The (ğºî…ž/𝐺)-expansion method is proposed to construct the exact traveling solutions to Hirota-Ramani equation: 𝑢𝑡−𝑢𝑥𝑥𝑡+ğ‘Žğ‘¢ğ‘¥(1−𝑢𝑡)=0, where ğ‘Žâ‰ 0. Our work is motivated by the fact that the (ğºî…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 “dispersive 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 “Wave of Translation” [1]. Many exactly solvable models have soliton solutions, including the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger 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′0𝑥02044𝐺)-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′0𝑥02044𝐺)-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′0𝑥02044𝐺)-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]:𝑢𝑡−𝑢𝑥𝑥𝑡+ğ‘Žğ‘¢ğ‘¥î€·1−𝑢𝑡=0,(1.1) where ğ‘Žâ‰ 0 is a real constant and 𝑢(𝑥,𝑡) 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 (ğºî…ž/𝐺)-Expansion Method

The objective of this section is to outline the use of the (𝐺′0𝑥02044𝐺)-expansion method for solving certain nonlinear partial differential equations (PDEs). Suppose we have a nonlinear PDE for 𝑢(𝑥,𝑡), in the form𝑃𝑢,𝑢𝑥,𝑢𝑡,𝑢𝑥𝑥,𝑢𝑥,𝑡,𝑢𝑡𝑡,…=0,(2.1) where 𝑃 is a polynomial in its arguments, which includes nonlinear terms and the highest-order derivatives. The transformation 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑘𝑥+𝜔𝑡, reduces (2.1) to the ordinary differential equation (ODE)𝑃𝑈,ğ‘˜ğ‘ˆî…ž,ğœ”ğ‘ˆî…ž,𝑘2ğ‘ˆî…žî…ž,ğ‘˜ğœ”ğ‘ˆî…žî…ž,𝜔2ğ‘ˆî…žî…žî€¸,…=0,(2.2) where 𝑈=𝑈(𝜉), and prime denotes derivative with respect to 𝜉. We assume that the solution of (2.2) can be expressed by a polynomial in (𝐺′0𝑥02044𝐺) as follows:𝑈(𝜉)=𝑚𝑖=1ğ›¼ğ‘–î‚µğºî…žğºî‚¶ğ‘–+𝛼0,𝛼𝑚≠0,(2.3) where 𝛼0, and 𝛼𝑖, are constants to be determined later, 𝐺(𝜉) satisfies a second-order linear ordinary differential equation (LODE):𝑑2𝐺(𝜉)𝑑𝜉2+𝜆𝑑𝐺(𝜉)𝑑𝜉+𝜇𝐺(𝜉)=0,(2.4) where 𝜆 and 𝜇 are arbitrary constants. Using the general solutions of (2.4), we have𝐺′(𝜉)𝐺=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩√(𝜉)𝜆2−4𝜇2âŽ›âŽœâŽœâŽğ¶1√sinh𝜆2𝜉−4𝜇/2+𝐶2√cosh𝜆2𝜉−4𝜇/2𝐶1√cosh𝜆2𝜉−4𝜇/2+𝐶2√sinh𝜆2î‚ğœ‰î‚âŽžâŽŸâŽŸâŽ âˆ’ğœ†âˆ’4𝜇/22,𝜆2√−4𝜇>0,4𝜇−𝜆22âŽ›âŽœâŽœâŽâˆ’ğ¶1√sin𝜆2𝜉−4𝜇/2+𝐶2√cos𝜆2𝜉−4𝜇/2𝐶1√cos𝜆2𝜉−4𝜇/2+𝐶2√sin𝜆2î‚ğœ‰î‚âŽžâŽŸâŽŸâŽ âˆ’ğœ†âˆ’4𝜇/22,𝜆2−4𝜇<0,(2.5) and it follows, from (2.3) and (2.4), that𝑈′=−𝑚𝑖=1ğ‘–ğ›¼ğ‘–îƒ©î‚µğºî…žğºî‚¶ğ‘–+1𝐺+ğœ†î…žğºî‚¶ğ‘–î‚µğº+ğœ‡î…žğºî‚¶ğ‘–âˆ’1,ğ‘ˆî…žî…ž=𝑚𝑖=1𝑖𝛼𝑖𝐺(𝑖+1)î…žğºî‚¶ğ‘–+2𝐺+(2𝑖+1)ğœ†î…žğºî‚¶ğ‘–+1𝜆+𝑖2𝐺+2ğœ‡î…žğºî‚¶ğ‘–+𝐺(2𝑖−1)ğœ†ğœ‡î…žğºî‚¶ğ‘–âˆ’1+(𝑖−1)𝜇2î‚µğºî…žğºî‚¶ğ‘–âˆ’2,(2.6) and so on, here the prime denotes the derivative with respect to 𝜉.

