Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications

Asghar Ali; Aly R. Seadawy; Dianchen Lu

doi:10.1515/phys-2018-0032

Article Open Access

Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications

Asghar Ali , Aly R. Seadawy and Dianchen Lu

Published/Copyright: May 24, 2018

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From the journal Open Physics Volume 16 Issue 1

Abstract

The aim of this article is to construct some new traveling wave solutions and investigate localized structures for fourth-order nonlinear Ablowitz-Kaup-Newell-Segur (AKNS) water wave dynamical equation. The simple equation method (SEM) and the modified simple equation method (MSEM) are applied in this paper to construct the analytical traveling wave solutions of AKNS equation. The different waves solutions are derived by assigning special values to the parameters. The obtained results have their importance in the field of physics and other areas of applied sciences. All the solutions are also graphically represented. The constructed results are often helpful for studying several new localized structures and the waves interaction in the high-dimensional models.

Keywords: Simple equation method; modified simple equation method; fourth-order nonlinear AKNS equation; exact solutions; solitary wave

PACS: 02.30.Jr; 05.45.Yv; 02.30.Ik

1 Introduction

Several dynamical nonlinear problems in the field of applied physics and other areas of natural science are generally characterized by nonlinear evolution of partial differential equations (PDEs) well-known as governing equations [1, 2, 3, 4, 5, 6]. Such nonlinear PDEs play a vital role in physical science to understand the nonlinear complex physical phenomena. The analytical solutions of nonlinear PDEs have their own importance in the various branches of mathematical physical sciences, applied sciences and other areas of engineering to understand their physical interpretation. Therefore, the search for an exact solution of nonlinear systems has been an interesting and important topic for mathematicians and physicists in nonlinear science. Many powerful and systemic methods have been developed to construct the solutions of nonlinear PDEs such as, Hirota bilinear method, tanh-coth method, Exp-function method, multi linear variable separation approach method, modified extended direct algebraic method, simple equation method, modified simple equation method, variable coefficients method, extended auxiliary equation method [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

In PDEs, the AKNS equations are very important and have many applications in the field of physics and other nonlinear sciences. These equations are reduced the some nonlinear evolution equations such as sine-Gordon equations, the nonlinear Schrödinger equation, KdV equation etc. Different methods have been used to acquire explicit solutions of the AKNS equations, inverse scattering transformation, the Bäcklund transformation, the Darboux transformation [19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

In the current work, we consider the well known fourth-Order nonlinear AKNS water wave equation [19] with a perturbation parameter β in the form of

4vxt+vxxxt+8vxvxy+4vxxvy−βvxx=0.(1)

we have employed proposed simple equation method and modified simple equation method on equation (1) to obtain new exact solitary wave solutions with different parameters. To the best of our knowledge, no work has been done in previous study by employing the current proposed methods. The obtained solutions are useful in physical sciences and help to understand the physical phenomena.

The article is structured as follows. The main steps of the proposed methods are given in Section 2. In Section 3 we apply the proposed methods to Eq. (1) for constructing solitary wave solution. Finally, the summary of this work is given in Section 4.

2 Discription of proposed methods

2.1 Simple equation method

In this section, simple equation method (SEM) will be applied to obtain the solitary wave solutions for Ablowitz-Kaup-Newell-Segur water wave equation. Consider the nonlinear PDE in form

Fu,ux,uy,ut,uxx,uyy,utt,...=0,(2)

where F is called a polynomial function of u(x, y, t) and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. The basic key steps of SEM are as follows:

Consider traveling wave transformation
u(x,y,t)=V(ξ),ξ=x+y+ωt,(3)
by utilizing the above transformation, the Eq. (2) is reduced into ODE as:
GV,V′,V″,V‴,...=0,(4)
where G is a polynomial in V(ξ) its derivatives with respect to ξ.
Let us assume that the solution of Eq. (4) has the form:
V(ξ)=∑i=−MMAiψi(ξ),(5)
where A_i (i = -M, -M+1,…,-1,0,1,…, M) is arbitrary constants which can be determined latter and M is a positive integer, which can be calculated by homogeneous balance principle on Eq. (4).
Let ψ satisfies the following equation:
ψ′(ξ)=b0+b1ψ+b2ψ2+b3ψ3,(6)
where b₀, b₁, b₂, b₃, are arbitrary constants.
Substituting Eq. (5) along with Eq. (6) into Eq. (4), and collecting the coefficients of (ψ)^j, then setting coefficients equal to zero, we obtained a system of algebraic equations in parameters b₀, b₁, b₂, b₃, ω and A_i. The system of algebraic equations is solved with the help of Mathematica and we get the values of these parameters.
By substituting all these values of parameters and ψ into Eq. (5), we obtained the required solutions of Eq. (2).

