A Further Improved Tanh Function Method Exactly Solving The (2+1)-Dimensional Dispersive Long Wave Equations

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1 Applied Mathematics E-Notes, 8(2008), c ISSN Available free at mirror sites of amen/ A Further Improved Tanh Function Method Exactly Solving The (2+1)-Dimensional Dispersive Long Wave Equations Sheng Zhang, Tie-cheng Xia Received 8 January 2007 Abstract In this paper, a further improved tanh function method is used to construct exact solutions of the (2+1)-dimensional dispersive long wave equations. As a result, many new and more general solutions are obtained including soliton-lie solutions, periodic formal solutions and rational function solutions. Compared with most existing tanh function methods, the proposed method gives new and more general exact solutions. More importantly, with the aid of symbolic computation, this method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. 1 Introduction It is well nown that nonlinear complex physics phenomena are related to nonlinear partial differential equations (NLPDEs) which are involved in many fields from physics to biology, chemistry, mechanics, etc. As mathematical models of the phenomena, the investigation of exact solutions of NLPDEs will help us to understand these phenomena better. Many effective methods for obtaining exact solutions of NLPDEs have been presented, such as Bäclund transformation [1], hyperbolic function method [2], sine-cosine method [3], Jacobi elliptic function expansion method [4], homotopy perturbation method [5], F-expansion method [6] and so on. One of the most effective direct methods to construct exact solutions of NLPDEs is tanh function method [7 9], the method was later extended in different manners [10 16]. Recently, Xie et al. [17] generalized the wor made in [10 16]. Very recently, by using a more general transformation we improved the method [17] and proposed a further improved tanh function method [18] to see more general exact solutions of NLPDEs. The present paper is motivated by the desire to extend the further improved tanh function method to the (2+1)-dimensional dispersive long wave (DLW) equations: u yt + H xx (u2 ) xy =0, (1) Mathematics Subject Classifications: 35Q51, 35Q53, 35Q99. Department of Mathematics, Bohai University, Jinzhou, Liaoning , P. R. China Department of Mathematics, Shanghai University, Shanghai , P. R. China 58

