ξ = 0 , (8) In tegrating Eq.(8) t w ice an d setting the integrat ion constan t to zero, we obtain u ′′ ( ξ ) + a 1 − v 2 v 2 u ( ξ ) + a 3 v 2 u 3 ( ξ ) + a 5 v 2 u 5 ( ξ ) = 0 . (9) Up to now, by means of the ansatz (7), we reduce the generalized PC equation (6) to the L i´ enard equation (1 ) for the case l = a 1 − v 2 v 2 , m = a 3 v 2 and n = a 5 v 2 . Su bstituting the solutions (4a)-(4e) and the solutions (5a)-(5c) of Eq.(1) into (7), we can obtain a series of exact trav elling wa ve solutions to Eq.(6) (where ǫ 1 = ± 1 and ǫ 2 = ± 1). When v 2 − a 1 > 0 and 3 a 2 3 − 16 a 5 ( a 1 − v 2 ) > 0, Eq.(6 ) has b ell-shap e solitary w a ve solution, u 1 ± ( x, t ) = ±     4( v 2 − a 1 ) a 3 + ǫ 1 p a 2 3 − 16 a 5 ( a 1 − v 2 ) / 3 cosh( 2 √ v 2 − a 1 v ξ )     1 2 . When v 2 − a 1 > 0 and 3 a 2 3 − 16 a 5 ( a 1 − v 2 ) < 0, Eq.(6 ) has the singular solitary wa v e solution, u 2 ± ( x, t ) = ±     4( v 2 − a 1 ) a 3 + ǫ 1 p 16 a 5 ( a 1 − v 2 ) / 3 − a 2 3 sinh( 2 √ v 2 − a 1 v ξ )     1 2 . 4 When a 3 > 0, a 5 < 0, v 2 − a 1 > 0 and 3 a 2 3 − 16 a 5 ( a 1 − v 2 ) = 0, E q.(6) has tw o kin k -sh ap e solitary wa ve solutions, u 3 ± ( x, t ) = ±  2( v 2 − a 1 ) a 3  1 + ǫ 1 tanh  √ v 2 − a 1 v ξ    1 2 , u 4 ± ( x, t ) = ±  2( v 2 − a 1 ) a 3  1 + ǫ 1 coth  √ v 2 − a 1 v ξ    1 2 . When v 2 − a 1 < 0 and 3 a 2 3 − 16 a 5 ( a 1 − v 2 ) > 0 , E q .(6 ) has the trigonometric fu nction solution, u 5 ± ( x, t ) = ±     4( v 2 − a 1 ) a 3 + ǫ 1 p a 2 3 − 16 a 5 ( a 1 − v 2 ) / 3 cos ( 2 √ a 1 − v 2 v ξ )     1 2 . When a 5 < 0 and a 3 > 0, Eq.(6) has th e Jacobian sine function solution, u 6 ± ( x, t ) = ± 1 2 " − 3 a 3 2 a 5 1 + ǫ 1 sn √ 3 a 3 4 r v √ − a 5 ξ ! ! # 1 2 , where v = ǫ 2 p a 5 (64 a 5 r 2 a 1 − 15 a 3 2 r 2 + 3 a 3 2 ) / (8 r a 5 ). When a 5 > 0 and a 3 < 0, Eq.(6) has tw o p erio d ic wa v e solutions. On e is u 7 ± ( x, t ) = ± 1 2 " − 3 a 3 2 a 5 1 + ǫ 1 cn √ 3 a 3 4 r v √ a 5 ξ ! ! # 1 2 , where v = ǫ 2 p a 5 (64 a 5 r 2 a 1 − 12 a 3 2 r 2 − 3 a 3 2 ) / (8 r a 5 ). An d another on e is u 8 ± ( x, t ) = ± 1 2 " − 3 a 3 2 a 5 1 + ǫ 1 dn √ 3 a 3 4 v √ a 5 ξ ! ! # 1 2 , where v = ǫ 2 p a 5 (64 a 5 a 1 − 3 a 3 2 r 2 − 12 a 3 2 ) / (8 a 5 ). Among the ab o ve solutions, only u 1 ± ( x, t ) with ǫ 1 = 1 and u 3 ± ( x, t ) repro d uce the results giv en in Refs.