Paper Info
Title
Applications of a further extended tanh method
Abstract
By using a further extended tanh method [Chaos Solitons Fract. 17 (2003) 669] and symbolic computation system Maple, we obtain abundant new explicit exact solutions of the (2 + 1)-dimensional Burgers equation and the (2 + 1)-dimensional Boussinesq equation. The new solutions including soliton-like solutions, period form solutions and some other solutions. Keywords Further extended tanh method (2 + 1)-dimensional Burgers equation (2 + 1)-dimensional Boussinesq equation Maple 1 Introduction Constructing exact solutions for nonlinear evolution equations (NLEEs) has long been a major concern for both mathematicians and physicists. Up to now, there exist many powerful methods to obtain exact solutions of NLEEs related to nonlinear problems [1] . For example, Bäcklund transformation [2] , Darboux transformation [3,4] , inverse scattering method [5] , Hirota's method [6] , tanh method and extended tanh method [7–12] , etc. Recently we presented a further extended tanh method [13,14] . The further extended tanh method [13,14] is more powerful than the methods presented in Refs. [7–12] . By using the further extended tanh method, we are now able to obtain soliton-like solutions, multi-soliton-like solutions or period form solutions which can not be obtained by using the methods in [7–12] . In this paper we would like to apply the further extended tanh method to the (2 + 1)-dimensional Burgers equation and the (2 + 1)-dimensional Boussinesq equation to seek more explicit exact solutions of this two equations. 2 Review of the further extended tanh method For a given NLEE with independent variables x =( t , x 1 , x 2 ,…, x m ), and dependent variable u , we seek its solutions in the form (1) u=∑ i=0 n a i (x)φ i (ω(x)) with (2) φ ′ =δ+φ 2 , where δ is a constant, prime denotes differentiation with respect to ω . To determine u explicitly, we take the following four steps. Step 1. Determine n by balancing the highest nonlinear terms and the highest order partial differential terms in the given NLEE. Step 2. Substituting (1) and (2) into the given NLEE and collecting coefficients of polynomials of φ , then setting each coefficient to zero to derive a set of partial differential equations of a i ( i =0,1,…, n ) and ω . Step 3. Solving the system of partial differential equations obtained in step 2 for a i and ω . Step 4. As (2) possesses the general solution (3) φ= − −δ tanh ( −δ ω) δ<0, − −δ coth ( −δ ω) δ<0, δ tan ( δ ω) δ>0, − δ cot ( δ ω) δ>0, −1/ω δ=0, substituting a i , ω and (3) into (1) to obtain the solutions of the NLEE in concern. 3 Application of the further extended tanh method to the (2 + 1)-dimensional Burgers equation The (2 + 1)-dimensional Burgers equation reads: (4) (u t +uu x −u xx ) x +u yy =0. This equation describes weakly nonlinear two-dimensional shocks in dissipative media. A general derivation of Eq. (4) has been given in Ref. [15] . It is of considerable interest as a 1 + 2D nonlinear wave equation [16] . Eq. (4) also can be referred to as the Zabolotskaya–Khoklov equation in nonlinear acoustics [17] , with the u yy term representing wave diffraction. Now we apply the further extended tanh method to Eq. (4) . The balancing procedure leads to n =1. Then we set (5) u=f+gφ(κx+q), where f , g and q are functions of ( y , t ) to be determined, κ is a nonzero constant. Substituting (5) and (2) into (4) and collecting coefficients of polynomials of φ with the aid of Maple, then setting each coefficient to zero, we have (6) −3κ 2 g(−g+2κ)=0, 2g(κ 2 f+q y 2 +κq t )=0, 2κgq t δ+g yy +2gκ 2 fδ+2gq y 2 δ=0, −8gκ 3 δ+2q y g y +κg t +gq yy +4g 2 κ 2 δ=0, g 2 κ 2 δ 2 +2q y g y δ+κg t δ+gq yy δ+f yy −2gκ 3 δ 2 =0. If we set (7) g=2κ, (6) can be reduced to (8) q yy =0, f yy =0, κq t +κ 2 f+q y 2 =0. Solving Eq. (8) we have (9) q=Φ(t)y+Ω(t), f=−(Φ 2 +κΦ t y+κΩ t )/κ 2 , where Φ ( t ) and Ω(t) are arbitrary functions of t . Thus we have the following solutions of Eq. (4) : for δ <0, (10) u 1 =−(Φ 2 +κΦ t y+κΩ t )/κ 2 −2κ −δ tanh ( −δ (κx+Φy+Ω)), u 2 =−(Φ 2 +κΦ t y+κΩ t )/κ 2 −2κ −δ coth ( −δ (κx+Φy+Ω)); for δ >0, (11) u 3 =−(Φ 2 +κΦ t y+κΩ t )/κ 2 +2κ δ tan ( δ (κx+Φy+Ω)), u 4 =−(Φ 2 +κΦ t y+κΩ t )/κ 2 −2κ δ cot ( δ (κx+Φy+Ω)); for δ =0, (12) u 5 =−(Φ 2 +κΦ t y+κΩ t )/κ 2 −2κ/(κx+Φy+Ω). Eq. (4) has been considered in Ref. [18] . The following exact solutions were obtained: (13) u=− b k − a 2 k 2 −k tanh kx+ay+bt+c 2 , u=−k 1+ tanh kx+k(k−B)t+c 1 2 +B, u=− m 2 k 2 −k tanh kx+my 2 , u=−k 1+ tanh kx+k 2 t+ ln cos B k y−c 1 +c 2 2 +B, u=−k 1+ tanh kx+k(k−B)t+ ln (y−c 1 )+c 2 2 +B. We should point out that the first three solutions of (13) are special cases of (10) , but the last two are incorrect. In fact, Eqs. (2.12) and (2.18) in [18] leads to R yy =0 and R yy =−1/ y 2 respectively, however, the last two solutions of (13) are based on R=k 2 t+ ln [ cos B k(y−c 1 )]+c 2 and R = k ( k − B ) t +ln( y 2 − c 1 y )+ c 2 respectively. 4 Application of the further extended tanh method to the (2 + 1)-dimensional Boussinesq equation Consider the (2 + 1)-dimensional Boussinesq equation [19] (and references therein) (14) u tt −u xx −u yy −(u 2 ) xx −u xxxx =0. This equation has recently been investigated by Senthilvelan [19] . Some solitary wave and periodic solutions were obtained by using Fan's method. We now apply the further extended tanh method to Eq. (14) . Balancing u xxxx with u x 2 or uu xx in Eq. (14) leads to n =2. Then we choose the following ansatz (15) u=f+gφ(κx+q)+hφ 2 (κx+q), with f = f ( y , t ), g = g ( y , t ), h = h ( y , t ) and q = q ( y , t ) to be determined. κ is a nonzero constant. Substituting (15) and (2) into (14) and collecting coefficients of polynomials of φ we have (16) −24gκ 2 (κ 2 +h)=0, −20hκ 2 (6κ 2 +h)=0, −6κ 2 g 2 +6q t 2 h−240κ 4 hδ−6κ 2 h−32κ 2 h 2 δ−12κ 2 fh−6q y 2 h=0, −4κ 2 fg+2q t 2 g−4q y h y −36κ 2 gδh−2q y 2 g−2q yy h +4q t h t −2κ 2 g−40κ 4 gδ+2q tt h=0, −f yy +f tt −q yy gδ+2q t g t δ−4κ 2 fhδ 2 +q tt gδ−16κ 4 hδ 3 −2q y g y δ−2q y 2 hδ 2 −2κ 2 hδ 2 +2q t 2 hδ 2 −2κ 2 g 2 δ 2 =0, g tt −2q yy hδ−g yy +4q t h t δ−2κ 2 gδ−4κ 2 fgδ−2q y 2 gδ −4q y h y δ+2q t 2 gδ+2q tt hδ−16κ 4 gδ 2 −12κ 2 gδ 2 h=0, q tt g+2q t g t −2q y g y +8q t 2 hδ−h yy −12κ 2 h 2 δ 2 −8κ 2 hδ −136κ 