2 S. Zhang and T. C. Xia 59 H t +(uh + u + u xy ) x =0. (2) Soliton-lie solutions, Jacobi elliptic function solutions and other exact solutions can be found in [19 21]. 2 A Further Improved Tanh Function Method Given a system of NLPDEs with independent variables x = (t, x 1,x 2,...,x m ) and dependent variables u, v: F (u, v, u t,v t,u x1,v x1,...,u x1t,v x1t...,u tt,v tt,...,u xmx m,v xmx m,...)=0, (3) G(u, v, u t,v t,u x1,v x1,...,u x1t,v x1t...,u tt,v tt,...,u xmx m,v xmx m,...)=0, (4) we see its solutions in the more general forms: N 1 { u = a 0 (x)+ a i (x)φ i (ξ)+a i (x)φ i (ξ) }, (5) i=1 N 2 { v = b 0 (x)+ b j (x)φ j (ξ)+b j (x)φ j (ξ) }, (6) j=1 with φ (ξ) =r + pφ(ξ)+qφ 2 (ξ), (7) where the prime denotes d/dξ, r, p and q are all real constants, while a 0 (x), a i (x), a i (x), b 0 (x), b j (x), b j (x) (i = 1, 2,...,N 1 ; j = 1, 2,...,N 2 ) and ξ = ξ(x) are all differentiable functions to be determined later. Given different values of r, p and q, equation (7) has twenty seven special solutions which are listed in [18]. To determine u explicitly, we tae the following four steps: Step 1. Determine the integers N 1 and N 2 by balancing the highest order nonlinear term(s) and the highest order partial derivative term(s) in equations (3) and (4). Step 2. Substitute (5) and (6) along with (7) into equations (3) and (4), then collect coefficients of the same order of φ l (ξ) (l = ±1, ±2,...) and set each coefficient to zero to derive a set of over-determined partial differential equations for a 0 (x), a i (x), a i (x), b 0 (x), b j (x), b j (x) and ξ. Step 3. Solve the over-determined partial differential equations obtained in Step 2 by use of Mathematica and using Wu elimination method. Step 4. Substitute a 0 (x), a i (x), a i (x), b 0 (x), b j (x), b j (x) and ξ along with one solution φ(ξ) of equation (7) into (5) and (6), we then obtain soliton-lie solutions, periodic formal solutions and rational function solutions of equations (3) and (4). 3 Exact Solutions of the DLW Equations According to Step 1, we get N 1 = 2 for H and N 2 = 1 for u. In order to search for explicit solutions of equations (1) and (2), we set a 0 = a 0 (y, t), a 2 = a 2 (y, t),

3 60 Tanh Method for Dispersive Long Wave Equations a 1 = a 1 (y, t), a 1 = a 1 (y, t), a 2 = a 2 (y, t), b 0 = b 0 (y, t), b 1 = b 1 (y, t), b 1 = b 1 (y, t), ξ = x + η, η = η(y, t), is a nonzero constant. Thus we have H = a 0 + a 2 φ 2 (ξ)+a 1 φ 1 (ξ)+a 1 φ(ξ)+a 2 φ 2 (ξ), (8) u = b 0 + b 1 φ 1 (ξ)+b 1 φ(ξ). (9) Substituting (8) and (9) along with (7) into equations (1) and (2), then collecting coefficients of the same order of φ(ξ) l (l = ±1, ±2,...) and setting each coefficient to zero, we derive a set of over-determined partial differential equations for a 0,a 2,a 1, a 1,a 2,b 0,b 1,b 1 and η as follows: 2 2 r 2 a pra 2 2rb 1 b 1,y +2r 2 b 0 b 1 η y +5prb 2 1η y +2r 2 b 1 η t η y =0, 6 2 r 2 a 2 +3r 2 b 2 1η y =0, 3 2 pra p 2 a qra 2 rb 1 b 0,y rb 0 b 1,y 2pb 1 b 1,y rη t b 1,y +3prb 0 b 1 η y +2p 2 b 2 1 η y +4qrb 2 1 η y rb 1,t η y +3prb 1 η t η y rb 1 η t,y =0, 3qa 2 b q 3 b 1 η y =0, 2 p 2 a qra pqa 2 pb 1 b 0,y pb 0 b 1,y 2qb 