# The Application of theexp⁡(-Φ(ξ))-Expansion Method for Finding the Exact Solutions of Two Integrable Equations

Authors: Naila Sajid; Ghazala Akram
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this article, theexp⁡-Φξ method is connected to search for new hyperbolic, periodic, and rational solutions of (1+1)-dimensional fifth-order nonlinear integrable equation and (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. The obtained solutions consist of trigonometric, hyperbolic, rational functions and W-shaped soliton. Furthermore, 3D and 2D graphs are plotted by choosing the suitable values of the parameters involved.


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## Body

## 1. Introduction

Firstly, consider the(1+1)-dimensional fifth-order nonlinear integrable equation. In [1], Wazwaz proposed a new (1+1)-dimensional fifth-order nonlinear integrable equation of the form(1)uttt-utxxxx-4uxutxx-4uxuxtx=0,where u=u(x,t) stands for wave propagation of physical quantity and subscripts represent partial differentiation with respect to the given variable and obtained multiple soliton solutions using the simplified Hirota’s method established by Hereman and Nuseir [2].Furthermore, Yuanet al. [3] consider the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. To study the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation, many researchers considered the following integrable equation:(2)uxxxxy+4uxxyux+2uxxxuy+6uxyuxx+uyyy-2uxxt=0,where u is the real function of the variables x, y, and t. With the help of the Hirota method and auxiliary variables, the bilinear Bäcklund transformation and N-soliton solutions are obtained.Nonlinear evolution equations (NLEEs) are one of the fastest developing zones of research in the field of science and engineering, especially in mathematical biology, nonlinear optics, optical fiber, fluid mechanics, solid state physics, biophysics, chemical physics, chemical kinetics,etc. Many effective methods have been proposed to solve the NLEEs, such as the Hirota method [1], Hereman-Nuseir method [2], inverse scattering transformation [4], Painlevé technique [5], Bäcklund transformation [6], Darboux transformation [7, 8], Binary-Bell-polynomial scheme [9], first integral method [10, 11], (G′/G)-expansion method [12], the exp⁡(-Φ(ξ)) expansion method [13], Exp-function method [14], ansatz method [15], sine-Gordon expansion method [16, 17], the trial equation method [18, 19], homotopy asymptotic [20], and so on.The present paper organized as follows: In Section2, description of the exp⁡-Φξ method for finding the exact traveling wave solutions of NLEEs is presented. Section 3 illustrates the method to solve the (1+1)-dimensional fifth-order nonlinear integrable equation and (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. Results and discussion are presented in Section 4. Finally, in Section 5, some conclusions are given.


## 2. Traveling Wave Hypothesis

Consider a NLEE in two independent variablesx and t of the form as(3)Pu,ux,ut,uxx,uxt,utt,…=0,where u=u(x,t) is an unknown function and P is the polynomial in u=u(x,t) which have various partial derivatives. Under the wave transformation(4)ux,t=uξ,ξ=νx-kt,where ν and k are the wave number and wave speed, respectively. Eq. (3) can be transformed into the following nonlinear ordinary differential equation as(5)Qu,νu′,-νku′,ν2u′′,-ν2ku′′,k2ν2u′′,…=0,where prime denotes the derivatives with respect to ξ.


