Kinks and travelling wave solutions for Burgers-like equations

Accepted Manuscript Kinks and travelling wave solutions for Burgers-like equations Abdul-Majid Wazwaz PII: DOI: Reference:

S0893-9659(14)00250-X http://dx.doi.org/10.1016/j.aml.2014.08.003 AML 4609

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Applied Mathematics Letters

Received date: 2 July 2014 Accepted date: 1 August 2014 Please cite this article as: A.-M. Wazwaz, Kinks and travelling wave solutions for Burgers-like equations, Appl. Math. Lett. (2014), http://dx.doi.org/10.1016/j.aml.2014.08.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Kinks and travelling wave solutions for Burgers-like equations Abdul-Majid Wazwaz

∗

Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA

Abstract In this work we develop a variety of Burgers-like equations. We show that these derived equations share some of the travelling wave solutions of the Burgers equation, and differ in some other solutions. Keywords: Burgers-like equation; travelling wave solutions.

1

Introduction

The standard Burgers equation reads ut + uux + uxx = 0.

(1)

This equation is the lowest order approximation for the one-dimensional propagation of weak shock waves in a fluid [1-2]. The Burgers equation model the coupling between dissipation effect uxx and the convection effect uux . It is completely integrable and admits multiple-kink solutions. The kink solution of the Burgers equation is given by
u(x, t) = 2k 1 + tanh(kx − 2k 2 t) . (2) Moreover, the singular solution


u(x, t) = 2k 1 + coth(kx − 2k 2 t) ,

(3)

satisfies the Burgers equation (1). Studies on nonlinear evolution equations are growing rapidly because these equations describe real features in scientific applications. Powerful methods have been used to determine the solutions with distinct physical structures [1–10]. Examples of the methods that have been used are the Hirota bilinear method, the B¨acklund transformation method, and the inverse scattering method. Moreover, the derivation of new nonlinear equations has attracted much attention recently [11–17]. The new derived equations may describe significant features that are related to the well known models. It is the aim of this work to formulate new Burgers-like equations, where some of these equation share the same solutions with Burgers equation.

2

Formulation of the Burgers-like equations

Following [3], a generalized form of an advection dissipation equation is considered as ut + V ux = δuxx , ∗ E-mail

address: [email protected]

1

(4)

2

A.M.Wazwaz /

where δ is an arbitrary dimensionless parameter and V (u, ux , uxx , · · ·) is an arbitrary function. We assume the travelling wave to be of the form u(x, t) = f (x − ct) = f (ξ),

(5)

that solves the Burgers equation (1), and also solves the advection dissipation equation (4) for the same speed c. Using ξ = x − ct transforms (1) and (4) to −cf ′ + f f ′ + f ′′ = 0,

(6)

−cf ′ + V f ′ − δf ′′ = 0.

(7)

V = (δ + 1)c − δf.

(8)

and Eliminating f from these two equations, and by noting that f 6= 0, we obtain ′′

′

The advection dissipation equation, or the Burgers-like equation can be simply obtained by using a variety of values of the speed c. From (6), we find f ′′ . f′

(9)

f′ 1 f+ . 2 f

(10)

c=f+ Integrating (6) one time and solving for c we find c=

To determine more values for the speed c, we can differentiate (6) as many times as we want [3]. For example, differentiating (6) once, two times, and three times and solving for c we find (f ′ )2 + f ′′′ , f ′′

(11)

3f ′ f ′′ + f (iv) , f ′′′

(12)

4f ′ f ′′′ + 3(f ′′ )2 + f (v) , f (iv)

(13)

c=f+

c=f+ and c=f+

respectively. By differentiating as many times as we want we can determine many values for c. Substituting (9) –(13) into (8) gives   f ′′ V1 = (δ + 1) f + ′ − δf, f V2 = (δ + 1)



1 f′ f+ 2 f



− δf,

  (f ′ )2 + f ′′′ − δf, V3 = (δ + 1) f + f ′′   3f ′ f ′′ + f (iv) V4 = (δ + 1) f + − δf, f ′′′

(14)

(15) (16) (17)

A.M.Wazwaz /

and

3

  4f ′ f ′′′ + 3(f ′′ )2 + f (v) − δf. V5 = (δ + 1) f + f (iv)

(18)

   4ux uxxx + 3u2xx + uxxxxx ux − δuxx = 0, ut + u + (1 + δ) uxxxx

(23)

Substituting the last results of Vi , 1 ≤ i ≤ 5 into (4), we obtain the following Burgers-like equations   uxx ut + u + (δ + 1) ux − δuxx = 0, (19) ux   ux 1 ux − δuxx = 0, (20) (1 − δ)u + (1 + δ) ut + 2 u   2  ux + uxxx ut + u + (1 + δ) ux − δuxx = 0, (21) uxx    3ux uxx + uxxxx ux − δuxx = 0, (22) ut + u + (1 + δ) uxxx and

