Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
VolumeΒ 2011, Article IDΒ 424801, 11 pages
http://dx.doi.org/10.1155/2011/424801
Research Article

Hyperbolic, Trigonometric, and Rational Function Solutions of Hirota-Ramani Equation via (𝐺′/𝐺)-Expansion Method

1Young Researchers Club, Ardabil Branch, Islamic Azad University, P.O. Box 56169-54184, Ardabil, Iran
2Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran

Received 13 November 2010; Accepted 22 February 2011

Academic Editor: PeterΒ Wolenski

Copyright Β© 2011 Reza Abazari and Rasoul Abazari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The (πΊξ…ž/𝐺)-expansion method is proposed to construct the exact traveling solutions to Hirota-Ramani equation: π‘’π‘‘βˆ’π‘’π‘₯π‘₯𝑑+π‘Žπ‘’π‘₯(1βˆ’π‘’π‘‘)=0, where π‘Žβ‰ 0. Our work is motivated by the fact that the (πΊξ…ž/𝐺)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.

1. Introduction

Nonlinear evolution equations (NLEEs) have been the subject of study in various branches of mathematical-physical sciences such as physics, biology, and chemistry. The analytical solutions of such equations are of fundamental importance since a lot of mathematical-physical models are described by NLEEs. Among the possible solutions to NLEEs, certain special form solutions may depend only on a single combination of variables such as solitons. In mathematics and physics, a soliton is a self-reinforcing solitary wave, a wave packet or pulse, that maintains its shape while it travels at constant speed. Solitons are caused by a cancelation of nonlinear and dispersive effects in the medium. The term β€œdispersive effects” refers to a property of certain systems where the speed of the waves varies according to frequency. Solitons arise as the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the β€œWave of Translation” [1]. Many exactly solvable models have soliton solutions, including the Korteweg-de Vries equation, the nonlinear SchrΓΆdinger equation, the coupled nonlinear SchrΓΆdinger equation, the sine-Gordon equation, and Gardner equation. The soliton solutions are typically obtained by means of the inverse scattering transform [2] and owe their stability to the integrability of the field equations. In the past years, many other powerful and direct methods have been developed to find special solutions of nonlinear evolution equations (NEE(s)), such as the Bcklund transformation [3], Hirota bilinear method [4], numerical methods [5], and the Wronskian determinant technique [6]. With the help of the computer software, many algebraic methods are proposed, such as tanh method [7], F-expanded method [8], homogeneous balance method [9], Jacobi elliptic function method [10], the Miura transformation [11], and some other new methods [12, 13].

Recently, the (𝐺′0π‘₯02044𝐺)-expansion method, firstly introduced by Wang et al. [14], has become widely used to search for various exact solutions of NLEEs [14–18]. The value of the (𝐺′0π‘₯02044𝐺)-expansion method is that one treats nonlinear problems by essentially linear methods. The method is based on the explicit linearization of NLEEs for traveling waves with a certain substitution which leads to a second-order differential equation with constant coefficients. Moreover, it transforms a nonlinear equation to a simple algebraic computation.

Our first interest in the present work is in implementing the (𝐺′0π‘₯02044𝐺)-expansion method to stress its power in handling nonlinear equations, so that one can apply it to models of various types of nonlinearity. The next interest is in the determination of new exact traveling wave solutions for the Hirota-Ramani equation [11–13]:π‘’π‘‘βˆ’π‘’π‘₯π‘₯𝑑+π‘Žπ‘’π‘₯ξ€·1βˆ’π‘’π‘‘ξ€Έ=0,(1.1) where π‘Žβ‰ 0 is a real constant and 𝑒(π‘₯,𝑑) is the amplitude of the relevant wave mode. This equation was first introduced by Hirota and Ramani in [11]. Ji obtained some travelling soliton solutions of this equation by using Exp-function method [13]. This equation is completely integrable by the inverse scattering method. Equation (1.1) is studied in [11–13] where new kind of solutions were obtained. Hirota-Ramani equation is widely used in various branches of physics, and such as plasma physics, fluid physics, and quantum field theory. It also describes a variety of wave phenomena in plasma and solid state [11].