### 2.1. Theexp⁡-ϕξ Expansion Method: Quick Recapitulation

The key steps ofexp⁡-Φξ method are given asStep 1.
According toexp⁡-Φξ method, the wave solution can be expressed as(6)Uξ=∑i=0Naiexp⁡-Φξi,where ai(aN≠0) are constants to be determined and Φ(ξ) satisfies the auxiliary ODE given as(7)Φ′ξ=exp-Φξ+μexp⁡Φξ+λ.The auxiliary Eq. (7) has the general solutions given as follows.
Case 1(hyperbolic function solutions). When λ2-4μ>0 and μ≠0,(8)Φ1ξ=ln⁡-λ2-4μtanh⁡λ2-4μ/2ξ+K-λ2μ.
Case 2(trigonometric function solutions). When λ2-4μ<0 and μ≠0,(9)Φ2ξ=ln⁡4μ-λ2tan⁡4μ-λ2/2ξ+K-λ2μ.
Case 3(hyperbolic function solutions). When λ2-4μ>0,μ=0, and λ≠0,(10)Φ3ξ=-ln⁡λcosh⁡λξ+K+sinh⁡λξ+K-1.
Case 4(rational function solutions). When λ2-4μ=0,μ≠0, and λ≠0,(11)Φ4ξ=ln⁡-2λξ+K+2λ2ξ+K.
Case 5. When λ2-4μ=0,μ=0, and λ=0,(12)Φ5ξ=ln⁡ξ+K,where K is integration constant.Step 2.
The positive integerN can be determined by balancing the highest derivative term with the highest order nonlinear term in Eq. (5). Substituting Eq. (6) into Eq. (5) yields an algebraic equation involving powers of exp⁡(-Φ(ξ))i(i=0,1,2,…). Equating the coefficients of each power of exp⁡(-Φ(ξ)) to zero gives a system of algebraic equations for ai(i=0,1,2,…,n), ν, k, λ, and μ.Step 3.
Substitutingai, ν, k, λ, and μ, a variety of exact solutions of Eq.(3) can be constructed.


## 2.1. Theexp⁡-ϕξ Expansion Method: Quick Recapitulation

The key steps ofexp⁡-Φξ method are given asStep 1.
According toexp⁡-Φξ method, the wave solution can be expressed as(6)Uξ=∑i=0Naiexp⁡-Φξi,where ai(aN≠0) are constants to be determined and Φ(ξ) satisfies the auxiliary ODE given as(7)Φ′ξ=exp-Φξ+μexp⁡Φξ+λ.The auxiliary Eq. (7) has the general solutions given as follows.
Case 1(hyperbolic function solutions). When λ2-4μ>0 and μ≠0,(8)Φ1ξ=ln⁡-λ2-4μtanh⁡λ2-4μ/2ξ+K-λ2μ.
Case 2(trigonometric function solutions). When λ2-4μ<0 and μ≠0,(9)Φ2ξ=ln⁡4μ-λ2tan⁡4μ-λ2/2ξ+K-λ2μ.
Case 3(hyperbolic function solutions). When λ2-4μ>0,μ=0, and λ≠0,(10)Φ3ξ=-ln⁡λcosh⁡λξ+K+sinh⁡λξ+K-1.
Case 4(rational function solutions). When λ2-4μ=0,μ≠0, and λ≠0,(11)Φ4ξ=ln⁡-2λξ+K+2λ2ξ+K.
Case 5. When λ2-4μ=0,μ=0, and λ=0,(12)Φ5ξ=ln⁡ξ+K,where K is integration constant.Step 2.
The positive integerN can be determined by balancing the highest derivative term with the highest order nonlinear term in Eq. (5). Substituting Eq. (6) into Eq. (5) yields an algebraic equation involving powers of exp⁡(-Φ(ξ))i(i=0,1,2,…). Equating the coefficients of each power of exp⁡(-Φ(ξ)) to zero gives a system of algebraic equations for ai(i=0,1,2,…,n), ν, k, λ, and μ.Step 3.
Substitutingai, ν, k, λ, and μ, a variety of exact solutions of Eq.(3) can be constructed.


