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  with a perturbation parameter β in the form of
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
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:
- Step 1.
Consider traveling wave transformation
by utilizing the above transformation, the Eq. (2) is reduced into ODE as:
where G is a polynomial in V(ξ) its derivatives with respect to ξ.
- Step 2.
Let us assume that the solution of Eq. (4) has the form:
where Ai (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:
where b0, b1, b2, b3, are arbitrary constants.
- Step 3.
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 Ai. The system of algebraic equations is solved with the help of Mathematica and we get the values of these parameters.
- Step 4.
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
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:
- Step 1.
Consider travelling wave transformation
where H is a polynomial in U(ξ) its derivatives with respect to ξ.
- Step 2.
Let us assume that the solution of Eq. (9) has the form:
where BM are arbitrary constants to be determined, such that BN ≠ 0 and Ψ(ξ) is to be determined.
- Step 3.
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).
- Step 4.
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 BM, Ψ(ξ) and Ψ′ (ξ).
- Step 5.
3 Applications of descriptions method on AKNS
3.1 Applications of SEM
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
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.
- Case 1.
b3 = 0,
- Case 2.
b0 = b3 = 0,
- Case 3.
b1 = b3 = 0,
- Case 4.
b0 = b2 = 0,
3.2 Application of MSEM
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
consequently, we obtain
where ξ = x + y + ωt.
Simplifying Eq. (43), we obtain
where ξ = x + y + ωt.
Similarly we can randomly choose parameter c1 and c2, by setting c1 = μc2, we obtain the following solitary solution
where ξ = x + y + ωt.
Now again we choose c1 = –c2μ, we obtain following solitary waves solution.
where ξ = x + y + ωt and B0 is left as a free parameter.
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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Published Online: 2018-05-24