Solitary wave solutions of few nonlinear evolution equations

1.   Introduction

Now it is well recognized that nonlinear evolution equations (NLEEs) and its solutions are the most embracing way to describe the physical significance of nonlinear phenomena appearing in the field of science and engineering. In particular, the soliton solutions are most remarkable in the study of the nonlinear physical sciences, as for instance the wave phenomena are observed in fluid mechanics, optical fibers, biophysics, high-energy physics, chemical kinematics etc. But, the nonlinear processes are one of the basic challenges and not easy to control, because the nonlinear characteristic of the system sharply changes due to small changes of valid parameters including time. Thus the issue becomes more intricate and hence ultimate solution is needed. Therefore, searching solitary wave solutions to NLEEs is becoming increasingly attractive field in nonlinear sciences day by day. There are lot of NLEEs that can be solved using different mathematical methods. For these physical problems, soliton solutions, compactons, singular solitons and the other solutions have been originated. However, not all equations posed of these models are solvable. Thus, the methods for deriving exact solutions for the governing equations have to be developed. As a result, significant improvements have been made for searching solitary wave solutions to NLEEs and many effective and powerful methods have been established to examine the NLEEs, such as the nonlinear transformation method ^[1], the Hirota’s bilinear transformation method ^[2,3], the first integral method ^[4], the sine-cosine method ^[5,6], the Jacobi-elliptic function expansion method ^[7,8], the functional variable method ^[9], the Adomian decomposition method ^[10], the modified Exp-function method ^[11], the generalized Riccati equation method ^[12], the Exp-function method ^[13,14], the bifurcation theory ^[15], the exp(-Φ(η))-expansion method ^[16,17], the extended direct algebraic method ^[18,19,20], the (G^'/G)-expansion method and its different variant ^{[21,22,23,24,25,26,27,28]}, the variational method ^[29,30,31], the generalized Kudryashov method ^[32], the ansatz method ^{[33,34,35,36,37,38]}, the modified simple equation (MSE) ^{[39,40,41,42]} method, the modified extended direct algebraic method ^[43,44], the modified extended auxiliary equation method ^[45], the modified auxiliary equation method ^[46,47], the generalized exponential rational function method ^[48], the bilinear forms ^[49], the generalized unified method ^[50,51], the extended unified method ^[52,53] etc. The modified simple equation method is a recently developed method and getting popularity in use because of its straight forward calculation procedure. The objective of this article is to contrivance the modified simple equation method to construct solitary wave solutions to the simplified modified Camassa-Holm (MCH) equation, the Pochhammer-Chree (PC) equation and the Schrödinger-Hirota equation. The rest of the article is organized as follows: In section 2, we summarize the description of the method. In section 3, the MSE method is applied to extract exact soliton solutions to the NLEEs stated earlier. In section 4, explanation and physical interpretation of the solutions are presented and in section 5, we have drawn our conclusions.

2.   The modified simple equation (MSE) method

To describe the MSE, let us consider a nonlinear evolution equation in two independent variables x and t in the form:

where $u = u(x, t)$ is an unknown function and ${\cal F}$ is a polynomial of $u(x, t)$ and its partial derivatives wherein the highest order derivatives and nonlinear terms are involved and the subscripts are used for partial derivatives. The essential steps of this method are presented as the following:

Step 1: Initiating a compound variable ξ, we combine the real variables x and t:

where c is the speed of the traveling wave.

The traveling wave transformations (2.2) allow us in reducing Eq. (2.1) into an ODE for $u = u\left(\xi \right)$ in the form:

where ${\cal R}$ is a polynomial in u(ξ) and its derivatives with respect to ξ.

Step 2: Affording the MSE method, it is assume that the solution of (2.3) can be expressed in the subsequent form:

where ${a_i}\left({i = 0, 1, 2, 3, \ldots N} \right)$ are arbitrary constants to be determined such that ${a_N} \ne 0$ and $\psi \left(\xi \right)$ is an unknown function to be evaluated later, such that $\psi '\left(\xi \right) \ne 0$. The attribute and uniqueness of this method is that, it is not possible to assumption in advance what kind of solutions one may obtain through this method. Thus, it might be possible to achieve some fresh solution by this method.

Step 3: The positive integer N arises in (2.4) can be found by the balancing principle of the highest order of linear and nonlinear terms appearing in (2.3).