It is to be noted that (19) is the Burgers equation. The Burgers–like equations (21)–(23) involve higher order derivatives than the dissipative term uxx of the Burgers equation. Moreover, the Burgers–like equations (20)–(23) share the same kink and singular solutions (2) and (3). This can be verified by substitution. In this work, our main focus will be on determining travelling wave solutions for the Burgers-like equations (20)–(23), will be referred to as form I, form II, form III, and form IV respectively. We will also show that these forms share some solutions with the Burgers equation and have other distinct solutions in other cases. To achieve our goals we will use several tools that will be applied in order to extract exact solutions.

3

The kink solutions

We first substitute the kink assumption u(x, t) =

R , 1 + ekx−ct

(24)

into (1) and solve the resulting equation to find that c R

= =

−k 2 , −2k.

(25)

Consequently, the kink solution

2k , (26) 1 + ekx+k2 t satisfies all four forms of the Burgers-like equations in addition to the Burgers equation. In a like manner, we can show that the kink solution 2k , (27) u(x, t) = 1 + e−(kx−k2 t) satisfies all four forms of the Burgers-like equations in addition to the Burgers equation. In a parallel manner, we can that these equations are also satisfied by the singular solutions u(x, t) = −

u(x, t) = −

2k , 1 − ekx+k2 t

(28)

4

A.M.Wazwaz /

and

2k . (29) 1 − e−(kx−k2 t) It is worth noting that the obtained solutions do not depend on the parameter δ and satisfy the Burgers-like equations for any δ. u(x, t) =

3.1

More kink solutions

To derive more kink solutions, we assume that the solution for (1) has the form u(x, t) = 1 + R tanh(kx − ct).

(30)

Substituting this assumption into (1), and solving the resulting equation for R and c, we find c = R =

k, 2k,

(31)

This in turn gives the kink solution u(x, t) = 1 + 2k tanh(kx − k t).

(32)

In a like manner, we can also derive the singular equation u(x, t) = 1 + 2k coth(kx − k t).

(33)

It is worth noting that, unlike the previous kink solutions (26) and (27) that satisfy all equations, the solutions (32) and (33) satisfy only the forms II, III, and IV for any δ in addition to the Burgers equation and do not satisfy the form I.

4

The periodic solutions

To derive periodic solutions, we assume that the solution for (1) has the form u(x, t) = 1 + R tan(kx − ct).

(34)

Substituting this assumption into (1), and solving the resulting equation for R and c, we find c R

= =

k, −2k,

(35)

This in turn gives the solution u(x, t) = 1 − 2k tan(kx − k t).

(36)

Similarly, we can derive the singular solution u(x, t) = 1 + 2k cot(kx − k t).

(37)

Notice that (36) and (37) satisfy the Burgers equation, and only the Burgers-like forms II, III and IV.

5

Solutions for Burgers-like equations only

It remains to show that there are exact solutions that satisfy the Burgers-like equations only but they do not satisfy the Burgers equation. We find that such solutions exist for specific values of the parameter δ.

A.M.Wazwaz /

5.1

5

The Burgers-like equation: form I

Using δ = 1, Eq. (20) becomes ut + We assume that the solution is of the form



2ux u



ux − uxx = 0.

u(x, t) = Rekx−rt ,

(38)

(39)

where by substituting this assumption into (38) and solving the resulting equation we find r = k2 ,

(40)

and R can be any real number. This gives the exact solution 2

u(x, t) = Rekx−k t .

(41)

In a like manner, we can also assume the solution u(x, t) = Re−(kx−rt) ,

(42)

r = −k 2 ,

(43)

and by proceeding as earlier we find and R can be any real number. This gives the exact solution u(x, t) = Re−(kx+k

2

)t

.

(44)

Notice that (41) and (44) satisfy form I and do not satisfy Burgers equatiom.

5.2

The Burgers-like equation: form II

Using δ = −2, Eq. (21) becomes

  2  ux + uxxx ut + u − ux + 2uxx = 0. uxx

(45)

We assume that this form has the solution u(x, t) = Rekx−rt ,

(46)

where by substituting this assumption into (45) and solving the resulting equation we find r = k2 ,

(47)

and R can be any real number. This gives the exact solution 2

u(x, t) = Rekx−k t .

(48)

In a like manner, we can derive the exact solution u(x, t) = Re−(kx+k

2

)t

.

(49)

We can observe that the derived solutions satisfy the form II for δ = −2, and do not satisfy the Burgers equation.