4 hδ 2 −8q y 2 hδ+h tt −q yy g−8κ 2 g 2 δ−16κ 2 fhδ=0. We do not consider the case g = h =0, so the first two equations of (16) leads to (17) g=0, h=−6κ 2 . Reducing (16) with (17) we can obtain (18) f=(−8κ 4 δ−κ 2 −q y 2 +q t 2 )/(2κ 2 ), q yy −q tt =0, (q yy −q ty )(q yy +q ty )=0. Solving (18) we get the following two systems of solutions: (19) q=Φ(t+y)+αy+βt, f=(−8κ 4 δ−κ 2 +(β−α)(α+2Φ ′ +β))/(2κ 2 ) and (20) q=Φ(t−y)+αy+βt, f=(−8κ 4 δ−κ 2 +(α+β)(β−α+2Φ′))/(2κ 2 ), in which Φ is an arbitrary function, α and β are arbitrary constants. Then we get the following solutions of Eq. (14) : (21) u= f+6κ 2 δ tanh 2 ( −δ (κx+q)) δ<0, f+6κ 2 δ coth 2 ( −δ (κx+q)) δ<0, f−6κ 2 δ tan 2 ( δ (κx+q)) δ>0, f−6κ 2 δ cot 2 ( δ (κx+q)) δ>0, f−6κ 2 /(κx+q) 2 δ=0, where q and f satisfy (19) or (20) . It is easy to prove that the solutions obtained in Ref. [19] are special cases of (21) . 5 Conclusion In summary, we apply the further extended tanh method to the (2 + 1)-dimensional Burgers equation and the (2 + 1)-dimensional Boussinesq equation. Abundant new explicit exact solutions including soliton-like solutions, period form solutions and other forms of solutions of the equations were obtained. Acknowledgements This work is supported by the National Key Basic Research Development of China (grant no. 1998030600) and the National Nature Science Foundation of China (grant no. 10072013). References [1] M.J. Ablowitz P.A. Clarkson Solitons, Nonlinear Evolution Equations and Inverse Scatting 1991 Cambridge University Press Cambridge [2] G.L. Lamb Rev. Mod. Phys 43 1971 99 [3] M. Wadati H. Sanuki K. Konno Prog. Theor. Phys 53 1975 419 [4] C.H. Gu Darboux Transformation in Soliton Theory and its Geometric Applications 1999 Shanghai Scientific and Technical Publishers Shanghai [5] C.S. Gardner Phys. Lett. Rev 19 1967 1905 [6] R. Hirota Phys. Rev. Lett 27 1971 1192 [7] L. Huibin W. Kelin J. Phys. A 23 1990 4097 [8] W. Malfliet Am. J. Phys 60 1992 650 [9] W.X. Ma Int. J. Non-Linear Mech 31 1996 329 [10] E.G. Fan H.Q. Zhang Phys. Lett. A 246 1998 403 [11] E.G. Fan Appl. Math. J. Chinese Univ. Ser. B 16 2001 149 [12] Y.T. Gao B. Tian Comput. Math. Appl 33 1997 115 [13] Z.S. Lü H.Q. Zhang Phys. Lett. A 307 2003 269 [14] Z.S. Lü H.Q. Zhang Chaos Solitons Fract 17 2003 669 [15] M. Bartucelli P. Pantano T. Brugarino Lett. Nuovo. Cimento 37 1983 433 [16] H. Segur Physica D 18 1986 1 [17] J.K. Hunter SIAM J. Appl. Math 48 1988 1 [18] Sirendaoreji J. Phys. A: Math. Gen 32 1999 6897 [19] M. Senthilvelan Appl. Math. Comput 123 2001 381
Year
DOI
Venue
2004
10.1016/j.amc.2003.10.021
Applied Mathematics and Computation
Keywords
Field
DocType
symbolic computation,boussinesq equation,burgers equation,exact solution
Maple,Exact solutions in general relativity,Soliton,Mathematical optimization,Mathematical analysis,Symbolic computation,Burgers' equation,Hyperbolic function,Numerical analysis,Mathematics,Boussinesq approximation (water waves)
Journal
Volume
Issue
ISSN
159
2
Applied Mathematics and Computation
Citations 
PageRank 
References 
2
2.14
1
Authors
2
Name
Order
Citations
PageRank
Zhuosheng Lü122.14
Hongqing Zhang213848.35