2. Description of the (πΊξ…ž/𝐺)-Expansion Method

The objective of this section is to outline the use of the (𝐺′0π‘₯02044𝐺)-expansion method for solving certain nonlinear partial differential equations (PDEs). Suppose we have a nonlinear PDE for 𝑒(π‘₯,𝑑), in the form𝑃𝑒,𝑒π‘₯,𝑒𝑑,𝑒π‘₯π‘₯,𝑒π‘₯,𝑑,𝑒𝑑𝑑,…=0,(2.1) where 𝑃 is a polynomial in its arguments, which includes nonlinear terms and the highest-order derivatives. The transformation 𝑒(π‘₯,𝑑)=π‘ˆ(πœ‰),πœ‰=π‘˜π‘₯+πœ”π‘‘, reduces (2.1) to the ordinary differential equation (ODE)π‘ƒξ€·π‘ˆ,π‘˜π‘ˆξ…ž,πœ”π‘ˆξ…ž,π‘˜2π‘ˆξ…žξ…ž,π‘˜πœ”π‘ˆξ…žξ…ž,πœ”2π‘ˆξ…žξ…žξ€Έ,…=0,(2.2) where π‘ˆ=π‘ˆ(πœ‰), and prime denotes derivative with respect to πœ‰. We assume that the solution of (2.2) can be expressed by a polynomial in (𝐺′0π‘₯02044𝐺) as follows:π‘ˆ(πœ‰)=π‘šξ“π‘–=1π›Όπ‘–ξ‚΅πΊξ…žπΊξ‚Άπ‘–+𝛼0,π›Όπ‘šβ‰ 0,(2.3) where 𝛼0, and 𝛼𝑖, are constants to be determined later, 𝐺(πœ‰) satisfies a second-order linear ordinary differential equation (LODE):𝑑2𝐺(πœ‰)π‘‘πœ‰2+πœ†π‘‘πΊ(πœ‰)π‘‘πœ‰+πœ‡πΊ(πœ‰)=0,(2.4) where πœ† and πœ‡ are arbitrary constants. Using the general solutions of (2.4), we have𝐺′(πœ‰)𝐺=⎧βŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺ⎩√(πœ‰)πœ†2βˆ’4πœ‡2βŽ›βŽœβŽœβŽπΆ1√sinhξ‚€ξ‚€πœ†2ξ‚πœ‰ξ‚βˆ’4πœ‡/2+𝐢2√coshξ‚€ξ‚€πœ†2ξ‚πœ‰ξ‚βˆ’4πœ‡/2𝐢1√coshξ‚€ξ‚€πœ†2ξ‚πœ‰ξ‚βˆ’4πœ‡/2+𝐢2√sinhξ‚€ξ‚€πœ†2ξ‚πœ‰ξ‚βŽžβŽŸβŽŸβŽ βˆ’πœ†βˆ’4πœ‡/22,πœ†2βˆšβˆ’4πœ‡>0,4πœ‡βˆ’πœ†22βŽ›βŽœβŽœβŽβˆ’πΆ1√sinξ‚€ξ‚€πœ†2ξ‚πœ‰ξ‚βˆ’4πœ‡/2+𝐢2√cosξ‚€ξ‚€πœ†2ξ‚πœ‰ξ‚βˆ’4πœ‡/2𝐢1√cosξ‚€ξ‚€πœ†2ξ‚πœ‰ξ‚βˆ’4πœ‡/2+𝐢2√sinξ‚€ξ‚€πœ†2ξ‚πœ‰ξ‚βŽžβŽŸβŽŸβŽ βˆ’πœ†βˆ’4πœ‡/22,πœ†2βˆ’4πœ‡<0,(2.5) and it follows, from (2.3) and (2.4), thatπ‘ˆβ€²=βˆ’π‘šξ“π‘–=1π‘–π›Όπ‘–ξƒ©ξ‚΅πΊξ…žπΊξ‚Άπ‘–+1𝐺+πœ†ξ…žπΊξ‚Άπ‘–ξ‚΅πΊ+πœ‡ξ…žπΊξ‚Άπ‘–βˆ’1ξƒͺ,π‘ˆξ…žξ…ž=π‘šξ“π‘–=1𝑖𝛼𝑖𝐺(𝑖+1)ξ…žπΊξ‚Άπ‘–+2𝐺+(2𝑖+1)πœ†ξ…žπΊξ‚Άπ‘–+1ξ€·πœ†+𝑖2𝐺+2πœ‡ξ…žπΊξ‚Άπ‘–+𝐺(2π‘–βˆ’1)πœ†πœ‡ξ…žπΊξ‚Άπ‘–βˆ’1+(π‘–βˆ’1)πœ‡2ξ‚΅πΊξ…žπΊξ‚Άπ‘–βˆ’2ξƒͺ,(2.6) and so on, here the prime denotes the derivative with respect to πœ‰.

