The (G′/G)-expansion method is proposed to construct the exact traveling solutions to Hirota-Ramani equation:
ut-uxxt+aux(1-ut)=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′⁄G)-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′⁄G)-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′⁄G)-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+aux(1-ut)=0,
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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-Expansion Method

The objective of this section is to outline the use of the (G′⁄G)-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,
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,
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′⁄G) as follows:U(ξ)=∑i=1mαi(G′G)i+α0,αm≠0,
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,
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)+C2sinh((λ2-4μ/2)ξ))-λ2,λ2-4μ>0,4μ-λ22(-C1sin((λ2-4μ/2)ξ)+C2cos((λ2-4μ/2)ξ)C1cos((λ2-4μ/2)ξ)+C2sin((λ2-4μ/2)ξ))-λ2,λ2-4μ<0,
and it follows, from (2.3) and (2.4), thatU′=-∑i=1miαi((G′G)i+1+λ(G′G)i+μ(G′G)i-1),U′′=∑i=1miαi((i+1)(G′G)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(G′G)i-2),
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+aux(1-ut)=0,
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,
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.
By setting U′=V,nonlinear ordinary differential equation (3.3) reduce to(ω+ak)V-k2ωV′′-akωV2=0.
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(G′G)2+α1(G′G)+α0,α2≠0,
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:(G′G)0:(-2ωα2μ2-ωα1λμ)k2-aα0(ωα0-1)k+ωα0=0,(G′G)1:(-ω(2μ+λ2)α1-6ωα2λμ)k2-a(-1+2ωα0)α1k+ωα1=0,(G′G)2:(-3ωα1λ-4ωα2(2μ+λ2))k2-(aα2(-1+2ωα0)+aωα12)k+ωα2=0,(G′G)3:(-10ωα2λ-2ωα1)k2-2akωα2α1=0,(G′G)4:-6k2ωα2-akωα22=0,
and solving by use of Maple, we get the following results.

Case 1.

λ=-16aα1k,μ=1144-36ak-36ω+a2ωα12k2ω,α0=124-36ak-36ω+a2ωα12akω,α2=-6ka,
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(G′G)2+α1(G′G)+124-36ak-36ω+a2ωα12akω}dξ.
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 Uℋ, of Hirota-Ramani equation (3.1) as follows:
UH(ξ)=6(ak+ω)(C12-C22)akωC1D1×sinh((1/4)D1ξ)cosh((1/4)D1ξ)2C2sinh((1/4)D1ξ)cosh((1/4)D1ξ)+2C1cosh2((1/4)D1ξ)-C1,
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 C12>C22, as follows: uH(x,t)=32(ak+ω)akωD1{(tanh((1/2)D1ξ+ηH)-1tanh((1/2)D1ξ+ηH)+1)2tanh(12D1ξ+ηH)+ln(tanh((1/2)D1ξ+ηH)-1tanh((1/2)D1ξ+ηH)+1)+D1ξ},
while at C12<C22, one can obtain
uH(x,t)=32(ak+ω)akωD1{coth((1/2)D1ξ+ηH)-1coth((1/2)D1ξ+ηH)+12coth(12D1ξ+ηH)+ln(coth((1/2)D1ξ+ηH)-1coth((1/2)D1ξ+ηH)+1)+D1ξ},where ξ=kx+ωt, ηℋ=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 U𝒯, of Hirota-Ramani equation (3.1) as follows:
UT(ξ)=-3(ak+ω)(C12+C22)akωC2-D11C2tan((1/2)-D1ξ)+C1,
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 C12>C22, and C12<C22, as follows: uT(x,t)=3(ak+ω)akωC2-D1tan(12-D1ξ+ηT),uT(x,t)=-3(ak+ω)akωC2-D1cot(12-D1ξ+ηT),respectively, where ξ=kx+ωt,η𝒯=tan-1(C1/C2), and k,ω, are arbitrary constants.

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

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=-16aα1k,μ=114436ak+36ω+a2ωα12k2ω,α0=-12412ak+12ω+a2ωα12akω,α2=-6ka,
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(G′G)2+α1(G′G)-12412ak+12ω+a2ωα12akω}dξ,
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:
UH(ξ)=2(ak+ω)(C22-C12)akωD2×{ln(tanh((1/4)D2ξ)+1tanh((1/4)D2ξ)-1)+3tanh((1/4)D2ξ)C12tanh2((1/4)D2ξ)+2C1C2tanh((1/4)D2ξ)+C12},UT(ξ)=(ak+ω)akωC2-D2{C2-D2ξ+3(C21+C22)C2tan((1/2)-D2ξ)+C1},
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 C12>C22, then hyperbolic and trigonometric function solutions (3.15)-(3.16) become
uH(x,t)=-12(ak+ω)akωD2{3ln(tanh((1/2)D2ξ+ηH)-1tanh((1/2)D2ξ+ηH)+1),+6tanh(12D2ξ+ηH)+D2ξcoth((1/2)D2ξ+ηH)-1coth((1/2)D2ξ+ηH)+1},uT(x,t)=-12(ak+ω)akω-D2{-6tan((1/2)-D2ξ+ηT)+6ηT},
while at C12<C22, the hyperbolic and trigonometric function solutions (3.15)-(3.16) become
uH(x,t)=-12(ak+ω)akωD2{3ln(coth((1/2)D2ξ+ηH)-1coth((1/2)D2ξ+ηH)+1)+6coth(12D2ξ+ηH)+D2ξcoth((1/2)D2ξ+ηH)-1coth((1/2)D2ξ+ηH)+1},uT(x,t)=-12(ak+ω)akω-D2{6cot(12-D2ξ+ηT)-3π+6ηT},
respectively, where ηℋ=tanh-1(C1/C2),η𝒯=tan-1(C1/C2),k and ω are arbitrary constants.

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:urat(x,t)=6kC2a(C1+C2(kx-akt)),
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.

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.

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