2.2 Modified simple equation method

In this section, we describe the algorithm of modified simple equation method (MSEM) to obtain the solitary wave solutions of nonlinear evolution equations. Consider the nonlinear evolution equation in the form

Gu,ux,uy,ut,uxx,uyy,utt,...=0,(7)

where G is a polynomial function of u(x, t) and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. The basics keysteps are:

Consider travelling wave transformation
u(x,y,t)=U(ξ), ξ=x+y+ωt,(8)
By utilizing the above transformation into Eq. (7), the Eq. (7) is reduced into ODE as
HU,U′,U″,U‴,...=0,(9)
where H is a polynomial in U(ξ) its derivatives with respect to ξ.
Let us assume that the solution of Eq. (9) has the form:
U(ξ)=∑M=0NBMΨ′(ξ)Ψ(ξ)M,(10)
where B_M are arbitrary constants to be determined, such that B_N ≠ 0 and Ψ(ξ) is to be determined.
The postive integer N can be determined by applying the homogeneous balance technique between the highest order derivatives and nonlinear terms as in Eq. (7).
We calculate all the required derivatives of U′, U″, U′″… and substitute into Eq. (10) and (9). We obtain a polynomial of Ψ^–j(ξ) with the derivatives of Ψ(ξ). We equate all the coefficients of Ψ^–j(ξ) to zero, where j ≥ 0. This procedure yields a system of equations which can be solved to find B_M, Ψ(ξ) and Ψ^′ (ξ).
We substitute the values of B_M, Ψ(ξ) and Ψ^′ (ξ) into Eq. (10) and (8) to complete the determination of exact solution of Eq. (1).

3 Applications of descriptions method on AKNS

3.1 Applications of SEM

In this section we apply the method which is described in Section 2.1 on Eq. (1). Consider the traveling waves transformation

v(x,y,t)=V(ξ), ξ=x+y+kt, (11)

where k is arbitrary constant, which can be determined later. By using the above transformation to the Eq. (1) into the following ordinary differential equation and integrated

(4k−β)V′+6V′2+kV‴=0.(12)

Now applying the homogeneous balance principle between (V′)² and V^′″ in Eq. (12), we get M = 2. We suppose the solution of Eq. (12) has the form:

V(ξ)=A−2ψ−2+A−1ψ−1+A0+A1ψ+A2ψ2.(13)

Substituting Eq. (13) along Eq. (6) into Eq. (12), we get system of algebraic equation in parameters b₀, b₁, b₂, b₃, β, k, A₀, A_–1, A_–2, A₁, A₂. The system of algebraic equations can be solved for these parameters, we have following solutions cases.

