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

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:

1. 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.

2. 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.

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 A_i. The system of algebraic equations is solved with the help of Mathematica and we get the values of these parameters.

4. 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

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:

1. 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.

2. 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.

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).

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 B_M, Ψ(ξ) and Ψ^′ (ξ).

5. 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).

3.1 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

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

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:

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.

1. 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

2. 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

3. 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

4. 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

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

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:

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:

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

consequently, we obtain

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

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

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.

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