## 3. Application of theexp⁡-Φξ Method

### 3.1. The(1+1)-Dimensional Fifth-Order Nonlinear Integrable Equation

The(1+1)-dimensional fifth-order nonlinear integrable equation is given as(13)uttt-utxxxx-4uxutxx-4uxuxtx=0.Applying the wave transformation u(x,t)=V(ξ), ξ=ν(x-kt) to Eq. (13) yields the following nonlinear ODE:(14)-k2V′′+ν2V4+12νV′V′′=0.Set U=V′ to get(15)-k2U′+ν2U′′′+12νUU′=0,where the integration constant is taken as zero. Balance the linear term of highest order U′′′ with the highest order nonlinear term UU′ in Eq. (15), to get balancing number as N=2. Thus the solution is(16)Uξ=a0+a1exp⁡-Φξ+a2exp⁡-Φξ2.Substituting Eq. (16) into Eq. (15) and collecting the coefficient of each power of exp⁡(-Φ(ξ))i and then setting each of coefficients to zero give a system of algebraic equations as (17)exp⁡-ϕξ0:k2a1μ-12νa0a1μ-ν2a1μλ2-6ν2a2μ2λ-2ν2a1μ2=0,exp⁡-ϕξ1:-24νa0a2μ-12νa0a1λ-14ν2a2μλ2-8ν2a1μλ+k2a1λ-16ν2a2μ2-ν2a1λ3+2k2a2μ-12νa12μ=0,exp⁡-ϕξ2:k2a1-24νa0a2λ-36νa1a2μ-52ν2a2μλ-8ν2a2λ3-7ν2a1λ2-8ν2a1μ+2k2a2λ-12νa0a1-12νa12λ=0,exp⁡-ϕξ3:-12νa12+2k2a2-36νa1a2λ-38ν2a2λ2-12ν2a1λ-40ν2a2μ-24νa22μ-24νa0a2=0,exp⁡-ϕξ4:-6ν2a1-36νa1a2-54ν2a2λ-24νa22λ=0,exp⁡-ϕξ5:-24νa22-24ν2a2=0.Solving the above system of equations yields(18)a0=a0,a1=-νλ,a2=-ν,k=±ν2λ2+8ν2μ+12νa0.Consequently, the following different cases are obtained for the exact solutions of (1+1)-dimensional fifth-order nonlinear integrable equation.Case 1(hyperbolic function solutions). When λ2-4μ>0 and μ≠0,(19)u1ξ=a0ξ+4νλ3μln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ2-4μλ2-4μ+λλ2-4μ-λ-16νλμ2ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ2-4μλ2-4μ+λλ2-4μ-λ-4νλμln⁡tanh⁡1/2λ2-4μξ+K+1λ2-4μ2λ2-4μ-2λ-4νλμln⁡tanh⁡1/2λ2-4μξ+K-1λ2-4μ2λ2-4μ+2λ+8νμ2λ2-4μ+λλ2-4μ-λλ2-4μtanh⁡1/2λ2-4μξ+K+λ+16νμ2ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ3λ2-4μλ2-4μ+λ2λ2-4μ-λ2-64νμ3ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλλ2-4μλ2-4μ+λ2λ2-4μ-λ2-4νμ2lntanh1/2λ2-4μξ+K+1λ2-4μλ2-4μ-λ2+4νμ2ln⁡tanh⁡1/2λ2-4μξ+K-1λ2-4μλ2-4μ+λ2.Case 2(trigonometric function solutions). When λ2-4μ<0 and μ≠0,(20)u2ξ=a0ξ-νλln⁡4μ-λ2tan⁡124μ-λ2ξ+K-λ+2μν4μ-λ2tan⁡1/24μ-λ2ξ+K-λ+2μνarctan⁡tan⁡1/24μ-λ2ξ+K4μ-λ2+4νλμln⁡4μ-λ2tan⁡1/24μ-λ2ξ+K-λ4μ-λ2-νλ3ln⁡4μ-λ2tan⁡1/24μ-λ2ξ+K-λ4μ-λ2.Case 3(hyperbolic function solutions). When λ2-4μ>0,μ=0, and λ≠0,(21)u3ξ=a0ξ-νln⁡tanh⁡12λξ+12λK+νln⁡tanh⁡12λξ+12λK+1+12νλtanh⁡1/2λξ+1/2λK+νln⁡tanh⁡1/2λξ+1/2λKλ-νln⁡tanh⁡1/2λξ+1/2λK+1λ.Case 4(rational function solutions). When λ2-4μ=0,μ≠0, and λ≠0,(22)u4ξ=a0ξ+14νλ2ξ+νλλξ+λK+2.Case 5. When λ2-4μ=0,μ=0, and λ=0,(23)u5ξ=a0-νλξ+K-νξ+K2,where ξ=ν(x-kt) and K is the integration constant.The solutions (u1,u2,u3,u4,u5) of the (1+1)-dimensional fifth-order nonlinear integrable equation are graphically presented by Figures 1, 2, 3, 4, and 5.Figure 1
3D and 2D plot ofu1 with μ=1, λ=2.5, and t=0.002 for 2D graphics.Figure 2
3D and 2D plot ofu2 with μ=2.5, λ=1, and t=0.002 for 2D graphics.Figure 3
3D and 2D plot ofu3 with μ=0, λ=1, and t=0.002 for 2D graphics.Figure 4
3D and 2D plot ofu4 with μ=0.25, λ=1, and t=0.002 for 2D graphics.Figure 5
3D and 2D-plot ofu5 with μ=0, λ=0, and t=0.002 for 2D graphics.