6

5.3

A.M.Wazwaz /

The Burgers-like equation: form III

Using δ = − 34 , Eq. (22) becomes

   1 3ux uxx + uxxxx 4 ut + u − ux + uxx = 0. 3 uxxx 3

(50)

Proceeding as before we obtain 2

u(x, t) = Rekx−k t . and u(x, t) = Re−(kx+k

2

)t

(51) ,

(52)

that satisfy form III and do not satisfy the Burgers equation.

5.4

The Burgers-like equation: form IV

Using δ = − 78 , Eq. (23) and following the same analysis we find 2

u(x, t) = Rekx−k t . and u(x, t) = Re−(kx+k

2

)t

(53)

,

(54)

that satisfy form IV and not the Burgers equation.

6

Solutions for Burgers equation only

In what follows we list some of the solutions that satisfy the Burgers equation (1) only and do not satisfy any of the Burgers-like equations 2 cosh x , (55) u(x, t) = sinh x ± et 2 sinh x , cosh x ± et

(56)

u(x, t) =

x x 2 + tanh , t t t

(57)

u(x, t) =

x 2 x + coth , t t t

(58)

u(x, t) =

x 2 x − tan , t t t

(59)

u(x, t) =

x 2 x + cot , t t t

(60)

u(x, t) =

7

Discussion

We formally derived a variety of Burgers-like equations. We showed that these derived equations share some of the travelling wave solutions of the Burgers equation, and differ in some other solutions. The study revealed that many nonlinear equations, even with terms of higher derivatives can give the same solutions.

A.M.Wazwaz /

7

References [1] A.M.Wazwaz, Partial Differential Equations and Solitary Waves Theorem, Springer and HEP, Berlin, (2009). [2] A. M. Wazwaz, A study on a (2+1)-dimensional and a (3+1)-dimensional generalized Burgers equations, Applied Mathematics Letters, 31 (2014) 41–45. [3] A. Sen, D.P. Ahalpara, A. Thyagaraja, G. S. Krishnaswami, A KdV–like advection-dispersion equation with some remarkable properties, Commun Nonlinear Sci Numer Simulat 17 (2012) 4115-4124. [4] S. Kumar, Le. Wei, X. Zhang, Numerical study based on an implicit fully discreate local discontinuous Galerkin method for time fractional coupled Schrodinger system, Computer and Mathematics with application, 64 (8), (2012) 2603-2615. [5] S. Kumar, L. Wei, Y. He, Numerical study based on an implicit fully discreate local discontinuous Galerkin method for time fractional KdV- Burgers Kuramoto equation, JAMM Journal of Applied Mathematics and Mechanics, 93 (1) (2013) 14–28 . [6] H. Leblond and D. Mihalache, Models of few optical cycle solitons beyond the slowly varying envelope approximation, Phys. Rep., 523 (2013) 61–126. [7] H. Leblond and D. Mihalache, Few–optical–cycle solitons: Modified Korteweg-de Vries sine-Gordon equation versus other non-slowly-varying–envelope–approximation models, Phys. Rev. A 79, (2009) 063835. [8] M. Dehghan and F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy 13, (2009) 53–59. [9] M. Dehghan, A. Ghesmati, Numerical simulation of two–dimensional sine–Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM), Computer Physics Communications, 181 (2010) 772–786 . [10] C. M. Khalique, F. M. Mahomed, B. Muatjetjeja, Lagrangian formulation of a generalized Lane– Emden equation and double reduction, J. Nonlinear Math. Phys. 15 (2008) 152–161. [11] C. M. Khalique, On the solutions and conservation laws of a coupled Kadomtsev-Petviashvili equation, Journal of Applied Mathematics, 2013 (2013) 1–7. [12] C. M. Khalique, K. R. Adem, Exact solutions of the (2+1 )-dimensional Zakharov-Kuznetsov modified equal width equation using Lie group analysis, Mathematical and Computer Modelling, 54(1/2) (2011) 184–189. [13] A. Ebaid and A.M.Wazwaz, On the generalized Exp-function method and its application to boundary layer flow at nano-scale, Journal of Computational and Theoretical Nanoscience, 11 (2014) 178–184. [14] A. M. Wazwaz, A new fifth order nonlinear integrable equation: multiple soliton solutions, Physica Scripta, 83 (2011) 015012. [15] A. M. Wazwaz, A new generalized fifth order nonlinear integrable equation, Physica Scripta, 83 (2011) 035003. [16] A.M. Wazwaz, A modified KdV–type equation that admits a variety of travelling wave solutions: kinks, solitons, peakons and cuspons, Physica Scripta, 86 (2012) 045501. [17] A. M. Wazwaz, Soliton solutions of the modified KdV6, modified (2+1)-dimensional Boussinesq equation, and (3+1)-dimensional KdV equation, Journal of Applied Nonlinear Dynamics, (2014) 95–104.