To determine 𝑒 explicitly, we take the following four steps.

Step 1. Determine the integer π‘š by substituting (2.3) along with (2.4) into (2.2), and balancing the highest-order nonlinear term(s) and the highest-order partial derivative.

Step 2. Substitute (2.3), given the value of π‘š determined in Step 1, along with (2.4) into (2.2) and collect all terms with the same order of (𝐺′/𝐺) together, the left-hand side of (2.2) is converted into a polynomial in (𝐺′/𝐺). Then set each coefficient of this polynomial to zero to derive a set of algebraic equations for π‘˜,πœ”,πœ†,πœ‡,𝛼0 and 𝛼𝑖 for 𝑖=1,2,…,π‘š.

Step 3. Solve the system of algebraic equations obtained in Step 2, for π‘˜,πœ”,πœ†,πœ‡,𝛼0 and 𝛼𝑖 by use of Maple.

Step 4. Use the results obtained in above steps to derive a series of fundamental solutions 𝑒(πœ‰) of (2.2) depending on (𝐺′/𝐺); since the solutions of (2.4) have been well known for us, then we can obtain exact solutions of (2.1).

3. Application on Hirota-Ramani Equation

In this section, we will use our method to find solutions to Hirota–Ramani equation [10–12]:π‘’π‘‘βˆ’π‘’π‘₯π‘₯𝑑+π‘Žπ‘’π‘₯ξ€·1βˆ’π‘’π‘‘ξ€Έ=0,(3.1) where π‘Žβ‰ 0. We would like to use our method to obtain more general exact solutions of (3.1) by assuming the solution in the following frame:𝑒=π‘ˆ(πœ‰),πœ‰=π‘˜π‘₯+πœ”π‘‘,(3.2) where π‘˜, πœ” are constants. We substitute (3.2) into (3.1) to obtain nonlinear ordinary differential equation(πœ”+π‘Žπ‘˜)π‘ˆβ€²βˆ’π‘˜2πœ”π‘ˆξ…žξ…žξ…žξ€·π‘ˆβˆ’π‘Žπ‘˜πœ”ξ…žξ€Έ2=0.(3.3) By setting π‘ˆξ…ž=𝑉,nonlinear ordinary differential equation (3.3) reduce to(πœ”+π‘Žπ‘˜)π‘‰βˆ’π‘˜2πœ”π‘‰ξ…žξ…žβˆ’π‘Žπ‘˜πœ”π‘‰2=0.(3.4) According to Step 1, we get π‘š+2=2π‘š, hence π‘š=2. We then suppose that (3.4) has the following formal solutions:𝑉=𝛼2ξ‚΅πΊξ…žπΊξ‚Ά2+𝛼1ξ‚΅πΊξ…žπΊξ‚Ά+𝛼0,𝛼2β‰ 0,(3.5) where 𝛼2,𝛼1, and 𝛼0, are unknown to be determined later.