### 3.2.(2+1)-Dimensional Date-Jimbo-Kashiwara-Miwa Equation

The(2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation is given as(24)uxxxxy+4uxxyux+2uxxxuy+6uxyuxx+uyyy-2uxxt=0.Applying the wave transformation u(x,y,t)=V(ξ), ξ=ν(x+ay-kt) to Eq. (24) yields the following nonlinear ODE:(25)aν2V4+6aνV′V′′+a3+2kV′′=0.Set U=V′ to get(26)aν2U′′′+6aνUU′+a3+2kU′=0,where the integration constant is taken as zero. Balance the linear term of highest order U′′′ with the highest order nonlinear term UU′ in Eq. (26), to get balancing number as N=2. Thus, the solution can be written as(27)Uξ=a0+a1exp⁡-Φξ+a2exp⁡-Φξ2.Substituting Eq. (27) into Eq. (26) and collecting the coefficient of each power of exp⁡(-Φ(ξ))i and then setting each of coefficient to zero give a system of algebraic equations as(28)exp⁡-ϕξ0:-aν2a1μλ2-6aν2a2μ2λ-2aν2a1μ2-a3a1μ-6aνa0a1μ-2ka1μ=0,exp⁡-ϕξ1:-a3a1λ-2ka1λ-2a3a2μ-4ka2μ-8aν2a1μλ-14aν2a2μλ2-12aνa0a2μ-6aνa0a1λ-6aνa12μ-aν2a1λ3-16aν2a2μ2=0,exp⁡-ϕξ2:-2a3a2λ-4ka2λ-52aν2a2μλ-12aνa0a2λ-18aνa1a2μ-6aνa0a1-6aνa12λ-2ka1-a3a1-7aν2a1λ2-8aν2a2λ3-8aν2a1μ=0,exp⁡-ϕξ3:-18aνa1a2λ-12aνa22μ-12aνa0a2-2a3a2-4ka2-12aν2a1λ-38aν2a2λ2-40aν2a2μ-6aνa12=0,exp⁡-ϕξ4:-6aν2a1-18aνa1a2-12aνa22λ-54aν2a2λ=0,exp⁡-ϕξ5:-24aν2a2-12aνa22=0.Solving the above system of equations yields(29)a0=-16ν2aλ2+8ν2aμ+a3+2kaν,a1=-2νλ,a2=-2ν,ν=ν,k=k.Consequently, the following different cases are obtained for the exact solutions of (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation.Case 1(hyperbolic function solutions). When λ2-4μ>0 and μ≠0,(30)u6ξ=-16νξλ2-43νξμ-16a2ξν-13ξkaν-8νλμln⁡tanh⁡1/2λ2-4μξ+K+1λ2-4μ2λ2-4μ-2λ-8νλμln⁡tanh⁡1/2λ2-4μξ+K-1λ2-4μ2λ2-4μ+2λ+8νλ3μln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ2-4μλ2-4μ+λλ2-4μ-λ-32νλμ2ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ2-4μλ2-4μ+λλ2-4μ-λ+16νμ2λ2-4μ+λλ2-4μ-λλ2-4μtanh⁡1/2λ2-4μξ+K+λ-8νμ2ln⁡tanh⁡1/2λ2-4μξ+K+1λ2-4μλ2-4μ-λ2+8νμ2ln⁡tanh⁡1/2λ2-4μξ+K-1λ2-4μλ2-4μ+λ2+32νμ2ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ3λ2-4μλ2-4μ+λ2λ2-4μ-λ2-128νμ3ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλλ2-4μλ2-4μ+λ2λ2-4μ-λ2.Case 2(trigonometric function solutions). When λ2-4μ<0 and μ≠0,(31)u7ξ=-16νξλ2-43νξμ-16b2ξν-13ξkbν-2νλln⁡4μ-λ2tan⁡124μ-λ2ξ+K-λ+4νμarctan⁡tan⁡1/24μ-λ2ξ+K4μ-λ2+4νμ4μ-λ2tan⁡1/24μ-λ2ξ+K-λ+8νλμln⁡4μ-λ2tan⁡1/24μ-λ2ξ+K-λ4μ-λ2-2νλ3ln4μ-λ2tan⁡1/24μ-λ2ξ+K-λ4μ-λ2.Case 3(hyperbolic function solutions). When λ2-4μ>0,μ=0, and λ≠0,(32)u8ξ=-16νξλ2-43νξμ-16b2ξν-13ξkbν+2νln⁡tanh⁡12λξ+12λK+1-2νln⁡tanh⁡12λξ+12λK+νλtanh⁡1/2λξ+1/2λK-2νln⁡tanh⁡1/2λξ+1/2λK+1λ+2νln⁡tanh⁡1/2λξ+1/2λKλ.Case 4(rational function solutions). When λ2-4μ=0,μ≠0, and λ≠0,(33)u9ξ=-16ν2bλ2+8ν2bμ+b3+2kξbν+12νλ2ξ+2νλλξ+λK+2.Case 5. When λ2-4μ=0,μ=0, and λ=0,(34)u10ξ=-16νξλ2-43νξμ-16b2ξν-13ξkbν-2νλln⁡ξ+K+2νξ+K,where ξ=ν(x+ay-kt) and K is the integration constant.The graphical representation of the solutions(u6,u7,u8,u9,u10) for the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation is shown by Figures 6–10.Figure 6
W-shaped soliton 3D and 2D plot ofu6 with μ=1, λ=2.5, and y=0.002 for 2D graphics.Figure 7
3D and 2D plot ofu7 with μ=2.5, λ=1, and y=0.002 for 2D graphics.Figure 8
3D and 2D plot ofu8 with μ=0, λ=1, and y=0.002 for 2D graphics.Figure 9
3D and 2D plot ofu9 with μ=0.25, λ=1, and y=0.002 for 2D graphics.Figure 10
3D and 2D plot ofu10 with μ=0, λ=0, and y=0.002 for 2D graphics.