[32]-[35], and the other solutions hav e not b een found b efore. Example 2. Next we consider th e Ku ndu equ ation, iu t + u xx + β | u | 2 u + δ | u | 4 u + iα ( | u | 2 u ) x + i s ( | u | 2 ) x u = 0 , (10) where β , δ, α, s are real constan ts. E q .(10 ) was derived by Kundu [39 ] in the study of int egrability and it is an imp ortan t sp ecial case of the generalized complex Ginzbu rg-Laudau equ ation[40 ]. Mean while, Eq.(10) and its sp ecial c ases arise in v arious physica l and mec h anical applicatio n s, suc h as plasma physics, nonlinear fl uid mec hanics, n onlinear optics and quan tum physics. F en g[34] 5 deriv ed the explicit exact solitary wa v e solutions of Eq.(10) by usin g the alge braic curve metho d. Zhang e t al. [4 1] studied the o r bital s tability o f solitary w av es for Eq.(10) b y means of sp ectral analysis. Assume that Eq.(10) has solutions of the the form u ( x, t ) = φ ( ξ ) e i ( ψ ( ξ ) − ω t ) , ξ = x − v t, (11) where ω and v are constants to b e determined. Sub stituting Eq.(11) into Eq.(10 ) and then sepa- rating the real part and imaginary part yields, ( ω + v ψ ′ ( ξ )) φ ( ξ ) + φ ′′ ( ξ ) − φ ( ξ ) ψ ′ 2 ( ξ ) − α φ 3 ( ξ ) ψ ′ ( ξ ) + β φ 3 ( ξ ) + δ φ 5 ( ξ ) = 0 , (12a) − v φ ′ ( ξ ) + 2 φ ′ ( ξ ) ψ ′ ( ξ ) + φ ( ξ ) ψ ′′ ( ξ ) + (3 α + 2 s ) φ 2 ( ξ ) φ ′ ( ξ ) = 0 . (12b) Letting ψ ′ ( ξ ) = A + B φ 2 ( ξ ) . (13) Substituting Eq.(13) in to Eq.(12b ) and setting the co efficien ts of φ ′ ( ξ ), φ 2 ( ξ ) φ ′ ( ξ ) to zero, we ha v e A = v/ 2, B = − (3 α + 2 s ) / 4. Then Eq.(13) b ecomes, ψ ′ ( ξ ) = v 2 − 3 α + 2 s 4 φ 2 ( ξ ) . (14) Substituting Eq.(14) into Eq.(12a) yields the Li´ en ard equation of the form, φ ′′ ( ξ ) + l φ ( ξ ) + m φ 3 ( ξ ) + n φ 5 ( ξ ) = 0 , (15) where l , m, n are give n b y l = ω + v 2 4 , m = β − α v 2 , n = δ + ( α − 2 s )(3 α + 2 s ) 16 . By the transformations (11) and (14), the exact solutions of Eq.(10) can b e obtained by using the solutions of Eq.(1) giv en in Section 2. In the follo win g solutions, ψ ( ξ ) is giv en b y Eq.(14), ∆ 1 = (2 β − αv ) 2 − (4 ω + v 2 )(16 δ + ( α − 2 s )(3 α + 2 s )) / 3 . When ∆ 1 > 0, and v 2 + 4 ω < 0, Eq.(10) has the solitary wa v e solution, u 1 ( x, t ) = φ ( x − v t ) e i ( ψ ( x − v t ) − ω t ) , φ ( ξ ) = ± " − 2(4 ω + v 2 ) 2 β − αv + ǫ √ ∆ 1 cosh( p − ( v 2 + 4 ω ) ξ ) # 1 2 . When ∆ 1 < 0, and v 2 + 4 ω < 0, Eq.