A new method for finding exact traveling wave solutions to nonlinear partial differential equations

23 July 2001

Physics Letters A 286 (2001) 175–179 www.elsevier.com/locate/pla

A new method for finding exact traveling wave solutions to nonlinear partial differential equations Jianlan Hu, Hanlin Zhang ∗ College of Applied Science, Beijing Polytechnic University, Beijing 100022, PR China Received 10 January 2001; received in revised form 16 April 2001; accepted 17 April 2001 Communicated by A.R. Bishop

Abstract In this Letter, we find traveling wave solutions to some nonlinear partial differential equations via using a new ansatz.  2001 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear partial differential equation; Ansatz method; Traveling wave solution

1. Introduction It is well known that there are many nonlinear partial equations in the studying of physics, mechanics and biologics. The solution of these equations can make people know deeply the described process. But because of the complexity of nonlinear partial differential equations and the limitations of mathematics methods, it is difficult for us to get the exact solutions for the problems. So that it hinders its further applications. In the recent decades, there has been great development in the exact solutions for nonlinear integrable and unintegrable systems. Up to now, there exist many methods of constructing exact solutions, for instance, inverse scattering method [1,2], Backlund transformation method [3–5], Painleve analytic method [6], Lie group method [7], direct algebra method [7], rank analytic method [8]. Of the above methods, the rank analytical method is a powerful tool for finding the exact solutions of unintegrable dynamic problems. The method is easy and has a strong operability. In this Letter, we have successfully found rich analytical traveling wave solutions for a few kinds of nonlinear physical models by introducing a new ansatz u = (Au − a)(Bu − b),

(1)

where A, B, a, b are constants to be decided. Of the solutions we got, there are soliton ones and many unlimited ones. * Corresponding author.

E-mail address: [email protected] (H. Zhang). 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 2 9 1 - 2

176

J. Hu, H. Zhang / Physics Letters A 286 (2001) 175–179

2. Nonlinear interaction model I [9] of chromatic dispersion and dissipation In this section, we discuss the nonlinear interaction model I of dispersion and dissipation which is Eq. (19) in [9] ut + E 2 ux + (−E + ∂x )∂x [ux − µum + Eu] = 0,

(2)

where E, µ are constants. Here we consider only the case m = 2. Let u(x, t) = u(ξ ), ξ = x − λt, where λ is the speed of propagating waves, Eq. (2) is thus transformed to a nonlinear ordinary differential equation as −λu + Eµu2 + u − 2µuu = F,

(3)

where F is an integration constant. From the ansatz (1), we have
 u = 2A2 B 2 u3 − 3AB(Ab + Ba)u2 + 2ABab + (Ab + Ba)2 u − ab(Ab + Ba). Substituting

u ,

u

(4)

into (3), then comparing the order of u, we get

−ab(Ab + aB) = F, 2A2B 2 − 2µAB = 0, 
−λ + 2ABab + (Ab + aB)2 − 2µab = 0, Eµ − 3AB(Ab + Ba) + 2µ(Ab + aB) = 0. From the above equations, we have AB = µ,

Ab + aB = E,

A = 0,

B = 0,

λ = E2,

E = −2Ab,

F = −abE

(5)

or

λ = A b − 2µab, 2 2

F = −ab A. 2

(6)

(Remark: the case B = 0, A = 0 is analogous and omitted.) So in the case of (5), we have the following two kinds of solutions: (I) u =

c1 b exp[(aB − Ab)(x − E 2 t)] − c2 a , c1 B exp[(aB − Ab)(x − E 2 t)] − c2 A

where AB = µ, Ab + aB = E, λ = E 2 , aB = bA, c12 + c22 > 0, and c1 , c2 are integration constants. (II) u =

a A

or u =

a 1 − , A A[B(x − E 2 t) − c]

where AB = µ, λ = E 2 , aB = bA = E/2, and c is a constant. In the case of (6), the have the solution      2 c E a E − 2µab t , u = + exp x− A A 2 4 where A = 0, B = 0, E = −2Ab, λ = A2 b2 − 2µab.