Step 4: Compute the necessary derivatives $u', u'', \ldots$ and insert Eq. (2.4) into (2.3) and then we account the function $\psi \left(\xi \right)$. The above procedure yields a polynomial in ($1/\psi \left(\xi \right)$). Equating the coefficients of same power of this polynomial to zero delivers a system of algebraic and differential equations that can be solved to get ${a_i}\left({i = 0, 1, 2, 3, \ldots N} \right)$ and ψ(ξ). This completes the determination of solutions of Eq. (2.1).

3.   Determination of the solutions

In this section, we will investigate the solitary wave solutions leading to the simplified MCH equation, the PC equation and the Schrödinger-Hirota equation using the MSE method.

3.1.   The simplified MCH equation

In this subsection, the MSE method has been put in use to examine the closed form soliton solutions and then the solitary wave solution to the simplified MCH equation of the form ^[19]:

where α and β are constants. Camassa and Holm derived a completely integrable wave equation namely CH equation for water waves by retaining two terms that are usually neglected in the small amplitude, shallow water limit ^[54]. Tian and Song ^[55] has investigated MCH equation and obtained peaked solitary wave solutions. Wazwaz ^[56] investigated a modified form of the Camassa-Holm equation, which is simplified from of MCH equation and Eq. (3.1) is obtained by considering n = 2 is called the simplified MCH equation. More details can be found in references ^{[14,36,54,55]}.

The traveling wave transformation $u\left({x, t} \right) = u\left(\xi \right), \; \xi = kx - ct$, converts Eq. (3.1) to the following form

where c is the wave speed.

Integrating (3.2) with respect to ξ once and the setting the constant of integration to zero, we obtain the following result

Since, solitary waves are localized and they decay as ξ→±∞ and we are probing solitary wave solutions, therefore the boundary conditions must be $u\left(\xi \right) \to 0$, $u'\left(\xi \right) \to 0$, ${u^{''\left(\xi \right)}} \to 0, \ldots$ etc. as ξ→±∞ and these boundary conditions yield zero constant.

Balancing between the terms u" and u³ yield N = 1. Therefore, the solution of Eq. (3.3) becomes

where a₀ and a₁ are constants, such that a₁≠0 and ψ(ξ) is an unknown function to be calculated. Inserting (3.4) and its derivatives into (3.3) yield a polynomial and equating the coefficients of ψ⁰, ψ^-1, ψ^-2, ψ^-3 to zero, we achieve the successive algebraic and differential equations

From Eq. (3.5) and Eq. (3.8), we obtain ${a_0} = 0, \pm \frac{{\sqrt {3\left({c - 2\alpha k} \right)} }}{{\sqrt {\beta k} }}$ and ${a_1} = \pm \frac{{\sqrt { - 6kc} }}{{\sqrt {\beta } }},$ since a₁≠0.

From Eq. (3.7), it can be deduced that

Integrating (3.9) with respect to ξ, yields

where c₁ and c₂ are constants of integration and $\theta = \frac{{\beta {a_0}{a_1}}}{{3kc}}$.

Case 1: When ${a_0} = \pm \frac{{\sqrt {3\left({c - 2\alpha k} \right)} }}{{\sqrt {\beta k} }}$ and ${a_1} = \pm \frac{{\sqrt { - 6kc} }}{{\sqrt {\beta } }}$, solving Eqs. (3.6) and (3.7) with (3.10) and (3.11), provides c = c and $\theta = \pm \frac{{\sqrt {2\left({2\alpha k - c} \right)} }}{{k\sqrt c }}$. Making use of the values of a₀, a₁ and c in (3.4), we found the subsequent general solution

where $\xi = \left({kx - ct} \right)$ and $\theta = \pm \frac{{\sqrt {2\left({2\alpha k - c} \right)} }}{{k\sqrt c }}$.

Thus, in (x, t) variables, the general closed form traveling wave solution of the simplified MCH equation is obtained as follows:

The exponential solution (3.13) can be transformed to the closed form hyperbolic function solution as

Since c₁ and c₂ are integral constants, we may generally pick their values, Therefore, if we put c₁ = θ and c₂ = 1 into solution (3.14), we attain the following kink shape soliton solution to the simplified MCH equation

On the other hand, if we put c₁ = -θ and c₂ = 1 into solution (3.14), we obtain the following closed form singular kink type solution to the simplified MCH equation

Using hyperbolic function identities, solutions (3.15) and (3.16) can be rewritten as

Case 2: When a₀ = 0 and ${a_1} = \pm \frac{{\sqrt { - 6kc} }}{{\sqrt {\beta } }}$, solving Eqs. (3.6) and (3.7) together with (3.10) and (3.11), we achieve $c = 2\alpha k$ and θ = 0. Introducing these values into solution (3.4), we achieve the next rational function solution

where $\xi = \left({kx - 2\alpha kt} \right)$.