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

Step 1. Determine the integer 𝑚 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 𝑚 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 𝑘,𝜔,𝜆,𝜇,𝛼0 and 𝛼𝑖 for 𝑖=1,2,…,𝑚.

Step 3. Solve the system of algebraic equations obtained in Step 2, for 𝑘,𝜔,𝜆,𝜇,𝛼0 and 𝛼𝑖 by use of Maple.

Step 4. Use the results obtained in above steps to derive a series of fundamental solutions 𝑢(𝜉) 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]:𝑢𝑡−𝑢𝑥𝑥𝑡+ğ‘Žğ‘¢ğ‘¥î€·1−𝑢𝑡=0,(3.1) where ğ‘Žâ‰ 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:𝑢=𝑈(𝜉),𝜉=𝑘𝑥+𝜔𝑡,(3.2) where 𝑘, 𝜔 are constants. We substitute (3.2) into (3.1) to obtain nonlinear ordinary differential equation(𝜔+ğ‘Žğ‘˜)𝑈′−𝑘2ğœ”ğ‘ˆî…žî…žî…žî€·ğ‘ˆâˆ’ğ‘Žğ‘˜ğœ”î…žî€¸2=0.(3.3) By setting ğ‘ˆî…ž=𝑉,nonlinear ordinary differential equation (3.3) reduce to(𝜔+ğ‘Žğ‘˜)𝑉−𝑘2ğœ”ğ‘‰î…žî…žâˆ’ğ‘Žğ‘˜ğœ”ğ‘‰2=0.(3.4) According to Step 1, we get 𝑚+2=2𝑚, hence 𝑚=2. We then suppose that (3.4) has the following formal solutions:𝑉=𝛼2î‚µğºî…žğºî‚¶2+𝛼1î‚µğºî…žğºî‚¶+𝛼0,𝛼2≠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 𝜆,𝜇,𝛼0,𝛼1, and 𝛼2, as follows:î‚µğºî…žğºî‚¶0∶−2𝜔𝛼2𝜇2−𝜔𝛼1𝑘𝜆𝜇2âˆ’ğ‘Žğ›¼0𝜔𝛼0−1𝑘+𝜔𝛼0𝐺=0,î…žğºî‚¶1∶−𝜔2𝜇+𝜆2𝛼1−6𝜔𝛼2𝑘𝜆𝜇2î€·âˆ’ğ‘Žâˆ’1+2𝜔𝛼0𝛼1𝑘+𝜔𝛼1𝐺=0,î…žğºî‚¶2∶−3𝜔𝛼1𝜆−4𝜔𝛼22𝜇+𝜆2𝑘2âˆ’î€·ğ‘Žğ›¼2−1+2𝜔𝛼0+ğ‘Žğœ”ğ›¼12𝑘+𝜔𝛼2𝐺=0,î…žğºî‚¶3∶−10𝜔𝛼2𝜆−2𝜔𝛼1𝑘2−2ğ‘Žğ‘˜ğœ”ğ›¼2𝛼1𝐺=0,î…žğºî‚¶4∶−6𝑘2𝜔𝛼2âˆ’ğ‘Žğ‘˜ğœ”ğ›¼22=0,(3.6) and solving by use of Maple, we get the following results.