(10) has the singular solitary wa v e solution, u 2 ( x, t ) = φ ( x − v t ) e i ( ψ ( x − v t ) − ω t ) , φ ( ξ ) = ± " − 2(4 ω + v 2 ) 2 β − αv + ǫ √ − ∆ 1 sinh( p − ( v 2 + 4 ω ) ξ ) # 1 2 . 6 When v 2 + 4 ω < 0 and α v − 2 β < 0 , Eq.(10) has t wo kink-shap e solitary wa ve solutions, u 3 ( x, t ) = φ ( x − v t ) e i ( ψ ( x − v t ) − ω t ) , φ ( ξ ) = ± " v 2 + 4 ω α v − 2 β 1 + ǫ tanh( p − (4 ω + v 2 ) 2 ξ ) !# 1 2 , u 4 ( x, t ) = φ ( x − v t ) e i ( ψ ( x − v t ) − ω t ) , φ ( ξ ) = ± " v 2 + 4 ω α v − 2 β 1 + ǫ coth( p − (4 ω + v 2 ) 2 ξ ) !# 1 2 , where ω is determined b y ∆ 1 = 0. When ∆ 1 > 0 and v 2 + 4 ω > 0, Eq.(10) has the p erio d ic solution of trigonometric fu nction, u 5 ( x, t ) = φ ( x − v t ) e i ( ψ ( x − v t ) − ω t ) , φ ( ξ ) = ±  − 2(4 ω + v 2 ) 2 β − αv + ǫ √ ∆ 1 cos( √ v 2 + 4 ω ξ )  1 2 . When 4 sα + 4 s 2 − 3 α 2 − 16 δ > 0, 2 β − α v > 0, Eq.(10) has the Jacobian elliptic sine f unction solution, u 6 ( x, t ) = φ ( x − v t ) e i ( ψ ( x − v t ) − ω t ) , φ ( ξ ) = ± " 3(2 β − α v ) 4 sα + 4 s 2 − 3 α 2 − 16 δ 1 + ǫ sn( √ 3(2 β − α v ) 2 r √ 4 sα + 4 s 2 − 3 α 2 − 16 δ ξ ) !# 1 2 , where ω is determined b y r 2 ( v 2 + 4 ω )(16 δ + (3 α + 2 s )( α − 2 s )) − 3( β − v α / 2) 2 (5 r 2 − 1) = 0. When 4 sα + 4 s 2 − 3 α 2 − 16 δ < 0, 2 β − α v < 0, Eq.(10) has tw o Jacobian elliptic f unction solutions. O ne is u 7 ( x, t ) = φ ( x − v t ) e i ( ψ ( x − v t ) − ω t ) , φ ( ξ ) = ± " 3(2 β − α v ) 4 sα + 4 s 2 − 3 α 2 − 16 δ 1 + ǫ cn( √ 3(2 β − α v ) 2 r √ 3 α 2 + 16 δ − 4 sα − 4 s 2 ξ ) !# 1 2 , where ω is determined b y r 2 ( v 2 + 4 ω )(16 δ + (3 α + 2 s )( α − 2 s )) − 3 ( β − v α/ 2) 2 (4 r 2 + 1) = 0. And another one is u 8 ( x, t ) = φ ( x − v t ) e i ( ψ ( x − v t ) − ω t ) , φ ( ξ ) = ± " 3(2 β − α v ) 4 sα + 4 s 2 − 3 α 2 − 16 δ 1 + ǫ dn( √ 3(2 β − α v ) 2 √ 3 α 2 + 16 δ − 4 sα − 4 s 2 ξ ) !# 1 2 , where ω is determined b y ( v 2 + 4 ω )(16 δ + (3 α + 2 s )( α − 2 s )) − 3 ( β − v α/ 2) 2 ( r 2 + 4) = 0. 7 The solutions u 1 ( x, t ) with ǫ = 1, u 3 ( x, t ), u 4 ( x, t ) are same as th e r esults rep orted in [34]. Other solutions ha v e not b een rep orted in [34]. In addition, the Kundu equation (10) con tains sev eral imp ortant nonlinear mo dels when taking differen t c hoices f or the parameters α , β , δ a nd s . F or example, if s = 0, Eq.(10) r educes to the deriv ativ e Sc hr¨ odinger equation[39] iu t + u xx + β | u | 2 u + δ | u | 4 u + iα ( | u | 2 u ) x = 0; (16) if δ = 2 σ 2 , α = − 2 σ , s = 4 σ , then E q.