1 b 1,y pη t b 1,y + p 2 b 0 b 1 η y +2qrb 0 b 1 η y +3pqb 2 1 η y pb 1,t η y + p 2 b 1 η t η y +2qrb 1 η t η y + b 1,t,y pb 1 η t,y =0, 3ra 2 b r 3 b 1 η y =0, 2 pra pqa q 2 a r 2 a 2 + rb 1 b 0,y qb 1 b 0,y + rb 0 b 1,y +rη t b 1,y qb 0 b 1,y qη t b 1,y + prb 0 b 1 η y + r 2 b 2 1η y + pqb 0 b 1 η y + rb 1 η t,y +rb 1,t η y qb 1,t η y + prb 1 η t η y qb 1 η t,y + pqb 1 η t η y + b 0,t,y + q 2 b 2 1η y =0, 2 p 2 a qra pra 2 + pb 1 b 0,y + pb 0 b 1,y +2rb 1 b 1,y + pη t b 1,y +p 2 b 0 b 1 η y +2qrb 0 b 1 η y +3prb 2 1η y + pb 1,t η y + p 2 b 1 η t η y +2qrb 1 η t η y +b 1,t,y + pb 1 η t,y =0, 6 2 q 2 a 2 +3q 2 b 2 1 η y =0, 3 2 pqa p 2 a qra 2 + qb 1 b 0,y + qb 0 b 1,y +2pb 1 b 1,y + qη t b 1,y +3pqb 0 b 1 η y +2p 2 b 2 1η y +4qrb 2 1η y + qb 1,t η y +3pqb 1 η t η y + qb 1 η t,y =0, 2 2 q 2 a pqa 2 +2qb 1 b 1,y +2q 2 b 0 b 1 η y +5pqb 2 1 η y +2q 2 b 1 η t η y =0, 2 2 r 2 b 1,y 2ra 2 b 0 2ra 1 b 1 3pa 2 b 1 2ra 2 η t 12 2 pr 2 b 1 η y =0, ra 1 b 0 2pa 2 b 0 ra 2 b 1 rb 1 ra 0 b 1 2pa 1 b 1 3qa 2 b 1 +a 2,t ra 1 η t 2pa 2 η t +3 2 prb 1,y 7 2 p 2 rb 1 η y 8 2 qr 2 b 1 η y =0, pa 1 b 0 2qa 2 b 0 pa 2 b 1 pb 1 pa 0 b 1 2qa 1 b 1 + a 1,t pa 1 η t 2qa 2 η t + 2 p 2 b 1,y +2 2 qrb 1,y 2 p 3 b 1 η y 8 2 pqrb 1 η y =0, ra 1 b 0 qa 1 b 0 + rb 1 + ra 0 b 1 qa 2 b 1 qb 1 qa 0 b 1 + ra 2 b 1 + a 0,t +ra 1 η t qa 1 η t + 2 prb 1,y + 2 pqb 1,y + 2 p 2 rb 1 η y +2 2 qr 2 b 1 η y

4 S. Zhang and T. C. Xia 61 2 p 2 qb 1 η y 2 2 q 2 rb 1 η y =0, pa 1 b 0 +2ra 2 b 0 + pa 2 b 1 + pb 1 + pa 0 b 1 +2ra 1 b 1 + pa 2 b 1 + a 1,t + pa 1 η t +2ra 2 η t + 2 p 2 b 1,y +2 2 qrb 1,y + 2 p 3 b 1 η y +8 2 pqrb 1 η y =0, qa 1 b 0 +2pa 2 b 0 + qb 1 + qa 0 b 1 +2pa 1 b 1 +3ra 2 b 1 + qa 2 b 1 + a 2,t +qa 1 η t +2pa 2 η t +3 2 pqb 1,y +7 2 p 2 qb 1 η y +8 2 q 2 rb 1 η y =0, 2qa 2 b 0 +2qa 1 b 1 +3pa 2 b 1 +2qa 2 η t +2 2 q 2 b 1,y pq 2 b 1 η y =0. Solving above over-determined partial differential equations by use of Mathematica, we get the following nontrivial results: Case 1: a 0 = 1 2qr(f 1 (y)t + f 2 (y)) ± f 1 (y), a 2 =0, a 1 =0, a 1 = 2pq(f 1(y)t + f 2(y)), a 2 = 2q 2 (f 1(y)t + f 2(y)), b 1 =0, b 0 = ±p f 1(y) +f 3(t), b 1 = ±2q, η = f 1 (y)t + f 2 (y) +f 3 (t), where f 1 (y), f 2 (y) and f 3 (t) are arbitrary functions, f 1 (y) = df 1(y)/dy, f 3 (t) = df 3 (t)/dt. The sign ± means that the same sign must be used in a 0,b 0 and b 1. Case 2: a 0 = 1 2qr(f 1(y)t + f 2(y)) ± f 1(y), a 1 =0, a 2 =0, a 1 = 2pr(f 1 (y)t + f 2 (y)), a 2 = 2r 2 (f 1 (y)t + f 2 (y)), b 1 =0, b 0 = p f 1(y)+f 3 (t), b 1 = 2r, η = f 1 (y)t + f 2 (y) +f 3 (t), where the signs ± and mean that different signs must be used in a 0 and b 1, furthermore the same sign must be used in b 0 and b 1. Case 3: a 0 = 1, a 2 = 2r 2 f 1 (y), a 1 = 2prf 1 (y), a 1 = 2pqf 1 (y), a 2 = 2q 2 f 1 (y), b 0 = ±p f 3 (t), b 1 = ±2r, b 1 = ±2q, η = f 1 (y) +f 3 (t), where the sign ± means that the same sign must be used in b 0,b 1 and b 1. For simplification, in the rest of this paper, we introduce the notations: p2 4qr 4qr p 2 M =, N =. (10) 2 2 From Cases 1 3, Appendix 1 in [18] and (8) (10), we can obtain many more general exact solutions of equations (1) and (2): Family 1. When p 2 4qr > 0 and pq 0 (or qr 0)