Case 1. 1𝜆=−6ğ‘Žğ›¼1𝑘,1𝜇=144−36ğ‘Žğ‘˜âˆ’36𝜔+ğ‘Ž2𝜔𝛼21𝑘2𝜔,𝛼0=124−36ğ‘Žğ‘˜âˆ’36𝜔+ğ‘Ž2𝜔𝛼21,ğ›¼ğ‘Žğ‘˜ğœ”2=−6ğ‘˜ğ‘Ž,(3.7) where 𝑘, 𝜔, and 𝛼1 are arbitrary constants. Therefore, substitute the above case in (3.5), and using the relationship ∫𝑈(𝜉)=𝑉(𝜉)𝑑𝜉, we get −𝑈=6ğ‘˜ğ‘Žî‚µğºî…žğºî‚¶2+𝛼1î‚µğºî…žğºî‚¶+124−36ğ‘Žğ‘˜âˆ’36𝜔+ğ‘Ž2𝜔𝛼21îƒ°ğ‘Žğ‘˜ğœ”ğ‘‘ğ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 𝜆2−4𝜇.
When 𝒟1=𝜆2−4𝜇=(ğ‘Žğ‘˜+𝜔)/𝑘2𝜔>0, using the integration relationship (3.8), we obtain hyperbolic function solution 𝑈ℋ, of Hirota-Ramani equation (3.1) as follows: 𝑈ℋ𝐶(𝜉)=6(ğ‘Žğ‘˜+𝜔)21−𝐶22î€¸ğ‘Žğ‘˜ğœ”ğ¶1√𝒟1×√sinh(1/4)𝒟1𝜉√cosh(1/4)𝒟1𝜉2𝐶2√sinh(1/4)𝒟1𝜉√cosh(1/4)𝒟1𝜉+2𝐶1cosh2√(1/4)𝒟1𝜉−𝐶1,(3.9) where 𝜉=𝑘𝑥+𝜔𝑡, and 𝐶1, 𝐶2, are arbitrary constants. This solution is shown in Figure 1 for ğ‘Ž=1, 𝑘=1, 𝜔=1/2, 𝐶1=2, and 𝐶2=1. It is easy to see that the hyperbolic solution (3.9) can be rewritten at 𝐶21>𝐶22, as follows: 𝑢ℋ3(𝑥,𝑡)=2(ğ‘Žğ‘˜+𝜔)âˆšğ‘Žğ‘˜ğœ”ğ’Ÿ1⎧⎪⎨⎪⎩12tanh2√𝒟1𝜉+ğœ‚â„‹î‚âŽ›âŽœâŽœâŽî‚€âˆš+lntanh(1/2)𝒟1𝜉+𝜂ℋ−1√tanh(1/2)𝒟1𝜉+ğœ‚â„‹î‚âŽžâŽŸâŽŸâŽ +√+1𝒟1ğœ‰âŽ«âŽªâŽ¬âŽªâŽ­,(3.10a) while at 𝐶21<𝐶22, one can obtain 𝑢ℋ3(𝑥,𝑡)=2(ğ‘Žğ‘˜+𝜔)âˆšğ‘Žğ‘˜ğœ”ğ’Ÿ1⎧⎪⎨⎪⎩12coth2√𝒟1𝜉+ğœ‚â„‹î‚âŽ›âŽœâŽœâŽî‚€âˆš+lncoth(1/2)𝒟1𝜉+𝜂ℋ−1√coth(1/2)𝒟1𝜉+ğœ‚â„‹î‚âŽžâŽŸâŽŸâŽ +√+1𝒟1ğœ‰âŽ«âŽªâŽ¬âŽªâŽ­,(3.10b)where 𝜉=𝑘𝑥+𝜔𝑡, 𝜂ℋ=tanh−1(𝐶1/𝐶2), and 𝑘, 𝜔, are arbitrary constants.
Now, when 𝒟1=𝜆2−4𝜇=((ğ‘Žğ‘˜+𝜔)/𝑘2𝜔)<0, using the integration relationship (3.8), we obtain trigonometric function solution 𝑈𝒯, of Hirota-Ramani equation (3.1) as follows: 𝑈𝒯𝐶(𝜉)=−3(ğ‘Žğ‘˜+𝜔)21+𝐶22î€¸ğ‘Žğ‘˜ğœ”ğ¶2√−𝒟11𝐶2√tan(1/2)−𝒟1𝜉+𝐶1,(3.11) where 𝜉=𝑘𝑥+𝜔𝑡, and 𝐶1, 𝐶2, are arbitrary constants. This solution is shown in Figure 2 for ğ‘Ž=1, 𝑘=−1, 𝜔=1/2, 𝐶1=2, and 𝐶2=1. Similarity, it is easy to see that the trigonometric solution (3.11) can be rewritten at 𝐶21>𝐶22, and 𝐶21<𝐶22, as follows: 𝑢𝒯(𝑥,𝑡)=3(ğ‘Žğ‘˜+𝜔)ğ‘Žğ‘˜ğœ”ğ¶2√−𝒟11tan2√−𝒟1𝜉+𝜂𝒯,(3.12a)𝑢𝒯(𝑥,𝑡)=−3(ğ‘Žğ‘˜+𝜔)ğ‘Žğ‘˜ğœ”ğ¶2√−𝒟1cot12√−𝒟1𝜉+𝜂𝒯,(3.12b)respectively, where 𝜉=𝑘𝑥+𝜔𝑡,𝜂𝒯=tan−1(𝐶1/𝐶2), and 𝑘,𝜔, are arbitrary constants.