(10) b ecomes th e Gerdjik ov-Iv ano v equation[42], iu t + u xx + β | u | 2 u + 2 σ 2 | u | 4 u + 2 iσ u 2 ¯ u x = 0 . (17) Ob viously , the explicit exact solutions of Eq.(16) and Eq.(17) can b e derived from the ab o v e solutions. Example 3. Finally we consider the generalized long-short wa ve resonance equations with strong nonlinear term, i S t + S xx = α LS + γ | S | 2 S + δ | S | 4 S, L t + β | S | 2 x = 0 , (18) where S is the env elop e of the short wa ve, and L is the a mplitude of th e long wa v e and is real. Th e parameters α, β , γ and δ are arbitrary real constan ts. Recen tly , Shang[43] obtained several kinds of explicit exact solutions of Eq.(18). In ord er to s eek the exact solutions of Eq.(18), w e in tro du ce the f ollo w ing transf ormation, S ( x, t ) = φ ( x, t ) e i ( k x + ω t + ξ 0 ) , (19) where φ ( x, t ) is a real-v alued function, and k and ω are constan ts to b e determined, ξ 0 is an arbitrary constant . Substituting Eq.(19) in to Eq.(18) and then separating the real and imaginary parts yields, φ xx − ( ω + k 2 ) φ − α L φ − γ φ 3 − δ φ 5 = 0 , (20a) φ t + 2 k φ x = 0 , (20b) L t + 2 β φφ x = 0 . (20c) In view of Eq.(20b) we s u pp ose φ ( x, t ) = φ ( ξ ) = φ ( x − 2 k t + ξ 1 ) , (21) where ξ 1 is an arbitrary constan t. Therefore w e also assume L ( x, t ) = ψ ( ξ ) = ψ ( x − 2 k t + ξ 1 ) . (22) 8 Substituting Eq.(21) into Eq.(20c) yields, ψ ( ξ ) = β φ 2 ( ξ ) 2 k + C , (23) where C is an in tegration constan t. Substituting Eqs.(21)-(23) in to Eq.(20a), w e ha ve, φ ′′ ( ξ ) + l φ ( ξ ) + m φ 3 ( ξ ) + n φ 5 ( ξ ) = 0 , (24) where the parameters l , m, n are giv en by l = − ( ω + k 2 + α C ) , m = − ( γ + α β 2 k ) , n = − δ. (25) Similar to E xample 1 , by means of the tran s formations (19), (2 1 )-(23), w e can also reduce the generalized long-short w av e r esonance equations (18) to the Li´ enard equation (1). T ogether with Eq.(21) and Eq.(23), su bstituting the solutions o f the Lienard equation give n in Sect ion 2 in to Eq.(19) and Eq.(22) yields abun dan t p erio dic w av e solutions of t he generalized long-short wa v e resonance equations (18). In the follo wing eigh t sets of solutions, ∆ 2 = ( γ + α β 2 k ) 2 − 16 δ ( ω + k 2 + α C ) / 3, ǫ = ± 1, and ξ = x − 2 k t + ξ 1 with k b eing nonzero arbitrary constan t. When ∆ 2 > 0 and ω + k 2 + α C > 0, Eqs.