5 62 Tanh Method for Dispersive Long Wave Equations 2rcosh( p For example, if we select φ 8 (ξ) = 2 4qrξ/2) from p2 4qrsinh( p 2 4qrξ/2) pcosh( p 2 4qrξ/2) Appendix 1 in [18], and use Case 1 and (8) (10), we then obtain soliton-lie solutions H 1.1 = 1 2qrΞ ± f 1(y) 2pqΞφ 8 (ξ) 2q 2 Ξφ 2 8(ξ) = 1 2qrΞ ± f 1 (y) 2qrΞcosh(Mξ)[4pMsinh(Mξ) (p2 +4M 2 )cosh(mξ)] [2Msinh(Mξ) pcosh(mξ)] 2, u 1.1 = ±p f 1(y) +f 3 (t) ± 2qφ 8 (ξ) = ±p f 1(y)+f 3(t) ± 4qrcosh(Mξ) 2Msinh(Mξ) pcosh(mξ), where ξ = x + f 1 (y)t + f 2 (y)+f 3 (t), Ξ=f 1(y)t + f 2(y). Family 2. When p 2 4qr < 0 and pq 0 (or qr 0) For example, if we select φ 22 (ξ), from Case 1 we obtain periodic formal solutions H 2.1 = 1 2qrΞ± f 1(y) 4N 2 )cos(2nξ) ± 4pN] +2qrΞcos(2Nξ)[4pNsin(2Nξ)+(p2 [2Nsin(2Nξ)+pcos(2Nξ) ± 2N] 2, u 2.1 = ±p f 1(y) +f 3(t) 4qrcos(2Nξ) 2Nsin(2Nξ)+pcos(2Nξ) ± 2N, where ξ = x + f 1 (y)t + f 2 (y)+f 3 (t), Ξ=f 1(y)t + f 2(y). Family 3. When r = 0 and pq 0 For example, if we select φ 25 (ξ), from Case 1 we obtain soliton-lie solutions H 3.1 = 1 ± f 1(y) u 3.1 = ±p f 1(y) +f 3(t) + 2p2 dξ[cosh(pξ) sinh(pξ)] [d + cosh(pξ) sinh(pξ)] 2, 2pd d + cosh(pξ) sinh(pξ), where ξ = x + f 1 (y)t + f 2 (y)+f 3 (t), Ξ = f 1(y)t + f 2(y), dis an arbitrary constant. Family 4. When q 0 and r = p =0 If we select φ 27 (ξ), from Case 1 we obtain rational function solutions H 4.1 = 1 ± f 1(y) 2q2 Ξ (qξ + c) 2, u 4.1 = f 1(y)+f 3(t) 2q (qξ + c), where ξ = x + f 1 (y)t + f 2 (y)+f 3 (t), Ξ=f 1 (y)t + f 2 (y), cis an arbitrary constant. Family 5. When p 2 4qr > 0 and pq 0 (or qr 0) For example, if we select φ 1 (ξ), from Case 2 we obtain soliton-lie solutions H 5.1 = 1 2qrΞ ± f 1 (y) u 5.1 = p f 1(y)+f 3(t) 2qrΞ[4pMtanh(Mξ)+p2 +4M 2 ] [p +2Mtanh(Mξ)] 2, ± 4qr p +2Mtanh(Mξ),

6 S. Zhang and T. C. Xia 63 where ξ = x + f 1 (y)t + f 2 (y)+f 3 (t), Ξ=f 1(y)t + f 2(y). Family 6. When p 2 4qr < 0 and pq 0 (or qr 0) For example, if we select φ 15 (ξ), from Case 2 we obtain periodic formal solutions H 6.1 = 1 2qrΞ ± f 1(y) u 6.1 = p f 1(y)+f 3(t) 2qrΞ[4pN(tan(2Nξ) ± sec(2nξ)) p2 +4N 2 ] [ p +2N(tan(2Nξ) ± sec(2nξ))] 2, 4qr p +2N[tan(2Nξ) ± sec(2nξ)], where ξ = x + f 1 (y)t + f 2 (y)+f 3 (t), Ξ=f 1(y)t + f 2(y). Family 7. When r = 0 and pq 0 From Case 2 we obtain rational function solutions H 7.1 = 1 ± f 1(y), u 7.1 = p f 1(y) +f 3(t). Family 8. When q 0 and r = p =0 From Case 2 we obtain rational function