Thus, in (x, t) variables, the general closed form traveling wave solution of the simplified MCH equation is obtained as follows:

3.2.   The Pochhammer-Chree equation

In this subsection, we will put in use the method described in section 2 to extract the closed form solutions of the Pochhammer-Chree (PC) equation of the form ^[27]:

where α, β and γ are constants and the equation describes the nonlinear model for longitudinal wave propagation in elastic rods. Li et al. ^[15] and Zhang et al. ^[56] derived some explicit solitary wave solution to the generalized PC equation of the form ${u_{tt}} - {u_{ttxx}} - {\left({\alpha u + \beta {u^{n + 1}} + \gamma {u^{2n + 1}}} \right)_{xx}} = 0, n \ge 1$, considering n = 1. In this article, we will study Eq. (3.21) considering n = 2. For details see the references ^[15,56,57].

The traveling wave transformation $u\left({x, t} \right) = u\left(\xi \right), \xi = x - ct$, where c is the wave speed to be determined latter, converts (3.21) to the ODE in the form

here ${u^{\left({iv} \right)}}$ stands for the fourth derivative and u" indicate the second derivative of u with respect to ξ. Eq. (3.22) is integrable, therefore integrating twice and setting constant of integration to zero, we obtain

Taking homogeneous balance between u" and u⁵ yields $n = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$2$}}$. To establish a closed form analytic solution through an ansatz method n should be an integer. This requires the use of the transformation $u\left(\xi \right) = v{\left(\xi \right)^{\frac{1}{2}}}$. This transformation converts Eq. (3.23) to the following equation:

Balancing vv" and v⁴ gives N = 1. Therefore, the solution structure of Eq. (3.24) is identical to solution (3.4). Substituting solution (3.4) and its derivatives into Eq. (3.24) and completing the analogous process described in subsection 3.1, we achieve the successive algebraic and differential equations,

From Eqs. (3.25) and (3.29), we achieve ${a_0} = 0, \frac{{ - \beta \pm \sqrt {{\beta ^2} + 4\gamma ({c^2} - \alpha)} }}{{2\gamma }}$ and ${a_1} = \pm \frac{{c\sqrt { - 3} }}{{2\sqrt \gamma }}$, since a₁≠0. From Eq. (3.26) it can be figure out that

Integrating (3.30) with respect to ξ, yields

where $\lambda = \frac{{{a_0}{c^2} + a_1^2\left({\beta + 4\gamma {a_0}} \right)}}{{{a_1}{c^2}}}$ and c₁, c₂ are integral constants.

Case 1: When ${a_0} = \frac{{ - \beta \pm \sqrt {{\beta ^2} + 4\gamma ({c^2} - \alpha)} }}{{2\gamma }}$ and ${a_1} = \pm \frac{{c\sqrt { - 3} }}{{2\sqrt \gamma }}$, solving Eqs. (3.26) and (3.27) with (3.31) and (3.32), we achieve $c = \pm \frac{{\sqrt {16\gamma \alpha - 3{\beta ^2}} }}{{4\sqrt \gamma }}$ and $\lambda = \frac{{2\beta \sqrt { - 3} }}{{\sqrt {16\gamma \alpha - 3{\beta ^2}})}}$. Embedding the values of a₀, a₁ and c into (3.4) provides

where $\xi = x - ct$ and $\lambda = \frac{{2\beta \sqrt { - 3} }}{{\sqrt {16\gamma \alpha - 3{\beta ^2}})}}$.

Thus, in (x, t) variables, the general solitary wave solution of the PC equation is obtained as follows:

Simplifying the exponential solution to the hyperbolic function, the solitary wave solution of the Eq. (3.34) turns into

Since c₁ and c₂ are integral constants, we might spontaneously choose their values, Therefore, if we choose c₁ = λ and c₂ = 1, from solution (3.35) we attain the following closed form solution to the PC equation:

Setting c₁ = -λ and c₂ = 1 into solution (3.35), we arrive at the following solitary wave solution to the PC equation:

Using hyperbolic function identities, solutions (3.36) and (3.37) can be rewritten as

where $c = \pm \frac{{\sqrt {16\gamma \alpha - 3{\beta ^2}} }}{{4\sqrt \gamma }}$ and $\lambda = \frac{{2\beta \sqrt { - 3} }}{{\sqrt {16\gamma \alpha - 3{\beta ^2}})}}$.

