The Application of the Homotopy Perturbation Method and the Homotopy Analysis Method to the Generalized Zakharov Equations

Abstract

We introduce two powerful methods to solve the generalized Zakharov equations; one is the homotopy perturbation method and the other is the homotopy analysis method. The homotopy perturbation method is proposed for solving the generalized Zakharov equations. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions; the homotopy analysis method is applied to solve the generalized Zakharov equations. HAM is a strong and easy-to-use analytic tool for nonlinear problems. Computation of the absolute errors between the exact solutions of the GZE equations and the approximate solutions, comparison of the HPM results with those of Adomian’s decomposition method and the HAM results, and computation the absolute errors between the exact solutions of the GZE equations with the HPM solutions and HAM solutions are presented.

1. Introduction

Nonlinearpartial differential equations are useful in describing the various phenomena in disciplines. Apart from a limited number of these problems, most of them do not have a precise analytical solution, so these nonlinear equations should be solved using approximate methods.

The application of the homotopy perturbation method (HPM) [1, 2] in nonlinear problems has been devoted by scientists and engineers, because this method is continuously.

Deform a simple problem which is easy to solve into the under study problem which is difficult to solve. The homotopy perturbation method was first proposed by He [3–6]. For solving differential and integral equations, linear and nonlinear has been the subject of extensive analytical and numerical studies. The method is a coupling of the traditional perturbation method and homotopy in topology. This method, which does not require a small parameter in an equation, has a significant advantage in that it provides an analytical approximate solution to a wide range of nonlinear problems in applied sciences. This HPM has already been applied successfully to solve the Laplace equation, nonlinear dispersive K(mp) equations, heat radiation equations, nonlinear integral equations, nonlinear heat conduction and convection equations, nonlinear oscillators, nonlinear Schrodinger equations, nonlinear wave equations, nonlinear chemistry problems, and other fields [7]. This HPM yields a very rapid convergence of the solution series in most cases, usually only a few iterations leading to very accurate solutions. Thus He’s HPM is a universal one which can solve various kinds of nonlinear equations. The HPM yields a very rapid convergence of the solution series in the most cases. The method does not depend on a small parameter in the equation. Using homotopy technique in topology, a homotopy is constructed with an embedding parameter 𝑝∈[0,1] which is considered as a “small parameter.” No need to linearization or discretization; large computational work and round-off errors are avoided. It has been used to solve effectively, easily, and accurately a large class of nonlinear problems with approximations. These approximations converge rapidly to accurate solutions [7–10].

The HPM was successfully applied to nonlinear oscillators with discontinuities [4] and bifurcation of nonlinear problem [11]. In [6], a comparison of HPM and homotopy analysis method was made.

In [12] the homotopy perturbation method is applied to compute the Laplace transform and construct solitary wave solutions for a generalized Hirota-Satsuma-coupled KdV equation [13]. In [14] the HPM is employed to compute an approximation to the solution of the epidemic model. As well, in [15] is applied the homotopy perturbation method for solving the Lane-Emden-type singular IVPs problem, in [16] using the homotopy perturbation method to find exact solutions of nonlinear differential-difference equations.

In 1992, Liao employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely, homotopy analysis method (HAM) [17–20]. This method has been successfully applied to solve many types of nonlinear problems by others [21–25].

In this paper, we consider the generalized Zakharov equations (GZE) which are a set of coupled equations and can be written as [26–29]𝑖𝜕𝐸+𝜕𝜕𝑡²𝐸𝜕𝑥²−2𝛽|𝐸|²𝜕𝐸+2𝑛𝐸=0,₂𝑛𝜕𝑡²−𝜕²𝑛𝜕𝑥²+𝜕²|𝐸|²𝜕𝑥²=0, (1.1) where 𝛽 is an arbitrary constant and 𝐸 is the envelope of the high-frequency electric field, and 𝑛 is the plasma density measured from its equilibrium value. When 𝛽=0, this system is reduced to the classical Zakharov equations of plasma physics. Because the GZE is much closer to the realistic model in plasma, it is meaningful for us to study the solitary wave solutions of the GZE. The motivation of this paper is to apply the Homotopy perturbation method and the homotopy analysis method to the problem mentioned above. When implementing the homotopy perturbation method (HPM) and the homotopy analysis method (HAM), we get the explicit solutions of the GZE equations without using any transformation method. Furthermore, we will show that considerably better approximations related to the accuracy level would be obtained. Comparing the HPM results for the study problem with the Adomian decomposition method (ADM) results takes six terms in evaluating the approximate solutions and HAM results which take eight terms in evaluating the approximate solutions of the generalized Zakharov equations.

2. Basic Idea of He’s Homotopy Perturbation Method

The homotopy perturbation method is a combination of the classical perturbation technique and homotopy technique, which has eliminated the limitations of the traditional perturbation methods. This technique can have full advantage of the traditional perturbation techniques. To illustrate the basic idea of the homotopy perturbation method for solving nonlinear differential equations, we consider the following nonlinear differential equation:𝐴(𝑢)−𝑓(𝑟)=0,𝑟∈Ω, (2.1) subject to boundary condition𝐵𝑢,𝜕𝑢𝜕𝑛=0,𝑟∈Γ, (2.2) where 𝐴 is a general differential operator, 𝐵 is a boundary operator, 𝑓(𝑟) is a known analytic function, and Γ is the boundary of the domain Ω.