solutions H 8.1 = 1 ± f 1(y), u 8.1 = f 1(y) +f 3(t). Family 9. When p 2 4qr > 0 and pq 0 (or qr 0) For example, if we select φ 2 (ξ), from Case 3 we get soliton-lie solutions H 9.1 = 1 8q 2 r 2 f 1(y) [p +2Mcoth(Mξ)] 2 + 4pqrf 1(y) p +2Mcoth(Mξ) f 1(y)[p 2 4M 2 coth 2 (Mξ)], u 9.1 = f 3(t) 4qr p +2Mcoth(Mξ) 2Mcoth(Mξ), where ξ = x + f 1 (y)+f 3 (t) Family 10. When p 2 4qr < 0 and pq 0 (or qr 0) For example, if we select φ 13 (ξ), from Case 3 we get periodic formal solutions 8q 2 r 2 f H 10.1 = 1 1(y) [ p +2Ntan(Nξ)] 2 4pqrf 1(y) p +2Ntan(Nξ) 1 2 f 1(y)[ p 2 +4N 2 tan 2 (Nξ)], u 10.1 = f 3(t) ± 4qr p +2Ntan(Nξ) ± 2Ntan(Nξ), where ξ = x + f 1 (y)+f 3 (t). Family 11. When r = 0 and pq 0 For example, if we select φ 25 (ξ), from Case 3 we get soliton-lie solutions H 11.1 = 1+ 2p2 df 1(y)[cosh(pξ) sinh(pξ)] [d + cosh(pξ) sinh(pξ)] 2, u 11.1 = ±p f 3(t) 2pd d + cosh(pξ) sinh(pξ),

7 64 Tanh Method for Dispersive Long Wave Equations where ξ = x + f 1 (y)+f 3 (t), dis an arbitrary constant. Family 12. When q 0 and r = p =0 If we select φ 27 (ξ), from Case 3 we get rational function solutions H 12.1 = 1 2q2 f 1(y) (qξ + c) 2, u 12.1 = f 3(t) 2q (qξ + c), where ξ = x + f 1 (y)+f 3 (t), cis an arbitrary constant. From Cases 1 3, Appendix 1 in [18] and (8) (10), we can also obtain other more general exact solutions of equations (1) and (2), we omit them here for simplicity. These solutions obtained contain some arbitrary functions, which can mae us discuss the behaviors of solutions and also provide us enough freedom to construct solutions that may be related to real physical problem. As an illustrative example, one plot of H 1.1 is shown in Figure 1, from which we can see that H 1.1 possesses solitonic features. 2 H x y 10 Figure 1. Plot of H 1.1 is shown at f 1 (y) =sn(y 0.5), f 2 (y) =cn(y 0.5), f 3 (t) = sin(t), = q = r =1,p=3,t= π/2, and the sign ± selected by +. REMARK 1. These solutions presented above have not been obtained in [19 21]. Case 1 can be obtained by using the method [17]. However, Cases 2 and 3 can not be obtained by the methods [10 17]. It shows that our method is more powerful in constructing exact solutions of NLPDEs. All solutions reported in this paper have been checed with Mathematica by putting them bac into equations (1) and (2). 4 Conclusion By using a further improved tanh function method, we have constructed many more general exact solutions of the (2+1)-dimensional DLW equations. The arbitrary functions in the solutions imply that these solutions have rich local structures. It may be important to explain some physical phenomena. Acnowledgments. The project supported by the Natural Science Foundation of Educational Committee of Liaoning Province of China under Grant No

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