(18) has a set of b ell-shap e solitary wa v e solutions, L 1 ( x, t ) = 4 β ( ω + k 2 + α C ) − 2 k γ − α β + 2 k ǫ √ ∆ 2 cosh(2 √ ω + k 2 + α C ξ ) + C , S 1 ( x, t ) = ±    4 ( ω + k 2 + α C ) − ( γ + α β 2 k ) + ǫ √ ∆ 2 cosh(2 √ ω + k 2 + α C ξ )    1 2 e i ( k x + ω t + ξ 0 ) . When ∆ 2 < 0, ω + k 2 + α C > 0 Eqs.(18) h as a set of singular s olitary wa v e solutions, L 2 ( x, t ) = 4 β ( ω + k 2 + α C ) − 2 k γ − α β + 2 k ǫ √ − ∆ 2 sinh(2 √ ω + k 2 + α C ξ ) + C , S 2 ( x, t ) = ±    4 ( ω + k 2 + α C ) − ( γ + α β 2 k ) + ǫ √ − ∆ 2 sinh(2 √ ω + k 2 + α C ξ )    1 2 e i ( k x + ω t + ξ 0 ) . When ∆ 2 = 0, ω + k 2 + αC > 0, and 2 kγ + αβ < 0, Eqs.(18) has t w o sets of kink-sh ap e sol itary w av e solutions, L 3 ( x, t ) = − 2 β ( ω + k 2 + α C ) 2 k γ + αβ  1 + ǫ tanh( √ ω + k 2 + α C ξ )  + C , S 3 ( x, t ) = ±  − 4 k ( ω + k 2 + α C ) 2 k γ + αβ  1 + ǫ tanh( √ ω + k 2 + α C ξ )   1 2 e i ( k x + ω t + ξ 0 ) , 9 L 4 ( x, t ) = − 2 β ( ω + k 2 + α C ) 2 k γ + αβ  1 + ǫ coth( √ ω + k 2 + α C ξ )  + C , S 4 ( x, t ) = ±  − 4 k ( ω + k 2 + α C ) 2 k γ + αβ  1 + ǫ coth( √ ω + k 2 + α C ξ )   1 2 e i ( k x + ω t + ξ 0 ) , When ∆ 2 > 0 and ω + k 2 + α C < 0, Eqs.(18) has a set of trigonometric fun ction solutions, L 5 ( x, t ) = 4 β ( ω + k 2 + α C ) − 2 k γ − α β + 2 k ǫ √ ∆ 2 cos(2 p − ( ω + k 2 + α C ) ξ ) + C , S 5 ( x, t ) = ±    4 ( ω + k 2 + α C ) − ( γ + α β 2 k ) + ǫ √ ∆ 2 cos(2 p − ( ω + k 2 + α C ) ξ )    1 2 e i ( k x + ω t + ξ 0 ) . When δ > 0 a nd k ( αβ + 2 kγ ) < 0, Eqs.(18) h as a set of Jacobian elliptic sine f u nction so lutions, L 6 ( x, t ) = − 3 β ( αβ + 2 k γ ) 32 k 2 δ 1 + ǫ sn( − √ 3 ( αβ + 2 k γ ) 8 k r √ δ ξ ) ! + C , S 6 ( x, t ) = ± " − 3( αβ + 2 k γ ) 16 k δ 1 + ǫ sn( − √ 3 ( αβ + 2 k γ ) 8 k r √ δ ξ ) !# 1 2 e i ( k x + ω t + ξ 0 ) , where ω is determined b y 64 r 2 δ ( ω + k 2 + α C ) − 3 (5 r 2 − 1) ( γ + αβ 2 k ) 2 = 0. When δ < 0, k ( αβ + 2 k γ ) > 0, Eqs.(18 ) has tw o sets of Jacobian elliptic function solutions. One is L 7 ( x, t ) = − 3 β ( αβ + 2 k γ ) 32 k 2 δ 1 + ǫ cn( − √ 3 ( αβ + 2 k γ ) 8 k √ − δ ξ ) ! + C , S 7 ( x, t ) = ± " − 3( αβ + 2 k γ ) 16 k δ 1 + ǫ cn( − √ 3 ( αβ + 2 k γ ) 8 k √ − δ ξ ) !# 1 2 e i ( k x + ω t + ξ 0 ) , where ω is determined b y 64 r 2 δ ( ω + k 2 + α C ) − 3 (4 r 2 + 1) ( γ + αβ 2 k ) 2 = 0. And another one is L 8 ( x, t ) = − 3 β ( αβ + 2 k γ ) 32 k 2 δ 1 + ǫ dn( − √ 3 ( αβ + 2 k γ ) 8 k √ − δ ξ ) ! + C, S 8 ( x, t ) = ± " − 3( αβ + 2 k γ ) 16 k δ 1 + ǫ dn( − √ 3 ( αβ + 2 k γ ) 8 k √ − δ ξ ) !