## 3.1. The(1+1)-Dimensional Fifth-Order Nonlinear Integrable Equation

The(1+1)-dimensional fifth-order nonlinear integrable equation is given as(13)uttt-utxxxx-4uxutxx-4uxuxtx=0.Applying the wave transformation u(x,t)=V(ξ), ξ=ν(x-kt) to Eq. (13) yields the following nonlinear ODE:(14)-k2V′′+ν2V4+12νV′V′′=0.Set U=V′ to get(15)-k2U′+ν2U′′′+12νUU′=0,where the integration constant is taken as zero. Balance the linear term of highest order U′′′ with the highest order nonlinear term UU′ in Eq. (15), to get balancing number as N=2. Thus the solution is(16)Uξ=a0+a1exp⁡-Φξ+a2exp⁡-Φξ2.Substituting Eq. (16) into Eq. (15) and collecting the coefficient of each power of exp⁡(-Φ(ξ))i and then setting each of coefficients to zero give a system of algebraic equations as (17)exp⁡-ϕξ0:k2a1μ-12νa0a1μ-ν2a1μλ2-6ν2a2μ2λ-2ν2a1μ2=0,exp⁡-ϕξ1:-24νa0a2μ-12νa0a1λ-14ν2a2μλ2-8ν2a1μλ+k2a1λ-16ν2a2μ2-ν2a1λ3+2k2a2μ-12νa12μ=0,exp⁡-ϕξ2:k2a1-24νa0a2λ-36νa1a2μ-52ν2a2μλ-8ν2a2λ3-7ν2a1λ2-8ν2a1μ+2k2a2λ-12νa0a1-12νa12λ=0,exp⁡-ϕξ3:-12νa12+2k2a2-36νa1a2λ-38ν2a2λ2-12ν2a1λ-40ν2a2μ-24νa22μ-24νa0a2=0,exp⁡-ϕξ4:-6ν2a1-36νa1a2-54ν2a2λ-24νa22λ=0,exp⁡-ϕξ5:-24νa22-24ν2a2=0.Solving the above system of equations yields(18)a0=a0,a1=-νλ,a2=-ν,k=±ν2λ2+8ν2μ+12νa0.Consequently, the following different cases are obtained for the exact solutions of (1+1)-dimensional fifth-order nonlinear integrable equation.Case 1(hyperbolic function solutions). When λ2-4μ>0 and μ≠0,(19)u1ξ=a0ξ+4νλ3μln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ2-4μλ2-4μ+λλ2-4μ-λ-16νλμ2ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ2-4μλ2-4μ+λλ2-4μ-λ-4νλμln⁡tanh⁡1/2λ2-4μξ+K+1λ2-4μ2λ2-4μ-2λ-4νλμln⁡tanh⁡1/2λ2-4μξ+K-1λ2-4μ2λ2-4μ+2λ+8νμ2λ2-4μ+λλ2-4μ-λλ2-4μtanh⁡1/2λ2-4μξ+K+λ+16νμ2ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ3λ2-4μλ2-4μ+λ2λ2-4μ-λ2-64νμ3ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλλ2-4μλ2-4μ+λ2λ2-4μ-λ2-4νμ2lntanh1/2λ2-4μξ+K+1λ2-4μλ2-4μ-λ2+4νμ2ln⁡tanh⁡1/2λ2-4μξ+K-1λ2-4μλ2-4μ+λ2.Case 2(trigonometric function solutions). When λ2-4μ<0 and μ≠0,(20)u2ξ=a0ξ-νλln⁡4μ-λ2tan⁡124μ-λ2ξ+K-λ+2μν4μ-λ2tan⁡1/24μ-λ2ξ+K-λ+2μνarctan⁡tan⁡1/24μ-λ2ξ+K4μ-λ2+4νλμln⁡4μ-λ2tan⁡1/24μ-λ2ξ+K-λ4μ-λ2-νλ3ln⁡4μ-λ2tan⁡1/24μ-λ2ξ+K-λ4μ-λ2.Case 3(hyperbolic function solutions). When λ2-4μ>0,μ=0, and λ≠0,(21)u3ξ=a0ξ-νln⁡tanh⁡12λξ+12λK+νln⁡tanh⁡12λξ+12λK+1+12νλtanh⁡1/2λξ+1/2λK+νln⁡tanh⁡1/2λξ+1/2λKλ-νln⁡tanh⁡1/2λξ+1/2λK+1λ.Case 4(rational function solutions). When λ2-4μ=0,μ≠0, and λ≠0,(22)u4ξ=a0ξ+14νλ2ξ+νλλξ+λK+2.Case 5. When λ2-4μ=0,μ=0, and λ=0,(23)u5ξ=a0-νλξ+K-νξ+K2,where ξ=ν(x-kt) and K is the integration constant.The solutions (u1,u2,u3,u4,u5) of the (1+1)-dimensional fifth-order nonlinear integrable equation are graphically presented by Figures 1, 2, 3, 4, and 5.Figure 1
3D and 2D plot ofu1 with μ=1, λ=2.5, and t=0.002 for 2D graphics.Figure 2
3D and 2D plot ofu2 with μ=2.5, λ=1, and t=0.002 for 2D graphics.Figure 3
3D and 2D plot ofu3 with μ=0, λ=1, and t=0.002 for 2D graphics.Figure 4
3D and 2D plot ofu4 with μ=0.25, λ=1, and t=0.002 for 2D graphics.Figure 5
3D and 2D-plot ofu5 with μ=0, λ=0, and t=0.002 for 2D graphics.


