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

Abstract

The (𝐺^/𝐺)-expansion method is proposed to construct the exact traveling solutions to Hirota-Ramani equation: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑎𝑢_𝑥(1−𝑢_𝑡)=0, where 𝑎≠0. Our work is motivated by the fact that the (𝐺^/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]:𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑎𝑢_𝑥1−𝑢_𝑡=0,(1.1) where 𝑎≠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 (𝐺^/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,𝑢_𝑥,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑥,𝑡,𝑢_𝑡𝑡,…=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,𝑘𝑈^,𝜔𝑈^,𝑘²𝑈^,𝑘𝜔𝑈^,𝜔²𝑈^,…=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=1𝛼_𝑖𝐺𝐺^^𝑖+𝛼₀,𝛼_𝑚≠0,(2.3) where 𝛼₀, and 𝛼_𝑖, 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)=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩√𝜆²−4𝜇2⎛⎜⎜⎝𝐶¹sinh√𝜆²−4𝜇/2𝜉+𝐶₂cosh√𝜆²−4𝜇/2𝜉𝐶₁cosh√𝜆²−4𝜇/2𝜉+𝐶₂sinh𝜆₂√𝜆²−4𝜇/2𝜉⎞⎟⎟⎠−𝜆2,√−4𝜇>0,4𝜇−𝜆²2⎛⎜⎜⎝−𝐶¹sin√𝜆²−4𝜇/2𝜉+𝐶₂cos√𝜆²−4𝜇/2𝜉𝐶₁cos√𝜆²−4𝜇/2𝜉+𝐶₂sin𝜆₂√𝜆²−4𝜇/2𝜉⎞⎟⎟⎠−𝜆2,−4𝜇<0,(2.5) and it follows, from (2.3) and (2.4), that𝑈′=−^𝑚_𝑖=1𝑖𝛼_𝑖𝐺𝐺^^𝑖+1+𝜆𝐺^𝐺^𝑖+𝜇𝐺^𝑈_𝐺^𝑖−1,=^𝑚_𝑖=1𝑖𝛼_𝑖(𝑖+1)𝐺𝐺^^𝑖+2+(2𝑖+1)𝜆𝐺^𝐺^𝑖+1+𝑖𝜆²+2𝜇𝐺^𝐺^𝑖+(2𝑖−1)𝜆𝜇𝐺^𝐺^𝑖−1+(𝑖−1)𝜇²𝐺𝐺^^𝑖−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 𝛼_𝑖 for 𝑖=1,2,…,𝑚.

Step 3. Solve the system of algebraic equations obtained in Step 2, for 𝑘,𝜔,𝜆,𝜇,𝛼₀ and 𝛼_𝑖 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]:𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑎𝑢_𝑥1−𝑢_𝑡=0,(3.1) where 𝑎≠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+𝑎𝑘)𝑈′−𝑘²𝜔𝑈^−𝑎𝑘𝜔𝑈^²=0.(3.3) By setting 𝑈^=𝑉,nonlinear ordinary differential equation (3.3) reduce to(𝜔+𝑎𝑘)𝑉−𝑘²𝜔𝑉^−𝑎𝑘𝜔𝑉²=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=𝛼₂𝐺𝐺^²+𝛼₁𝐺𝐺^+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:𝐺𝐺^⁰𝐺∶−2𝜔𝛼₂𝜇²−𝜔𝛼₁𝜆𝜇𝑘²−𝑎𝛼₀𝜔𝛼₀−1𝑘+𝜔𝛼₀=0,_𝐺¹𝐺∶−𝜔2𝜇+𝜆²𝛼₁−6𝜔𝛼₂𝜆𝜇𝑘²−𝑎−1+2𝜔𝛼₀𝛼₁𝑘+𝜔𝛼₁=0,_𝐺²𝐺∶−3𝜔𝛼₁𝜆−4𝜔𝛼₂2𝜇+𝜆²𝑘²−𝑎𝛼₂−1+2𝜔𝛼₀+𝑎𝜔𝛼₁₂𝑘+𝜔𝛼₂=0,_𝐺³𝐺∶−10𝜔𝛼₂𝜆−2𝜔𝛼₁𝑘²−2𝑎𝑘𝜔𝛼₂𝛼₁=0,_𝐺⁴∶−6𝑘²𝜔𝛼₂−𝑎𝑘𝜔𝛼²²=0,(3.6) and solving by use of Maple, we get the following results.