Case 2. 1𝜆=−6ğ‘Žğ›¼1𝑘,1𝜇=14436ğ‘Žğ‘˜+36𝜔+ğ‘Ž2𝜔𝛼21𝑘2𝜔,𝛼01=−2412ğ‘Žğ‘˜+12𝜔+ğ‘Ž2𝜔𝛼21,ğ›¼ğ‘Žğ‘˜ğœ”2=−6ğ‘˜ğ‘Ž,(3.13) where 𝑘,𝜔 and 𝛼1 is an arbitrary constant. Similar on the previous case, substitute the above case in (3.5), and using the relationship ∫𝑈(𝜉)=𝑉(𝜉)𝑑𝜉, we get −𝑈=6ğ‘˜ğ‘Žî‚µğºî…žğºî‚¶2+𝛼1î‚µğºî…žğºî‚¶âˆ’12412ğ‘Žğ‘˜+12𝜔+ğ‘Ž2𝜔𝛼21îƒ°ğ‘Žğ‘˜ğœ”ğ‘‘ğœ‰,(3.14) then for 𝒟2=𝜆2−4𝜇=−((ğ‘Žğ‘˜+𝜔)/𝑘2𝜔)>0, the hyperbolic and for 𝒟2=𝜆2−4𝜇=−((ğ‘Žğ‘˜+𝜔)/𝑘2𝜔)<0, trigonometric types of traveling wave solutions of Hirota-Ramani equation (3.1), are obtained as follows: 𝑈ℋ𝐶(𝜉)=2(ğ‘Žğ‘˜+𝜔)22−𝐶21î€¸âˆšğ‘Žğ‘˜ğœ”ğ’Ÿ2×⎧⎪⎨⎪⎩⎛⎜⎜⎝√lntanh(1/4)𝒟2𝜉+1√tanh(1/4)𝒟2ğœ‰î‚âŽžâŽŸâŽŸâŽ +√−13tanh(1/4)𝒟2𝜉𝐶21tanh2√(1/4)𝒟2𝜉+2𝐶1𝐶2√tanh(1/4)𝒟2𝜉+𝐶21⎫⎪⎬⎪⎭,(3.15)𝑈𝒯(𝜉)=(ğ‘Žğ‘˜+𝜔)ğ‘Žğ‘˜ğœ”ğ¶2√−𝒟2⎧⎪⎨⎪⎩𝐶2√−𝒟23𝐶𝜉+12+𝐶22𝐶2√tan(1/2)−𝒟2𝜉+𝐶1⎫⎪⎬⎪⎭,(3.16) respectively, where 𝜉=𝑘𝑥+𝜔𝑡, and 𝐶1,𝐶2, are arbitrary constants. The trigonometric function solution (3.16), for ğ‘Ž=−1, 𝑘=−1, 𝜔=1/2, 𝐶1=2 and 𝐶2=1 are shown in Figure 3. Similarly, to obtain some special forms of the solutions obtained above, we set 