Substituting (3.5) into (3.4) and collecting all terms with the same order of (𝐺′/𝐺), together, the left-hand sides of (3.4) are converted into a polynomial in (𝐺′/𝐺). Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for πœ†,πœ‡,𝛼0,𝛼1, and 𝛼2, as follows:ξ‚΅πΊξ…žπΊξ‚Ά0βˆΆξ€·βˆ’2πœ”π›Ό2πœ‡2βˆ’πœ”π›Ό1ξ€Έπ‘˜πœ†πœ‡2βˆ’π‘Žπ›Ό0ξ€·πœ”π›Ό0ξ€Έβˆ’1π‘˜+πœ”π›Ό0𝐺=0,ξ…žπΊξ‚Ά1βˆΆξ€·ξ€·βˆ’πœ”2πœ‡+πœ†2𝛼1βˆ’6πœ”π›Ό2ξ€Έπ‘˜πœ†πœ‡2ξ€·βˆ’π‘Žβˆ’1+2πœ”π›Ό0𝛼1π‘˜+πœ”π›Ό1𝐺=0,ξ…žπΊξ‚Ά2βˆΆξ€·βˆ’3πœ”π›Ό1πœ†βˆ’4πœ”π›Ό2ξ€·2πœ‡+πœ†2π‘˜ξ€Έξ€Έ2βˆ’ξ€·π‘Žπ›Ό2ξ€·βˆ’1+2πœ”π›Ό0ξ€Έ+π‘Žπœ”π›Ό12ξ€Έπ‘˜+πœ”π›Ό2𝐺=0,ξ…žπΊξ‚Ά3βˆΆξ€·βˆ’10πœ”π›Ό2πœ†βˆ’2πœ”π›Ό1ξ€Έπ‘˜2βˆ’2π‘Žπ‘˜πœ”π›Ό2𝛼1𝐺=0,ξ…žπΊξ‚Ά4βˆΆβˆ’6π‘˜2πœ”π›Ό2βˆ’π‘Žπ‘˜πœ”π›Ό22=0,(3.6) and solving by use of Maple, we get the following results.