## 3.2.(2+1)-Dimensional Date-Jimbo-Kashiwara-Miwa Equation

The(2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation is given as(24)uxxxxy+4uxxyux+2uxxxuy+6uxyuxx+uyyy-2uxxt=0.Applying the wave transformation u(x,y,t)=V(ξ), ξ=ν(x+ay-kt) to Eq. (24) yields the following nonlinear ODE:(25)aν2V4+6aνV′V′′+a3+2kV′′=0.Set U=V′ to get(26)aν2U′′′+6aνUU′+a3+2kU′=0,where the integration constant is taken as zero. Balance the linear term of highest order U′′′ with the highest order nonlinear term UU′ in Eq. (26), to get balancing number as N=2. Thus, the solution can be written as(27)Uξ=a0+a1exp⁡-Φξ+a2exp⁡-Φξ2.Substituting Eq. (27) into Eq. (26) and collecting the coefficient of each power of exp⁡(-Φ(ξ))i and then setting each of coefficient to zero give a system of algebraic equations as(28)exp⁡-ϕξ0:-aν2a1μλ2-6aν2a2μ2λ-2aν2a1μ2-a3a1μ-6aνa0a1μ-2ka1μ=0,exp⁡-ϕξ1:-a3a1λ-2ka1λ-2a3a2μ-4ka2μ-8aν2a1μλ-14aν2a2μλ2-12aνa0a2μ-6aνa0a1λ-6aνa12μ-aν2a1λ3-16aν2a2μ2=0,exp⁡-ϕξ2:-2a3a2λ-4ka2λ-52aν2a2μλ-12aνa0a2λ-18aνa1a2μ-6aνa0a1-6aνa12λ-2ka1-a3a1-7aν2a1λ2-8aν2a2λ3-8aν2a1μ=0,exp⁡-ϕξ3:-18aνa1a2λ-12aνa22μ-12aνa0a2-2a3a2-4ka2-12aν2a1λ-38aν2a2λ2-40aν2a2μ-6aνa12=0,exp⁡-ϕξ4:-6aν2a1-18aνa1a2-12aνa22λ-54aν2a2λ=0,exp⁡-ϕξ5:-24aν2a2-12aνa22=0.Solving the above system of equations yields(29)a0=-16ν2aλ2+8ν2aμ+a3+2kaν,a1=-2νλ,a2=-2ν,ν=ν,k=k.Consequently, the following different cases are obtained for the exact solutions of (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation.Case 1(hyperbolic function solutions). When λ2-4μ>0 and