𝐶21>𝐶22, then hyperbolic and trigonometric function solutions (3.15)-(3.16) become 𝑢ℋ1(𝑥,𝑡)=−2(ğ‘Žğ‘˜+𝜔)âˆšğ‘Žğ‘˜ğœ”ğ’Ÿ2⎧⎪⎨⎪⎩⎛⎜⎜⎝√3lntanh(1/2)𝒟2𝜉+𝜂ℋ−1√tanh(1/2)𝒟2𝜉+ğœ‚â„‹î‚âŽžâŽŸâŽŸâŽ ,1+1+6tanh2√𝒟2𝜉+𝜂ℋ+√𝒟2ğœ‰âŽ«âŽªâŽ¬âŽªâŽ­,𝑢𝒯1(𝑥,𝑡)=−2(ğ‘Žğ‘˜+𝜔)âˆšğ‘Žğ‘˜ğœ”âˆ’ğ’Ÿ2√−6tan(1/2)−𝒟2𝜉+𝜂𝒯+6𝜂𝒯,(3.17) while at 𝐶21<𝐶22, the hyperbolic and trigonometric function solutions (3.15)-(3.16) become 𝑢ℋ1(𝑥,𝑡)=−2(ğ‘Žğ‘˜+𝜔)âˆšğ‘Žğ‘˜ğœ”ğ’Ÿ2⎧⎪⎨⎪⎩⎛⎜⎜⎝√3lncoth(1/2)𝒟2𝜉+𝜂ℋ−1√coth(1/2)𝒟2𝜉+ğœ‚â„‹î‚âŽžâŽŸâŽŸâŽ î‚€1+1+6coth2√𝒟2𝜉+𝜂ℋ+√𝒟2ğœ‰âŽ«âŽªâŽ¬âŽªâŽ­,𝑢𝒯1(𝑥,𝑡)=−2(ğ‘Žğ‘˜+𝜔)âˆšğ‘Žğ‘˜ğœ”âˆ’ğ’Ÿ26cot12√−𝒟2𝜉+𝜂𝒯−3𝜋+6𝜂𝒯,(3.18) respectively, where 𝜂ℋ=tanh−1(𝐶1/𝐶2),𝜂𝒯=tan−1(𝐶1/𝐶2),𝑘 and 𝜔 are arbitrary constants.

3.1. Rational Solution

And finally, in both Cases 1 and 2, when 𝒟=𝜆2−4𝜇=0, we obtain rational solution:𝑢rat(𝑥,𝑡)=6𝑘𝐶2ğ‘Žî€·ğ¶1+𝐶2,(ğ‘˜ğ‘¥âˆ’ğ‘Žğ‘˜ğ‘¡)(3.19) where 𝐶1, 𝐶2, 𝑘 are arbitrary constants. This solution is shown in Figure 4, for ğ‘Ž=1, 𝑘=−1, 𝜔=1/2, 𝐶1=2, and 𝐶2=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.


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