Case 1. 𝜆=−1𝜇=16𝑎𝑘𝛼¹,144−36𝑎𝑘−36𝜔+𝑎²𝜔𝛼²¹𝛼₀𝑘²𝜔,=124−36𝑎𝑘−36𝜔+𝑎²𝜔𝛼²¹𝛼𝑎𝑘𝜔₂,=−6𝑘𝑎,(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=−6𝑘𝑎𝐺𝐺^²+𝛼₁𝐺𝐺^+214−36𝑎𝑘−36𝜔+𝑎²𝜔𝛼²¹𝑎𝑘𝜔𝑑𝜉.(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=(𝑎𝑘+𝜔)/𝑘²𝜔>0, using the integration relationship (3.8), we obtain hyperbolic function solution 𝑈_ℋ, of Hirota-Ramani equation (3.1) as follows: 𝑈_ℋ(𝜉)=6(𝑎𝑘+𝜔)𝐶²¹−𝐶²²𝑎𝑘𝜔𝐶₁√𝒟₁×sinh(1/4)√𝒟₁𝜉cosh(1/4)√𝒟₁𝜉2𝐶₂sinh(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 𝑎=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: 𝑢_ℋ(𝑥,𝑡)=32𝑎(𝑘𝑎𝑘𝜔√+𝒟𝜔)₁⎧⎪⎨⎪⎩2tanh12√+ln⎛⎜⎜⎝tanh(1/2𝒟)¹√𝜉+𝜂_ℋ𝒟₁𝜉+𝜂_ℋ−1tanh(1/2)√𝒟₁𝜉+𝜂_ℋ+1⎞⎟⎟⎠+√𝒟₁𝜉⎫⎪⎬⎪⎭,(3.10a) while at 𝐶²¹<𝐶²², one can obtain 𝑢_ℋ(𝑥,𝑡)=32𝑎(𝑘𝑎𝑘𝜔√+𝒟𝜔)₁⎧⎪⎨⎪⎩2coth12√+ln⎛⎜⎜⎝coth(1/2𝒟)¹√𝜉+𝜂_ℋ𝒟₁𝜉+𝜂_ℋ−1coth(1/2)√𝒟₁𝜉+𝜂_ℋ+1⎞⎟⎟⎠+√𝒟₁𝜉⎫⎪⎬⎪⎭,(3.10b)where 𝜉=𝑘𝑥+𝜔𝑡, 𝜂_ℋ=tanh⁻¹(𝐶₁/𝐶₂), and 𝑘, 𝜔, are arbitrary constants.
Now, when 𝒟₁=𝜆²−4𝜇=((𝑎𝑘+𝜔)/𝑘²𝜔)<0, using the integration relationship (3.8), we obtain trigonometric function solution 𝑈_𝒯, of Hirota-Ramani equation (3.1) as follows: 𝑈_𝒯(𝜉)=−3(𝑎𝑘+𝜔)𝐶²¹+𝐶²²𝑎𝑘𝜔𝐶₂√−𝒟₁1𝐶₂tan(1/2)√−𝒟₁𝜉+𝐶₁,(3.11) where 𝜉=𝑘𝑥+𝜔𝑡, and 𝐶₁, 𝐶₂, are arbitrary constants. This solution is shown in Figure 2 for 𝑎=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: 𝑢_𝒯(𝑥,𝑡)=3(𝑎𝑘+𝜔)𝑎𝑘𝜔𝐶₂√−𝒟₁tan12√−𝒟₁𝜉+𝜂_𝒯,(3.12a)𝑢_𝒯(𝑥,𝑡)=−3(𝑎𝑘+𝜔)𝑎𝑘𝜔𝐶₂√−𝒟₁cot12√−𝒟₁𝜉+𝜂_𝒯,(3.12b)respectively, where 𝜉=𝑘𝑥+𝜔𝑡,𝜂_𝒯=tan⁻¹(𝐶₁/𝐶₂), and 𝑘,𝜔, are arbitrary constants.

Figure 1 

Hyperbolic function solution (3.9) of Hirota-Ramani equation, for 𝑎=1, 𝑘=1, 𝜔=1/2, 𝐶1=2, and 𝐶2=1.

Figure 2 

Trigonometric function solution (3.11) of Hirota-Ramani equation, for 𝑎=1, 𝑘=−1, 𝜔=1/2, 𝐶1=2, and 𝐶2=1.