# 1 2 e i ( k x + ω t + ξ 0 ) , where ω is determined b y 64 δ ( ω + k 2 + α C ) − 3 (4 + r 2 ) ( γ + αβ 2 k ) 2 = 0. With the aid of Maple, w e ha ve c heck ed all solutions by pu tting them bac k into the original Equation. 10 4. Conclusions The Li ´ enard equation is used to describ e fluid-mec hanical and n onlinear elasti c m ec hanical phenomena. Moreo v er, a num b er of NLPDEs with strong nonlinear terms can b e reduced to the Li ´ enard equation b y so me pr op er transformations. Therefore, T o search for new sp ecial s olutions of the Li´ enard equation is a very imp ortan t job. In this letter, we obtain eight kinds of exp licit exact solutions for the Li´ enard equation, w hic h include solitary w a ve solutions, p erio dic w a ve solutions in terms of trigonometric function and Jacobian elliptic fu n ction. By means of these solutions, we obtain a v ariet y of explicit exact solutions for the generalized P C equation, the Kun du equation and the generalized long-short w a ve r esonance equations. These solutions ma y b e imp ortant explain some ph ysical ph en omena. Th e method presented here is also applicable to solv e other nonlinear equations with strong nonlinear terms. F or example, the Ablowitz equation[44], i u tt = u xx − 4 i u 2 ¯ u x + 8 | u | 4 u ; the third-order generalized NLS equation(also called RKL mo d el)[45], iu z + u tt + 2 | u | 2 u + iαu ttt + iβ ( | u | 2 u ) t + iγ ( | u | 4 u ) t + δ | u | 4 u = 0 , and the nonlinear equations which we re considered in Ref.[32] and Ref.[34]. Ac kno wledgemen t This work w as supp orted by the Natural Science F oundation of China und er Grant No. 1080 1037. References [1] M.J.Ablo witz, P .A.Clarkson, S olitons, n onlinear evolution equations and in verse scattering, Cambridge: Cam- bridge Univ ersity Press, 19 91. [2] C. R ogers and W.F. Shadwick, B¨ ac k lund transf ormations, New Y ork: Academic Pres s, 19 82. [3] V.A. Matveev and M.A. Salle, Darb oux t ransformation and solitons, Berl in: Springer, 1991. [4] R. H irota, The Direct Metho d in Soliton Theory, Cambridge: Cam bridge Universit y Press, 2004. [5] J.W eiss, M.T ab or and G.Carnev ale, J. Math. Phys. 24 (1983) 522. [6] S.Y. Lou, Ph ys. Rev. Lett. 80 (1998) 5027. [7] G.Q. X u and Z.B. Li, Comput. Phys. Commun. 