Case 1. 1πœ†=βˆ’6π‘Žπ›Ό1π‘˜,1πœ‡=144βˆ’36π‘Žπ‘˜βˆ’36πœ”+π‘Ž2πœ”π›Ό21π‘˜2πœ”,𝛼0=124βˆ’36π‘Žπ‘˜βˆ’36πœ”+π‘Ž2πœ”π›Ό21,π›Όπ‘Žπ‘˜πœ”2=βˆ’6π‘˜π‘Ž,(3.7) where π‘˜, πœ”, and 𝛼1 are arbitrary constants. Therefore, substitute the above case in (3.5), and using the relationship βˆ«π‘ˆ(πœ‰)=𝑉(πœ‰)π‘‘πœ‰, we get ξ€œξƒ―βˆ’π‘ˆ=6π‘˜π‘Žξ‚΅πΊξ…žπΊξ‚Ά2+𝛼1ξ‚΅πΊξ…žπΊξ‚Ά+124βˆ’36π‘Žπ‘˜βˆ’36πœ”+π‘Ž2πœ”π›Ό21ξƒ°π‘Žπ‘˜πœ”π‘‘πœ‰.(3.8) Substituting the general solutions (2.5) into (3.8), we obtain three types of traveling wave solutions of (3.1) in view of the positive, negative, or zero of πœ†2βˆ’4πœ‡.
When π’Ÿ1=πœ†2βˆ’4πœ‡=(π‘Žπ‘˜+πœ”)/π‘˜2πœ”>0, using the integration relationship (3.8), we obtain hyperbolic function solution π‘ˆβ„‹, of Hirota-Ramani equation (3.1) as follows: π‘ˆβ„‹ξ€·πΆ(πœ‰)=6(π‘Žπ‘˜+πœ”)21βˆ’πΆ22ξ€Έπ‘Žπ‘˜πœ”πΆ1βˆšπ’Ÿ1Γ—ξ‚€βˆšsinh(1/4)π’Ÿ1πœ‰ξ‚ξ‚€βˆšcosh(1/4)π’Ÿ1πœ‰ξ‚2𝐢2ξ‚€βˆšsinh(1/4)π’Ÿ1πœ‰ξ‚ξ‚€βˆšcosh(1/4)π’Ÿ1πœ‰ξ‚+2𝐢1cosh2ξ‚€βˆš(1/4)π’Ÿ1πœ‰ξ‚βˆ’πΆ1,(3.9) where πœ‰=π‘˜π‘₯+πœ”π‘‘, and 𝐢1, 𝐢2, are arbitrary constants. This solution is shown in Figure 1 for π‘Ž=1, π‘˜=1, πœ”=1/2, 𝐢1=2, and 𝐢2=1. It is easy to see that the hyperbolic solution (3.9) can be rewritten at 𝐢21>𝐢22, as follows: 