Case 2: When a₀ = 0 and ${a_1} = \pm \frac{{c\sqrt { - 3} }}{{2\sqrt \gamma }}$, solving Eq. (3.26) and (3.27) with (3.31) and (3.32), we achieve $c = \pm \frac{{\sqrt {16\gamma \alpha - 3{\beta ^2}} }}{{4\sqrt \gamma }}$ and $\lambda = \frac{{2\beta \sqrt { - 3} }}{{\sqrt {16\gamma \alpha - 3{\beta ^2}})}}$. Replace the values of a₀, a₁, c and λ into solution (3.4) provides

where $\xi = x - ct$ and $\lambda = \frac{{2\beta \sqrt { - 3} }}{{\sqrt {16\gamma \alpha - 3{\beta ^2}})}}$.

In (x, t) variables, the general solitary wave solution to the PC equation becomes:

Changing the exponential solution into the hyperbolic function solution, the solitary wave solution of the Eq. (3.41) turns into

Inasmuch as c₁ and c₂ are arbitrary constants, we may arbitrarily select their values, Therefore, if we select c₁ = λ and c₂ = 1, solution (3.42) to the PC equation turns into the stable kink type solution as follows:

Setting c₁ = -λ and c₂ = -1 into solution (3.42), we arrive to the following solution to the PC equation:

Using hyperbolic functions identities, solutions (3.43) and (3.44) can be rewritten as

where $c = \pm \frac{{\sqrt {16\gamma \alpha - 3{\beta ^2}} }}{{4\sqrt \gamma }}$ and $\lambda = \frac{{2\beta \sqrt { - 3} }}{{\sqrt {16\gamma \alpha - 3{\beta ^2}})}}$.

3.3.   The nonlinear Schrödinger-Hirota equation

Let us consider the nonlinear Schrödinger-Hirota Equation ^[6]:

The Eq. (3.47) analyzes the propagation of optical soliton in a dispersive optical fiber. Here q represents the wave profile and λ is the third order dispersion coefficient. The first term represents the evolution, while the second term is the group velocity dispersion and the fourth term is the Kerr law of nonlinearity that arises when the intensity of the light is dependent on the refractive index of the material. The third order dispersion term is taken into account when the group velocity dispersion is small so that there is performance enhancement during pulse propagation across transoceanic and trans-continental distances ^[37].

The complex transformation $q\left({x, t} \right) = {e^{i\left({\alpha x + \beta t} \right)}}u\left(\xi \right), \xi = k(x - 2\alpha t$, where α, β, k and ω are real constants reduces Eq. (3.47) to an ordinary differential equation of the form:

From the above Eq. (3.48), we obtain $\alpha = - 1/3{\rm{ \mathsf{ λ} }}$ and $u\left(\xi \right)$ satisfy the differential equation

Eq. (3.49) can be rewritten as

where ${\mu _1} = \frac{2}{{3{k^2}}}$ and ${\mu _2} = \frac{2}{{3{k^2}}}\left({\beta + \frac{5}{{54{\lambda ^2}}}} \right)$.

Taking homogeneous balance between linear term u" and nonlinear term u³ yields N = 1. Therefore, the solution structure of Eq. (3.50) is same as the solution (3.4). Hence substituting solution (3.4) and its derivatives into Eq. (3.50), and completing the similar procedure described in subsection 3.1, we achieve the successive algebraic and differential equations

From Eq. (3.51) and Eq. (3.54), we obtain ${a_0} = 0, \pm \sqrt {\frac{{{\mu _2}}}{{{\mu _1}}}}$ and ${a_1} = \pm \sqrt {\frac{{ - 2}}{{{\mu _1}}}}$, since a₁≠0. And from Eq. (3.53), we attain

Integrating (3.55) with respect to ξ, yields

where c₁ and c₂ are arbitrary constants and θ = μ₁a₀a₁.