161 (2004) 65. [8] G.Q. X u, Phys. Rev. E. 74 (2006) 027602. [9] G.Q. X u, Comput. Phys. Commun. 178 (2008) 505. [10] G.Q. Xu, Comput. Ph y s. Co mmun. 18 0 (2009) 11 37. 11 [11] M.L.W ang, B.Y.Zhou, Z.B.Li, Phys. Lett. A 216 (1996) 67. [12] M.A. Abdou , A.A. Soliman, Phys. D 211 (20 05) 1. [13] J.H. He, Phys. Lett. A 350 (2006) 87. [14] W.Malfliet, Am J. Phys. 60 (1992) 650. [15] S.K.Liu, Z.T .F u, S.D.Liu, Q.Zhao, Phys. Lett. A 289 (2001) 69. [16] Z.T.F u, S.K.Liu, S.D.Liu, Q.Zhao, Phys. Lett. A 290 (2001) 72. [17] G.Q. Xu, Z.B. Li, Chaos, Solitons and F ractals 26 (2005) 1363. [18] G.Q. Xu, Z.B. Li, Commun. Theor. Ph y s.(Beijing, China) 43 (2005) 385 [19] Z.Y. Y an, Phys. Lett. A 373 (2009) 2432. [20] E.G.F an, Ph ys. Lett. A 27 7 (2000) 21 2. [21] E.G. F an, J. Phys. A: Ma t h. Gen. 35 (2002) 685 3. [22] R.Sabry , M.A. Z ah ran , E.G.F an, Ph y s. Lett. A 326 (2004) 93. [23] Sirendaoreji, Jiong S, Ph y s. Lett. A 309 (2003) 387. [24] G.Q. Xu, Chaos, Solitons and F ractals 29 (2006) 942. [25] Y. Chen, Q. W ang, Appl. Math. Comput. 177 (2006) 396. [26] Y.B. Zhou, M.L. W ang, Y.M. W ang, Phys. Lett . A 308 (2003) 31. [27] E. Y om ba, Phys. Lett . A 340 (2005) 149. [28] M.L. W ang, X.Z. Li, J.L. Zhang, Phys. Lett. A 363 (2007) 96. [29] S.N. Behera, A. K hare, Lett. Math. Phys. 4 (1980) 153. [30] D. Dey , A. K hare, C.N. Kuma, Avai lable from: arXiv:hep-t h/951005 4. [31] D. Kong, Phys. Lett. A 196 (1995) 30 1. [32] W. Zh ang, A ct a Math. Appl. Sinica 21 (1998 ) 249. [33] W. Zh ang, W.X. Ma, Appl. Math. M ec h. 20 (1999) 666. [34] Z. F eng, X . W ang, Physica S cript a 64 (2001) 7. [35] Z. F eng, Phys. Lett. A 293 (2002) 50. [36] Z. F eng, Chaos, Solitons and F ractals 21 (2004 ) 343. [37] P .A. Cla rkson, R.J. Leveque, R. Saxton, Stud. Appl. Math. 75 ( 1986) 95 . [38] J. Li, L. Zh ang, Chaos, Solitons and F ractals 14 (2002) 581. [39] A. Kundu, J. Math. Phys. 25 (1984) 343 3. [40] W.V. Saar loos, P .C. Hohenberg, Phys. D 56 (199 2) 303. [41] W. Zh ang, Y . Qin, Y. Zhao, B. Guo, J. differen tial equations 247 (2009) 1591. [42] V.S. Gerdji ko v, I. Iv ano v, Bulg. J. Phys. 10 (1983) 130 . [43] Y.D. Shang, Chaos, Solitons and F ractals 26 (2005) 527. [44] M.J. A blo witz, A.Ramani and H . S egur, J. Math. Phys. 21 (1980) 1006. [45] R.Radhakrishnan, A.Kundu and M.Lakshmanan, Phys. R ev. E 60 (1999) 3314. 12