𝑒ℋ3(π‘₯,𝑑)=2(π‘Žπ‘˜+πœ”)βˆšπ‘Žπ‘˜πœ”π’Ÿ1⎧βŽͺ⎨βŽͺβŽ©ξ‚€12tanh2βˆšπ’Ÿ1πœ‰+πœ‚β„‹ξ‚βŽ›βŽœβŽœβŽξ‚€βˆš+lntanh(1/2)π’Ÿ1πœ‰+πœ‚β„‹ξ‚βˆ’1ξ‚€βˆštanh(1/2)π’Ÿ1πœ‰+πœ‚β„‹ξ‚βŽžβŽŸβŽŸβŽ +√+1π’Ÿ1πœ‰βŽ«βŽͺ⎬βŽͺ⎭,(3.10a) while at 𝐢21<𝐢22, one can obtain 𝑒ℋ3(π‘₯,𝑑)=2(π‘Žπ‘˜+πœ”)βˆšπ‘Žπ‘˜πœ”π’Ÿ1⎧βŽͺ⎨βŽͺβŽ©ξ‚€12coth2βˆšπ’Ÿ1πœ‰+πœ‚β„‹ξ‚βŽ›βŽœβŽœβŽξ‚€βˆš+lncoth(1/2)π’Ÿ1πœ‰+πœ‚β„‹ξ‚βˆ’1ξ‚€βˆšcoth(1/2)π’Ÿ1πœ‰+πœ‚β„‹ξ‚βŽžβŽŸβŽŸβŽ +√+1π’Ÿ1πœ‰βŽ«βŽͺ⎬βŽͺ⎭,(3.10b)where πœ‰=π‘˜π‘₯+πœ”π‘‘, πœ‚β„‹=tanhβˆ’1(𝐢1/𝐢2), and π‘˜, πœ”, are arbitrary constants.
Now, when π’Ÿ1=πœ†2βˆ’4πœ‡=((π‘Žπ‘˜+πœ”)/π‘˜2πœ”)<0, using the integration relationship (3.8), we obtain trigonometric function solution π‘ˆπ’―, of Hirota-Ramani equation (3.1) as follows: π‘ˆπ’―ξ€·πΆ(πœ‰)=βˆ’3(π‘Žπ‘˜+πœ”)21+𝐢22ξ€Έπ‘Žπ‘˜πœ”πΆ2βˆšβˆ’π’Ÿ11𝐢2ξ‚€βˆštan(1/2)βˆ’π’Ÿ1πœ‰ξ‚+𝐢1,(3.11) where πœ‰=π‘˜π‘₯+πœ”π‘‘, and 𝐢1, 𝐢2, are arbitrary constants. This solution is shown in Figure 2 for π‘Ž=1, π‘˜=βˆ’1, πœ”=1/2, 𝐢1=2, and 𝐢2=1. Similarity, it is easy to see that the trigonometric solution (3.11) can be rewritten at 𝐢21>𝐢22, and 𝐢21<𝐢22, as follows: 𝑒𝒯(π‘₯,𝑑)=3(π‘Žπ‘˜+πœ”)π‘Žπ‘˜πœ”πΆ2βˆšβˆ’π’Ÿ1ξ‚€1tan2βˆšβˆ’π’Ÿ1πœ‰+πœ‚π’―ξ‚,(3.12a)𝑒𝒯(π‘₯,𝑑)=βˆ’3(π‘Žπ‘˜+πœ”)π‘Žπ‘˜πœ”πΆ2βˆšβˆ’π’Ÿ1cotξ‚€12βˆšβˆ’π’Ÿ1πœ‰+πœ‚π’―ξ‚,(3.12b)respectively, where πœ‰=π‘˜π‘₯+πœ”π‘‘,πœ‚π’―=tanβˆ’1(𝐢1/𝐢2), and π‘˜,πœ”, are arbitrary constants.