3. Nonlinear interaction model II [9] of chromatic dispersion and dissipation In this section, we discuss the nonlinear interaction model II of chromatic dispersion and dissipation which is described by Eq. (25) in [9] ut − αux + αu3x + uxxx = µu2xx ,

(7)

J. Hu, H. Zhang / Physics Letters A 286 (2001) 175–179

177

where α = 4µ2 /9, and µ is an integration constant. Let u(x, t) = u(ξ ), ξ = x − λt; by integrating Eq. (7), we have −λu − αu + αu3 + u = 2µuu + E,

(8)

where E is an integration constant. Substituting (1), (4) into (8), and comparing the order of u, we have −ab(Ab + Ba) = E, α − 2µAB + 2A2 B 2 = 0, 
−(α + λ) − 2µab + 2ABab + (Ab + Ba)2 = 0, 2µ(Ab + Ba) − 3AB(Ab + Ba) = 0. From the above equations, we get 2 4 AB = µ, Ab + aB = 0, E = 0, λ = − µ2 − 3 9 2 Ab + aB = 0, E = −ab(Ab + aB)0, AB = µ, 3 4 1 Ab + aB = 0, E = 0, λ = − µ2 − AB = µ, 3 9 So, we have the following traveling wave solutions: 2 AB = µ, 3 η u=

, 2 1 − 3 µ x + 49 µ2 t − δ 1 (II) a = b = 0, AB = µ, 3 η u=

, 1 − 13 µ x + 49 µ2 t − δ (I) a = b = 0,

Ab + Ba = 0,

2 µab, 3 4 2 λ = − µ2 − µab + (Ab + aB)2 , 9 3 4 µab. 3

4 λ = − µ2 , 9

u(δ) = η; Ab + Ba = 0,

4 λ = − µ2 , 9

u(δ) = η;

2 4 2 (III) abAB < 0, Ab + Ba = 0, AB = µ, λ = − µab − µ2 , 3 3 9  



η − 23 µab + ab tanh − 23 µab x + 23 µab + 49 µ2 t − δ u(δ) = η; u=  



, − 23 µab − 23 µη tanh − 23 µab x + 23 µab + 49 µ2 t + δ 4 4 1 λ = − µab − µ2 , (IV) abAB < 0, Ab + Ba = 0, AB = µ, 3 3 9  



2 1 1 4 2 η − 3 µab + ab tanh − 3 µab x + 3 µab + 9 µ t − δ u=  u(δ) = η; 



, − 13 µab − 13 µη tanh − 13 µab x + 23 µab + 49 µ2 t + δ 4 2 2 λ = − µ2 − µab + 4a 2B 2 , AB = µ, 3 9 3 a 1 a or u = − , where C is a constant, u= A A A[B(x − λt) − C] 2 4 2 Ab = aB, AB = µ, λ = − µ2 − µab + (Ab + Ba)2 , 3 9 3 c1 b exp[(aB − Ab)(x − λt)] − c2 a , where c12 + c22 > 0, c1 , c2 are constants; u= c1 B exp[(aB − Ab)(x − λt)] − c2 A

(V) Ab = Ba = 0,

(9) (10) (11)

178

J. Hu, H. Zhang / Physics Letters A 286 (2001) 175–179

4 1 4 λ = − µ2 − µab, AB = µ, 3 9 3 c1 b exp[(aB − Ab)(x − λt)] − c2 a 2 2 , where c1 + c2 > 0, c1 , c2 are constants. u= c1 B exp[(aB − Ab)(x − λt)] − c2 A

(VI) Ab = aB,

Ab + aB = 0,

Comparing with Ref. [9], which got the exact solution of Eqs. (2) and (7) via a complicated method of parameter variation and stated that they could get confined solutions of twisted Kink form by using the form u = u0 + sF (s), s = eaξ , our method to solve the problem is direct and easy. What is more, we got more kinds of solutions.

4. Complex KdV–Burgers Model [10] In this section, we consider the complex KdV–Burgers model which is described as the following equation: ut + puux + qu2ux + ruxx − suxxx = 0. After transformation  6 u→r u, sq

s2 t → 3 t, r

(12)  p σ= r

s x → x, r

6s , q

Eq. (12) became ut + σ uux + 6u2 ux + uxx − uxxx = 0.