Case 2. 𝜆=−1𝜇=16𝑎𝑘𝛼¹,14436𝑎𝑘+36𝜔+𝑎²𝜔𝛼²¹𝛼₀𝑘²𝜔,=−12412𝑎𝑘+12𝜔+𝑎²𝜔𝛼²¹𝛼𝑎𝑘𝜔₂,=−6𝑘𝑎,(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=−6𝑘𝑎𝐺𝐺^²+𝛼₁𝐺𝐺^−21412𝑎𝑘+12𝜔+𝑎²𝜔𝛼²¹𝑎𝑘𝜔𝑑𝜉,(3.14) then for 𝒟₂=𝜆²−4𝜇=−((𝑎𝑘+𝜔)/𝑘²𝜔)>0, the hyperbolic and for 𝒟₂=𝜆²−4𝜇=−((𝑎𝑘+𝜔)/𝑘²𝜔)<0, trigonometric types of traveling wave solutions of Hirota-Ramani equation (3.1), are obtained as follows: 𝑈_ℋ(𝜉)=2(𝑎𝑘+𝜔)𝐶²²−𝐶²¹𝑎𝑘𝜔√𝒟₂×⎧⎪⎨⎪⎩ln⎛⎜⎜⎝tanh(1/4)√𝒟₂𝜉+1+tanh(1/4)√𝒟₂3ta𝜉nh−(11⎞⎟⎟⎠/4)√𝒟₂𝜉𝐶²¹tanh²(1/4)√𝒟₂𝜉+2𝐶₁𝐶₂tanh(1/4)√𝒟₂𝜉+𝐶²¹⎫⎪⎬⎪⎭,(3.15)𝑈_𝒯(𝜉)=(𝑎𝑘+𝜔)𝑎𝑘𝜔𝐶₂√−𝒟₂⎧⎪⎨⎪⎩𝐶₂√−𝒟₂𝜉+3𝐶¹²+𝐶²²𝐶₂tan(1/2)√−𝒟₂𝜉+𝐶₁⎫⎪⎬⎪⎭,(3.16) respectively, where 𝜉=𝑘𝑥+𝜔𝑡, and 𝐶₁,𝐶₂, are arbitrary constants. The trigonometric function solution (3.16), for 𝑎=−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 𝑢_ℋ(𝑥,𝑡)=−12𝑎(𝑘𝑎𝑘𝜔√+𝒟𝜔)₂⎧⎪⎨⎪⎩3ln⎛⎜⎜⎝tanh(1/2)√𝒟₂𝜉+𝜂_ℋ−1+6tanh1tanh(1/2)√𝒟₂𝜉+𝜂_ℋ+1⎞⎟⎟⎠,2√𝑢_𝒯𝒟₂𝜉+𝜂_ℋ+√𝒟₂𝜉⎫⎪⎬⎪⎭,(𝑥,𝑡)=−12𝑎𝑘(𝑎𝜔𝑘√+−𝜔𝒟)₂−6tan(1/2)√−𝒟₂𝜉+𝜂_𝒯+6𝜂^𝒯,(3.17) while at 𝐶²¹<𝐶²², the hyperbolic and trigonometric function solutions (3.15)-(3.16) become 𝑢_ℋ(𝑥,𝑡)=−12𝑎(𝑘𝑎𝑘𝜔√+𝒟𝜔)₂⎧⎪⎨⎪⎩3ln⎛⎜⎜⎝coth(1/2)√𝒟₂𝜉+𝜂_ℋ−1+6cothc1oth(1/2)√𝒟₂𝜉+𝜂_ℋ+1⎞⎟⎟⎠2√𝑢_𝒯𝒟₂𝜉+𝜂_ℋ+√𝒟₂𝜉⎫⎪⎬⎪⎭,(𝑥,𝑡)=−12𝑎𝑘(𝑎𝜔𝑘√+−𝜔𝒟)₂6cot12√−𝒟₂𝜉+𝜂_𝒯−3𝜋+6𝜂^𝒯,(3.18) respectively, where 𝜂_ℋ=tanh⁻¹(𝐶₁/𝐶₂),𝜂_𝒯=tan⁻¹(𝐶₁/𝐶₂),𝑘 and 𝜔 are arbitrary constants.

Figure 3 

Trigonometric function solution (3.16) of Hirota-Ramani equation, for 𝑎=−1, 𝑘=−1, 𝜔=1/2, 𝐶1=2, and 𝐶2=1.

3.1. Rational Solution

And finally, in both Cases 1 and 2, when 𝒟=𝜆²−4𝜇=0, we obtain rational solution:𝑢_rat(𝑥,𝑡)=6𝑘𝐶²𝑎𝐶₁+𝐶₂(𝑘𝑥−𝑎𝑘𝑡),(3.19) where 𝐶₁, 𝐶₂, 𝑘 are arbitrary constants. This solution is shown in Figure 4, for 𝑎=1, 𝑘=−1, 𝜔=1/2, 𝐶₁=2, and 𝐶₂=1.

Figure 4 

Rational function solution (3.19) of Hirota-Ramani equation, for 𝑎=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.

Acknowledgment

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