424801.fig.001
Figure 1: Hyperbolic function solution (3.9) of Hirota-Ramani equation, for π‘Ž=1, π‘˜=1, πœ”=1/2, 𝐢1=2, and 𝐢2=1.
424801.fig.002
Figure 2: Trigonometric function solution (3.11) of Hirota-Ramani equation, for π‘Ž=1, π‘˜=βˆ’1, πœ”=1/2, 𝐢1=2, and 𝐢2=1.

Case 2. 1πœ†=βˆ’6π‘Žπ›Ό1π‘˜,1πœ‡=14436π‘Žπ‘˜+36πœ”+π‘Ž2πœ”π›Ό21π‘˜2πœ”,𝛼01=βˆ’2412π‘Žπ‘˜+12πœ”+π‘Ž2πœ”π›Ό21,π›Όπ‘Žπ‘˜πœ”2=βˆ’6π‘˜π‘Ž,(3.13) where π‘˜,πœ” and 𝛼1 is an arbitrary constant. Similar on the previous case, substitute the above case in (3.5), and using the relationship βˆ«π‘ˆ(πœ‰)=𝑉(πœ‰)π‘‘πœ‰, we get ξ€œξƒ―βˆ’π‘ˆ=6π‘˜π‘Žξ‚΅πΊξ…žπΊξ‚Ά2+𝛼1ξ‚΅πΊξ…žπΊξ‚Άβˆ’12412π‘Žπ‘˜+12πœ”+π‘Ž2πœ”π›Ό21ξƒ°π‘Žπ‘˜πœ”π‘‘πœ‰,(3.14) then for π’Ÿ2=πœ†2βˆ’4πœ‡=βˆ’((π‘Žπ‘˜+πœ”)/π‘˜2πœ”)>0, the hyperbolic and for π’Ÿ2=πœ†2βˆ’4πœ‡=βˆ’((π‘Žπ‘˜+πœ”)/π‘˜2πœ”)<0, trigonometric types of traveling wave solutions of Hirota-Ramani equation (3.1), are obtained as follows: π‘ˆβ„‹ξ€·πΆ(πœ‰)=2(π‘Žπ‘˜+πœ”)22βˆ’πΆ21ξ€Έβˆšπ‘Žπ‘˜πœ”π’Ÿ2Γ—βŽ§βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽξ‚€βˆšlntanh(1/4)π’Ÿ2πœ‰ξ‚+1ξ‚€βˆštanh(1/4)π’Ÿ2πœ‰ξ‚βŽžβŽŸβŽŸβŽ +ξ‚€βˆšβˆ’13tanh(1/4)π’Ÿ2πœ‰ξ‚πΆ21tanh2ξ‚€βˆš(1/4)π’Ÿ2πœ‰ξ‚+2𝐢1𝐢2ξ‚€βˆštanh(1/4)π’Ÿ2πœ‰ξ‚+𝐢21⎫βŽͺ⎬βŽͺ⎭,(3.15)π‘ˆπ’―(πœ‰)=(π‘Žπ‘˜+πœ”)π‘Žπ‘˜πœ”πΆ2βˆšβˆ’π’Ÿ2⎧βŽͺ⎨βŽͺ⎩𝐢2βˆšβˆ’π’Ÿ23ξ€·πΆπœ‰+12+𝐢22𝐢2ξ‚€βˆštan(1/2)βˆ’π’Ÿ2πœ‰ξ‚+𝐢1⎫βŽͺ⎬βŽͺ⎭,(3.16) respectively, where πœ‰=π‘˜π‘₯+πœ”π‘‘, and 𝐢1,𝐢2, are arbitrary constants. The trigonometric function solution (3.16), for π‘Ž=βˆ’1, π‘˜=βˆ’1, πœ”=1/2, 𝐢1=2 and 𝐢2=1 are shown in Figure 3. Similarly, to obtain some special forms of the solutions obtained above, we set 𝐢21>𝐢22, then hyperbolic and trigonometric function solutions (3.15)-(3.16) become 𝑒ℋ1(π‘₯,𝑑)=βˆ’2(π‘Žπ‘˜+πœ”)βˆšπ‘Žπ‘˜πœ”π’Ÿ2⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽξ‚€βˆš3lntanh(1/2)π’Ÿ2πœ‰+πœ‚β„‹ξ‚βˆ’1ξ‚€βˆštanh(1/2)π’Ÿ2πœ‰+πœ‚β„‹ξ‚βŽžβŽŸβŽŸβŽ ,ξ‚€1+1+6tanh2βˆšπ’Ÿ2πœ‰+πœ‚β„‹ξ‚+βˆšπ’Ÿ2πœ‰βŽ«βŽͺ⎬βŽͺ⎭,𝑒𝒯1(π‘₯,𝑑)=βˆ’2(π‘Žπ‘˜+πœ”)βˆšπ‘Žπ‘˜πœ”βˆ’π’Ÿ2ξ‚†ξ‚€βˆšβˆ’6tan(1/2)βˆ’π’Ÿ2πœ‰+πœ‚π’―ξ‚+6πœ‚π’―ξ‚‡,(3.17) while at 𝐢21<𝐢22, the hyperbolic and trigonometric function solutions (3.15)-(3.16) become 𝑒ℋ1(π‘₯,𝑑)=βˆ’2(π‘Žπ‘˜+πœ”)βˆšπ‘Žπ‘˜πœ”π’Ÿ2⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽξ‚€βˆš3lncoth(1/2)π’Ÿ2πœ‰+πœ‚β„‹ξ‚βˆ’1ξ‚€βˆšcoth(1/2)π’Ÿ2πœ‰+πœ‚β„‹ξ‚βŽžβŽŸβŽŸβŽ ξ‚€1+1+6coth2βˆšπ’Ÿ2πœ‰+πœ‚β„‹ξ‚+βˆšπ’Ÿ2πœ‰βŽ«βŽͺ⎬βŽͺ⎭,𝑒𝒯1(π‘₯,𝑑)=βˆ’2(π‘Žπ‘˜+πœ”)βˆšπ‘Žπ‘˜πœ”βˆ’π’Ÿ26cotξ‚€12βˆšβˆ’π’Ÿ2πœ‰+πœ‚π’―ξ‚βˆ’3πœ‹+6πœ‚π’―ξ‚‡,(3.18) respectively, where πœ‚β„‹=tanhβˆ’1(𝐢1/𝐢2),πœ‚π’―=tanβˆ’1(𝐢1/𝐢2),π‘˜ and πœ” are arbitrary constants.