(13)

Let u(x, t) = u(ξ ), ξ = x − λt; by integrating Eq. (12) once, we have σ −λu + u2 + 2u3 + u − u = E, (14) 2 where E is an integration constant. Substituting (1) and (3) into (14), and comparing the order of u, we got the following algebraic equation system: σ ab + ab(Ab + aB) = E, A2 B = 1, + AB + 3AB(Ab + aB) = 0, 2 −λ − (Ab + aB) − 2ABab − (Ab + aB)2 = 0. (15) The system (15) has two groups of solutions: Ab = 1, Ab = −1,

1 σ Ab + aB = − − , 6 3 Ab + aB =

σ 1 − , 6 3



 2 σ E= − ab, 3 6   2 σ E= + ab, 3 6

λ = −2ab − λ = 2ab −

σ 2 σ2 + + ; 36 18 9

σ2 σ 2 − + . 36 18 9

So, we have the following traveling wave solutions: (I) a = b = 0, (II) a = b = 0,

AB = 1, AB = −1,

σ = −2, σ = −2,

λ = 0,

u(δ) = η,

λ = 0,

(III) AB = 1, σ = −2, Ab = −Ba, λ = −2ab, √ √ η −ab + ab tanh[ −ab (x + 2abt − δ)] u= √ √ ; −ab + η tanh[ −ab (x + 2abt − δ)] (IV) AB = −1, σ = 2, Ab = −Ba, λ = 2ab,

u(δ) = η, ab < 0,

ab > 0,

η ; 1 − (x − δ) η u= ; 1 + (x − δ) u(δ) = η,

u=

u(δ) = η,

(16) (17)

J. Hu, H. Zhang / Physics Letters A 286 (2001) 175–179

179

√ √ η ab + ab tanh[ ab (x − 2abt − δ)] u= √ ; √ ab + η tanh[ ab (x − 2abt − δ)] (V) Ab = aB = 0, u=

AB = 1,

1 a − , A A[B(x − λt) − c]

(VI) Ab = aB = 0,

AB = −1,

σ = −2,

λ = −2ab −

σ 2 σ2 + + , 36 18 9

where c is a constant; σ = 2,

λ = −2ab −

σ2 σ 2 − + , 36 18 9

1 a − , where c is a constant; A A[B(x − λt) − c] (VII) Ab = aB, a = 0 or b = 0 or ab = 0, AB = 1, σ2 σ 2 σ 1 λ = −2ab − + + , Ab + Ba = − − , 36 18 9 6 3 c1 b exp[(aB − Ab)(x − λt)] − c2 a , where c12 + c22 > 0, c1 , c2 are constants; u= c1 B exp[(aB − Ab)(x − λt)] − c2 A (VIII) Ab = aB, a = 0 or b = 0 or ab = 0, AB = −1, 2 σ σ 2 σ 1 λ = 2ab − − + , Ab + Ba = − , 36 18 9 6 3 c1 b exp[(aB − Ab)(x − λt)] − c2 a u= , where c12 + c22 > 0, c1 , c2 are constants. c1 B exp[(aB − Ab)(x − λt)] − c2 A u=

From the expression of the solutions, we know that the solutions of papers [10,11] are special cases of our solutions (III) and (IV). It is evident that our method can be used into the following (2 + 1)D complex KdV– Burgers equation:

ut + puux + qu2 ux + ruxx − suxxx x + εuyy = 0, where p, q, r, s, ε are constants. We have successfully found the solutions for three kinds of nonlinear partial differential equations. The method is very evident and easy. And it is hopeful to solve other nonlinear problems.

References [1] M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 (1993) 851. [2] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems study, Appl. Phys. 53 (1974) 249. [3] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. [4] C. Rogers, W.F. Shadwich, Backlund Transformations and their Application, Academic Press, New York, 1982. [5] A.H. Khater, O.H. El-kalauwy, Two new classes of exact solutions for the KdV equation via Backlund Transformation, Chaos, Solitons and Fractals 8 (1997) 1901. [6] J. Weiss, M. Tabor, G. Carnevale, The Painleve property for partial differential equations, J. Math. Phys. 24 (1983) 522. [7] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, Berlin, 1986. [8] X. Feng, Exploratory approach to explicit solution of nonlinear evolution equations, Int. J. Theor. Phys. 39 (1) (2000) 207–222. [9] P. Rosenau, On a class of nonlinear dispersive-dissipative interactions, Physica D 123 (1998) 525–546. [10] E.J. Parkes, B.R. Duffy, Traveling solitary wave solutions to a compound KdV–Burgers equation, Phys. Lett. A 229 (1997) 217–220. [11] M. Wang, Phys. Lett. A 213 (1996) 279.