b₃ = 0,
Family-I
k=βb12−4b0b2+4,A−1=βb0b12−4b0b2+4,A2=0,A1=0,A−2=0.(14)
Substituting Eq. (14) into Eq. (6), then the solution of Eq. (1) becomes:
v1(x,y,t)⋅−2b2b0β(b12−4b0b2+4)(b1−4b0b2−b12tan⁡(4b0b2−b122(ξ+ξ0)))+A0,4b0b2>b12,(15)
where ξ=x+y+βb12−4b0b2+4t.
Family-II
k=βb12−4b0b2+4,A1=βb2−b12+4b0b2−4,A−1=0,A−2=0,A2=0.(16)
Substituting Eq. (16) into Eq. (6), then the solution of Eq. (1) becomes:
v2(x,y,t)=A0+β(b1−4b0b2−b12tan⁡(4b0b2−b122(ξ+ξ0)))(2b12−8b0b2+8),4b0b2>b12,(17)
where ξ=x+y+βb12−4b0b2+4t.
Figure 1
Exact solitary wave solutions of Eq. (15) and Eq. (17) are plotted by choosing the values of parameters A₀ = 1.5, b₁ = 0.5, β = 1, ξ₀ = 1: (a) periodic solitary wave of v₁ at b₂ = 1, b₀ = 1 and (b) periodic solitary of v₂ at b₂ = –2, b₀ = –1
b₀ = b₃ = 0,
k=βb12+4,A1=−βb2b12+4,A−1=0,A−2=0,A2=0.(18)
Substituting Eq. (18) into Eq. (6) the solution of Eq. (1) becomes:
v31(x,y,t)=A0+−βb2b1eb1(ξ+ξ0)(b12+4)(1−b2eb1(ξ+ξ0)),b1>0;(19)
v32(x,y,t)=A0+βb2b1eb1(ξ+ξ0)(b12+4)(1+b2eb1(ξ+ξ0)),b1<0;(20)
where ξ=x+y+βb12+4t.
Figure 2
Exact solitary wave solutions of Eq. (19) and Eq. (20) are plotted by choosing the values of parameters as: A₀ = 0.5, b₂ = 0.5, β = –0.5, ξ₀ = –0.5: (a) solitary wave of v₃₁ at b₁ = 0.5 and (b) solitary wave of v₃₂ at b₁ = –0.5
b₁ = b₃ = 0,
Family-I
k=−β4b0b2−1,A1=0,A−1=−βb04b0b2−1,A−2=0,A2=0.(21)
Substituting Eq. (21) into Eq. (6) the solution of Eq. (1) becomes:
v41(x,y,t)=b2b0β(−4b0b2+4)b0b2tan⁡(b0b2(ξ+ξ0)+A0,b0b2>0;(22)
v42(x,y,t)=b2b0β(4b0b2−4)−b0b2tanh⁡(−b0b2(ξ+ξ0)+A0,b0b2<0;(23)
where ξ=x+y−β4b0b2−1t.
Figure 3
Exact solitary wave solutions of Eq. (22) and Eq. (23) are plotted by choosing these values of parameters at: A₀ = –1.5, β = 1, ξ₀ = 1: (a) periodic solitary wave of v₄₁ at b₀ = –0.5, b₂ = –1 and (b) solitary wave of v₄₂ at b₀ = –0.5, b₂ = 1
Family-II
k=−β4b0b2−1,A1=βb24b0b2−1,A−1=0,A−2=0,A2=0.(24)
Substituting Eq. (24) into Eq. (6) the solution of Eq. (1) becomes:
v51(x,y,t)=A0+βb0b2tan⁡(b0b2(ξ+ξ0))4b0b2−1,b0b2>0;(25)
V52(x,y,t)=A0+β−b0b2tanh⁡(−b0b2(ξ+ξ0))−4b0b2+4,b0b2<0;(26)
where ξ=x+y−β4b0b2−1t.
Figure 4
Exact solutions of Eq. (25) and Eq. (26) are plotted at A₀ = 1.5, β = –0.75, ξ₀ = 1: (a) periodic solitary wave of v₅₁ at b₀ = 0.75, b₂ = 0.5 and (b) solitary wave of v₅₂ at b₀ = 0.75, b₂ = –0.5
Family-III
k=−β44b0b2−1,A1=βb244b0b2−1,A−1=−βb044b0b2−1,A−2=0,A2=0.(27)
Substituting Eq. (27) into Eq. (6) the solution of Eq. (1) becomes:
v61(x,y,t)=b2b0β(−16b0b2+4)b0b2tan⁡(b0b2(ξ+ξ0)+A0+βb0b2tan⁡(b0b2(ξ+ξ0))44b0b2−1,b0b2>0;(28)
v62(x,y,t)=b2b0β(16b0b2−4)−b0b2tanh⁡(−b0b2(ξ+ξ0)+A0+β−b0b2tanh⁡(−b0b2(ξ+ξ0))−16b0b2+4,b0b2<0;(29)
where ξ=x+y−β4b0b2−1t.
Figure 5
Exact solutions of Eq. (28) and Eq. (29) are plotted at A₀ = 1, β = 2, ξ₀ = –0.5: (a) periodic solitary wave of v₆₁ at b₀ = 1, b₂ = 0.5 and (b) solitary wave of v₆₂ at b₀ = –1, b₂ = 0.5
b₀ = b₂ = 0,
k=β4+4b12,A2=−βb32+2b12,A−1=0,A−2=0,A1=0.(30)
where ξ=x+yβ4+4b12t. Substituting Eq. (30) into Eq. (6) the solution of Eq. (1) becomes:
v71(x,y,t)=−b1b3β(2+2b12)(e−2(ξ+ξ0)b1+b3)+A0,b1<0,(31)
v72(x,y,t)=−βb3b1e2(ξ+ξ0)b1(2+2b12)1−e2(ξ+ξ0)b1b3+A0,b1>0.(32)
where ξ=x+y+β4+4b12t.
Figure 6
Exact solutions of Eq. (31) and Eq. (32) are plotted at A₀ = 0.5, β = 3, ξ₀ = –0.5, b₁ = –0.5, b₃ = –0.5 at (a) and A₀ = –0.5, β = 3, ξ₀ = –0.5, b₁ = 1, b₃ = –1 at (b), respectively are solitary waves of v₇₁ and v₇₂