424801.fig.003
Figure 3: Trigonometric function solution (3.16) of Hirota-Ramani equation, for π‘Ž=βˆ’1, π‘˜=βˆ’1, πœ”=1/2, 𝐢1=2, and 𝐢2=1.
3.1. Rational Solution

And finally, in both Cases 1 and 2, when π’Ÿ=πœ†2βˆ’4πœ‡=0, we obtain rational solution:𝑒rat(π‘₯,𝑑)=6π‘˜πΆ2π‘Žξ€·πΆ1+𝐢2ξ€Έ,(π‘˜π‘₯βˆ’π‘Žπ‘˜π‘‘)(3.19) where 𝐢1, 𝐢2, π‘˜ are arbitrary constants. This solution is shown in Figure 4, for π‘Ž=1, π‘˜=βˆ’1, πœ”=1/2, 𝐢1=2, and 𝐢2=1.

424801.fig.004
Figure 4: Rational function solution (3.19) of Hirota-Ramani equation, for π‘Ž=1, π‘˜=βˆ’1, πœ”=1/2, 𝐢1=2, and 𝐢2=1.

4. Conclusions

This study shows that the (𝐺′/𝐺)-expansion method is quite efficient and practically well suited for use in finding exact solutions for the Hirota-Ramani equation. Our solutions are in more general forms, and many known solutions to these equations are only special cases of them. With the aid of Maple, we have assured the correctness of the obtained solutions by putting them back into the original equation. We hope that they will be useful for further studies in applied sciences.

Acknowledgment

The authors would like to thank the Young Researchers Club, Islamic Azad University, Ardabil Branch for its financial support.

References

  1. J. S. Russell, β€œReport on waves,” in Proceedings of the 14th Meeting of the British Association for the Advancement of Science, 1844.
  2. M. J. Ablowitz and J. F. Ladik, β€œOn the solution of a class of nonlinear partial difference equations,” Studies in Applied Mathematics, vol. 57, no. 1, pp. 1–12, 1977. View at Google Scholar Β· View at Zentralblatt MATH
  3. M. Wadati, β€œTransformation theories for nonlinear discrete systems,” Progress of Theoretical Physics, vol. 59, supplement, pp. 36–63, 1976. View at Publisher Β· View at Google Scholar
  4. H.-W. Tam and X.-B. Hu, β€œSoliton solutions and Bäcklund transformation for the Kupershmidt five-field lattice: a bilinear approach,” Applied Mathematics Letters, vol. 15, no. 8, pp. 987–993, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  5. C. E. Elmer and E. S. Van Vleck, β€œA variant of Newton's method for the computation of traveling waves of bistable differential-difference equations,” Journal of Dynamics and Differential Equations, vol. 14, no. 3, pp. 493–517, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  6. H. Wu and D.-J. Zhang, β€œMixed rational-soliton solutions of two differential-difference equations in Casorati determinant form,” Journal of Physics, vol. 36, no. 17, pp. 4867–4873, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  7. A.-M. Wazwaz, β€œThe tanh method for travelling wave solutions to the Zhiber-Shabat equation and other related equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 3, pp. 584–592, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  8. Y. Zhou, M. Wang, and Y. Wang, β€œPeriodic wave solutions to a coupled KdV equations with variable coefficients,” Physics Letters A, vol. 308, no. 1, pp. 31–36, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  9. F. En-gui and Z. Hong-qing, β€œA note on the homogeneous balance method,” Physics Letters A, vol. 246, pp. 403–406, 1998. View at Publisher Β· View at Google Scholar
  10. Z. Yan, β€œAbundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey-Stewartson-type equation via a new method,” Chaos, Solitons and Fractals, vol. 18, no. 2, pp. 299–309, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  11. R. Hirota and A. Ramani, β€œThe Miura transformations of Kaup's equation and of Mikhailov's equation,” Physics Letters A, vol. 76, no. 2, pp. 95–96, 1980. View at Publisher Β· View at Google Scholar
  12. G.-C. Wu and T.-C. Xia, β€œA new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations,” Physics Letters A, vol. 372, no. 5, pp. 604–609, 2008. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  13. J. Ji, β€œA new expansion and new families of exact traveling solutions to Hirota equation,” Applied Mathematics and Computation, vol. 204, no. 2, pp. 881–883, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  14. M. Wang, X. Li, and J. Zhang, β€œThe (G/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  15. M. Wang, J. Zhang, and X. Li, β€œApplication of the (G/G)-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 321–326, 2008. View at Publisher Β· View at Google Scholar
  16. R. Abazari, β€œApplication of (G/G)-expansion method to travelling wave solutions of three nonlinear evolution equation,” Computers & Fluids, vol. 39, no. 10, pp. 1957–1963, 2010. View at Publisher Β· View at Google Scholar
  17. R. Abazari, β€œThe (G/G)-expansion method for Tzitzéica type nonlinear evolution equations,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1834–1845, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  18. M. M. Kabir, A. Borhanifar, and R. Abazari, β€œApplication of (G/G)-expansion method to Regularized Long Wave (RLW) equation,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 2044–2047, 2011. View at Publisher Β· View at Google Scholar