μ≠0,(30)u6ξ=-16νξλ2-43νξμ-16a2ξν-13ξkaν-8νλμln⁡tanh⁡1/2λ2-4μξ+K+1λ2-4μ2λ2-4μ-2λ-8νλμln⁡tanh⁡1/2λ2-4μξ+K-1λ2-4μ2λ2-4μ+2λ+8νλ3μln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ2-4μλ2-4μ+λλ2-4μ-λ-32νλμ2ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ2-4μλ2-4μ+λλ2-4μ-λ+16νμ2λ2-4μ+λλ2-4μ-λλ2-4μtanh⁡1/2λ2-4μξ+K+λ-8νμ2ln⁡tanh⁡1/2λ2-4μξ+K+1λ2-4μλ2-4μ-λ2+8νμ2ln⁡tanh⁡1/2λ2-4μξ+K-1λ2-4μλ2-4μ+λ2+32νμ2ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλ3λ2-4μλ2-4μ+λ2λ2-4μ-λ2-128νμ3ln⁡λ2-4μtanh⁡1/2λ2-4μξ+K+λλλ2-4μλ2-4μ+λ2λ2-4μ-λ2.Case 2(trigonometric function solutions). When λ2-4μ<0 and μ≠0,(31)u7ξ=-16νξλ2-43νξμ-16b2ξν-13ξkbν-2νλln⁡4μ-λ2tan⁡124μ-λ2ξ+K-λ+4νμarctan⁡tan⁡1/24μ-λ2ξ+K4μ-λ2+4νμ4μ-λ2tan⁡1/24μ-λ2ξ+K-λ+8νλμln⁡4μ-λ2tan⁡1/24μ-λ2ξ+K-λ4μ-λ2-2νλ3ln4μ-λ2tan⁡1/24μ-λ2ξ+K-λ4μ-λ2.Case 3(hyperbolic function solutions). When λ2-4μ>0,μ=0, and λ≠0,(32)u8ξ=-16νξλ2-43νξμ-16b2ξν-13ξkbν+2νln⁡tanh⁡12λξ+12λK+1-2νln⁡tanh⁡12λξ+12λK+νλtanh⁡1/2λξ+1/2λK-2νln⁡tanh⁡1/2λξ+1/2λK+1λ+2νln⁡tanh⁡1/2λξ+1/2λKλ.Case 4(rational function solutions). When λ2-4μ=0,μ≠0, and λ≠0,(33)u9ξ=-16ν2bλ2+8ν2bμ+b3+2kξbν+12νλ2ξ+2νλλξ+λK+2.Case 5. When λ2-4μ=0,μ=0, and λ=0,(34)u10ξ=-16νξλ2-43νξμ-16b2ξν-13ξkbν-2νλln⁡ξ+K+2νξ+K,where ξ=ν(x+ay-kt) and K is the integration constant.The graphical representation of the solutions(u6,u7,u8,u9,u10) for the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation is shown by Figures 6–10.Figure 6
W-shaped soliton 3D and 2D plot ofu6 with μ=1, λ=2.5, and y=0.002 for 2D graphics.Figure 7
3D and 2D plot ofu7 with μ=2.5, λ=1, and y=0.002 for 2D graphics.Figure 8
3D and 2D plot ofu8 with μ=0, λ=1, and y=0.002 for 2D graphics.Figure 9
3D and 2D plot ofu9 with μ=0.25, λ=1, and y=0.002 for 2D graphics.Figure 10
3D and 2D plot ofu10 with μ=0, λ=0, and y=0.002 for 2D graphics.