3.2 Application of MSEM

In this section we apply method which is described in Section 2.2, on Eq. (1). Consider the traveling waves transformation

v(x,y,t)=U(ξ),ξ=x+y+ωt,(33)

where ω is arbitrary constant which can be determined later. By using the above transformation to the Eq. (1) into the following ordinary differential equation and integrated we have

(4ω−β)U′+6U′2+ωU‴=0.(34)

Now applying the homogeneous balance principle between (U′)² and U′″ in Eq. (34), we get N = 1. We suppose the solution of Eq. (34) has the form as:

U=B0+B1Ψ′(ξ)Ψ(ξ).(35)

Where B₀, B₁ are constant such that B₁ ≠ 0, substituting Eq. (35) into Eq. (34) and equating the coefficients of Ψ^–4, Ψ^–3, Ψ^–2, Ψ^–1 we get the following equations:

(−ω+B1)B1(Ψ′)4=0,(36)

(ω−B1)B1(Ψ′)2Ψ″=0,(37)

(βB1−4ωB1)(Ψ′)2+(6B12−3ωB1)(Ψ″)2−4ωB1Ψ′Ψ‴=0,(38)

(−βB1+4ωB1)Ψ″+ωB1Ψ(4)=0.(39)

From Eq. (36) and Eq. (37) we obtain, B₁ = ω, Now integrating Eq. (39) and substituting into Eq. (38), we get:

Ψ″Ψ′=μ, where μ=±3β−16ω3ω,(40)

consequently, we obtain

Ψ′(ξ)=c1eμξ,(41)

Ψ=c2+c1μc1eμξ,(42)

where c₁ and c₂ are constant of integration. Now substituting the values of B₁, Ψ and Ψ′ into Eq. (35), we get the exact solution of Eq. (1), as follows

v(x,y,t)=B0+ωμc1eμξμc2+c1eμξ,(43)

where ξ = x + y + ωt.

Figure 7

Exact solitary wave solution of Eq. (43) is plotted in different shapes at :B₀ = 1, β = 2, c₁ = 0.5, c₂ = –0.5, ω = 0.25:(a) solitary wave and (b) one dimensional solitary wave

Simplifying Eq. (43), we obtain

v(x,y,t)=B0+ωμc1coshμξ2+sinhμξ2(c1+μc2)coshμξ2+(c1−c2μ)sinhμξ2,(44)

where ξ = x + y + ωt.

Similarly we can randomly choose parameter c₁ and c₂, by setting c₁ = μc₂, we obtain the following solitary solution

v(x,y,t)=B0+3βω−16ω212tanh3β−16ω12ωξ±1,(45)

where ξ = x + y + ωt.

Now again we choose c₁ = –c₂μ, we obtain following solitary waves solution.

v(x,y,t)=B0+3βω−16ω212coth3β−16ω12ωξ±1,(46)

where ξ = x + y + ωt and B₀ is left as a free parameter.

Figure 8

Exact solitary wave solution of Eq. (46) is plotted in different shapes at :B₀ = 1, β = 1, ω = 0.5:(a) periodic solitary wave and (b) one-dimensional solitary wave

4 Conclusion

In the current paper, proposed methods such as simple equation method (SEM) and modified simple equation method (MSEM) have been successfully employed to obtain the exact solitary wave solutions for the fourth-order nonlinear ablowitz-Kaup-newell-segur water equation. The AKNS equations have a wide application in field of physical sciences and the obtained solitary wave solutions in different form help to understand the physical phenomenon in various aspects. All the solutions are also prescribed graphically by assigning special values to the parameters. Mathematica facilitate us to handle all the calculations. As the extensive applications of the solitary wave theory, it is valuable to study further the localized excitation and its applications in the future.

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Received: 2017-09-21

Accepted: 2017-12-27

Published Online: 2018-05-24

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Articles in the same Issue

https://doi.org/10.1515/phys-2018-0032

Keywords for this article

Simple equation method; modified simple equation method; fourth-order nonlinear AKNS equation; exact solutions; solitary wave

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