Case 1: When ${a_0} = \pm \sqrt {\frac{{{\mu _2}}}{{{\mu _1}}}}$ and ${a_1} = \pm \sqrt {\frac{{ - 2}}{{{\mu _1}}}}$, and $\alpha = - 1/3{\rm{ \mathsf{ λ} }}$, substitute the values of a₀, a₁ and α into solution (3.4), we determine

where $\xi = k\left({x - \frac{2}{{3\lambda }}t} \right)$ and $\theta = \pm \sqrt { - 2{\mu _2}}$.

Forasmuch as c₁ and c₂ are arbitrary constants, we may freely accept their values. Therefore, if we set c₁ = θ and c₂ = 1 into solution (3.58) and simplifying the exponential solution to hyperbolic function, we attain the following solitary wave solution of the Eq. (3.47)

Moreover, setting c₁ = -θ and c₂ = 1 into solution (3.58), we arrive at the following solitary wave solution of the nonlinear Schrödinger-Hirota equation:

Thus, in (x, t) variables, the general solitary wave solution to the nonlinear Schrödinger-Hirota equation is obtained as follows:

Case 2: When a₀ = 0 and ${a_1} = \pm \sqrt {\frac{{ - 2}}{{{\mu _1}}}}$, and $\alpha = - 1/3\lambda$, making use of the values of a₀, a₁ and α into (3.4), it does not satisfy the Eq. (3.53) and hence the solution must be rejected.

4.   Explanation and physical interpretation of the solutions

In this section, we have discussed about the obtained solution of the simplified MCH equation, the PC equation and the Schrödinger-Hirota equation. Using the MSE method, we get the traveling wave solutions assembled from Eqs. (3.12) to (3.20) to the simplified MCH equation. The solutions are general solitary wave solutions which are periodic wave solution, kink shape soliton and singular kink shape soliton respectively. From the above solution, it has been detected that the solutions (3.12) and (3.13) provides periodic wave solution where the solutions (3.15), (3.17) and (3.19) gives kink shape wave solution. The solutions (3.14), (3.16) and (3.20) present singular kink solutions. The kink shape wave solution (3.15) is represented in Figure 1 for α = 1, β = 1, k = 1 and ω = 4. The singular kink solutions (3.16) and (3.20) for α = 1, β = 1, k = 1, ω = 4 and for α = 1, β = 1 are plotted in the Figures 2 and 3 respectively. From the solutions to the PC equation, it is observed that the solutions (3.34) and (3.35) show the nature of singular kink, solutions (3.36), (3.38), (3.43) and (3.45) represent the kink shape soliton and solutions (3.37), (3.39), (3.44) and (3.46) are singular solution. Singular solitons can be connected to solitary waves when the center position of the solitary wave is imaginary ^[58]. This solution has spike and therefore it can probably provide an explanation to the formation of Rogue waves ^[38]. The kink shape solution (3.43) for $\alpha = \frac{1}{2}, \beta = 2$ and γ = -1 is represented in Figure 4. From the solutions of the Schrödinger-Hirota equation, the solutions (3.59)-(3.62) are categorized to the character of singular periodic solution and (3.58) represents periodic solution. Periodic traveling waves play an important role in numerous physical phenomena, including reaction-diffusion-advection systems, self-reinforcing systems, impulsive systems etc. Mathematical modeling of many intricate physical events, for instance physics, mathematical physics, engineering and many more phenomena resemble periodic traveling wave solutions. The singular periodic solutions (3.59) for β = 1, k = 1, λ = 1/3 and (3.60) for β = 1, k = 1, λ = 1/3 are given in Figures 5 and 6 respectively. The figures of other solutions are similar mentioned above and ignored these figures for simplicity.

5.   Conclusion

In this article, the modified simple equation method has successfully been used to establish the solitary wave solutions to the simplified MCH equation, the Pochhammer-Chree equation and the Schrödinger-Hirota equation. The attribute and uniqueness of this method is that the considered function ψ(ξ) is not an early known function. So is not possible to presume in advance what kind of solutions one may obtain through this method. Therefore, the obtained solutions are more general and fresh and important to analyze the inner mechanism of these nonlinear phenomena. The solutions are confirmed through checking the correctness by inserting them into the original equations and found correct. The results show that the method is reliable and effective. The used method has several advantages: it is straightforward and its calculation procedure is concise. Therefore this efficient method could be more effectively used to solve various NLEEs which regularly arise in science, engineering and other technical arenas.

Acknowledgments

The authors express their sincere thanks to the anonymous referees for their valuable comments and suggestions to improve the article.

Conflict of interest

The authors declare no conflict of interest.