## 4. Results and Discussion

In this manuscript, several traveling wave solutions are developed. A new kind of W-shaped soliton solution is demonstrated and the other obtained solutions consist of trigonometric, hyperbolic, rational functions which are also new. On comparing our results with the well-known results obtained in [1, 3], it may be concluded that the obtained results for Eq. (13) and Eq. (24) are newly constructed. Wazwaz [1] reported some multiple soliton solutions to Eq. (13) using simplified Hirota’s method and Yuanet al. [2] acquired N-soliton solutions to Eq. (24) using Hirota method and auxiliary variables. In both equations for case 1, the absolute behavior of the solution |u1(x,t)| and |u6(x,t)| is shown in Figures 1 and 6, respectively. The graphical representation of the other solutions (u2,u3,u4,u5) for the (1+1)-dimensional fifth-order nonlinear integrable equation and (u7,u8,u9,u10) for the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation is shown by Figures 2–5 and 7–10, respectively.


## 5. Conclusion

Theexp⁡(-Φ(ξ)) method is used to investigate the exact solutions of the (1+1)-dimensional fifth-order nonlinear integrable equation and (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. With the implementation of the exp⁡(-Φ(ξ)) method, many exact traveling wave solutions are obtained including trigonometric, hyperbolic, rational, and W-shaped soliton. The reported solutions in this article may be useful in explaining the physical meaning of the studied models and other nonlinear models arising in the field of fiber optics and other related fields. These results indicate that the proposed method is very useful and effective in performing solution to the NLEEs.



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*Source: 5191736-2018-11-26.xml*