The operator 𝐴 can, generally speaking, be divided into two parts: a linear part 𝐿 and a nonlinear part 𝑁. Equation (2.1) therefore can be rewritten as follows:𝐿(𝑢)+𝑁(𝑢)−𝑓(𝑟)=0. (2.3) By the homotopy technique, we construct a homotopy 𝑉(𝑟,𝑝)∶Ω×[0,1]→ℜ, which satisfies𝐻𝑉,𝑝=1−𝑝𝐿(𝑉)−𝐿𝑢₀+𝑝𝐴(𝑉)−𝑓(𝑟)=0,𝑝∈[0,1],𝑟∈Ω, (2.4) or𝐻𝑉,𝑝=𝐿(𝑉)−𝐿𝑢₀+𝑝𝐿𝑢₀+𝑝𝑁(𝑉)−𝑓(𝑟)=0, (2.5) where 𝑝∈[0,1] is an embedding parameter and 𝑢₀ is an initial approximation of (2.1) which satisfies the boundary conditions. It follows from (2.4) and (2.5) that we will have,𝐻(𝑉,0)=𝐿(𝑉)−𝐿𝑢₀,𝐻(𝑉,1)=𝐴(𝑉)−𝑓(𝑟). (2.6) Thus, the changing process of 𝑝 from zero to unity is just that of 𝑣(𝑟,𝑝) from 𝑢₀(𝑟) to 𝑢(𝑟). In topology, this is called deformation and 𝐿(𝑣)−𝐿(𝑢₀),𝐴(𝑣)−𝑓(𝑟) are called homotopic.

According to the HPM, we can first use the embedding parameter 𝑝 as a “small parameter,” and assume that the solution of (2.4) and (2.5) can be written as a power series in 𝑝:𝑉=𝑉₀+𝑝𝑉₁+𝑝²𝑉₂+⋯. (2.7) Setting 𝑝=1 results in the approximate solution of (2.1):𝑢=lim_𝑝→1𝑉=𝑉₀+𝑉₁+𝑉₂+⋯. (2.8) The series (2.8) is convergent for most cases; however, the convergent rate depends upon the nonlinear operator 𝐴(𝑉) (the following opinions are suggested by [2, 30]).(1) The second derivative of 𝑁(𝑉) with respect to 𝑉 must be small because the parameter may be relatively large; that is, 𝑝→1.(2)The norm of 𝐿⁻¹𝜕𝑁/𝜕𝑉 must be smaller than one so that the series converges.

3. Basic Idea of Homotopy Analysis Method

In this paper, we apply the homotopy analysis method [17–20] to find the approximate solutions for the problem. Let us consider the following differential equation:𝑁[𝑧(𝑥,𝑡)]=0, (3.1) where 𝑁 is a nonlinear operator, 𝑥 and 𝑡 denote independent variables, and 𝑧(𝑥,𝑡) is an unknown function, respectively. For simplicity, we ignore all boundary or initial conditions, which can be treated in the similar way. By means of generalizing the traditional homotopy method, Liao [18] constructs the so-called zero-order deformation equation:1−𝑝𝐿𝜙𝑥,𝑡;𝑝−𝑧₀(𝑥,𝑡)=𝑝ℏ𝑁𝜙𝑥,𝑡;𝑝, (3.2) where 𝑝∈[0,1] is the embedding parameter, ℏ≠0 is a nonzero auxiliary parameter, 𝐿 is an auxiliary linear operator, 𝑧₀(𝑥,𝑡) is an initial guess of 𝑧(𝑥,𝑡), and 𝜙(𝑥,𝑡;𝑝) is an unknown function, respectively. It is important that one has great freedom to choose auxiliary things in homotopy analysis method. When 𝑝=0 and 𝑝=1, it holds𝜙(𝑥,𝑡;0)=𝑧₀(𝑥,𝑡),𝜙(𝑥,𝑡;1)=𝑧(𝑥,𝑡), (3.3) respectively. Thus as 𝑝 increase from 0 to 1, the solution 𝜙(𝑥,𝑡;𝑝) varies from the initial guess 𝑧₀(𝑥,𝑡) to the solution 𝑧(𝑥,𝑡). Expanding 𝜙(𝑥,𝑡;𝑝) in the Taylor series with respect to 𝑝, one has𝜙𝑥,𝑡;𝑝=𝑧₀(𝑥,𝑡)+^∞_𝑚=1𝑧_𝑚(𝑥,𝑡)𝑝^𝑚, (3.4) where𝑧_𝑚()=1𝑥,𝑡𝜕𝑚!^𝑚𝜙𝑥,𝑡;𝑝𝜕𝑝^𝑚_𝑝=0. (3.5) If the auxiliary linear operator, the initial guess, and the auxiliary parameter ℏ are so properly chosen and the series (3.4) converges at 𝑝=1, one has𝑧(𝑥,𝑡)=𝑧₀(𝑥,𝑡)+^∞_𝑚=1𝑧_𝑚(𝑥,𝑡), (3.6) which must be one of solutions of original nonlinear equation, as proved by Liao [18]. As ℏ=−1, (3.2) becomes1−𝑝𝐿𝜙𝑥,𝑡;𝑝−𝑧₀(𝑥,𝑡)+𝑝𝑁𝜙𝑥,𝑡;𝑝=0, (3.7) which is used mostly in the homotopy perturbation method, whereas the solution obtained directly, without using the Taylor series [31]. The comparison between HAM and HPM can be found in [6].

According to (3.5), the governing equation can be deduced from the zero-order deformation equation (3.2). Define the vector𝑧^→_𝑛=𝑧₀(𝑥,𝑡),𝑧₁(𝑥,𝑡),…,𝑧_𝑛(𝑥,𝑡). (3.8) Differentiating (3.2) 𝑚 times with respect to the embedding parameter 𝑝 and setting 𝑝=0 and finally dividing them by 𝑚!, we have the so-called 𝑚th-order deformation equation:𝐿𝑧_𝑚(𝑥,𝑡)−𝜒_𝑚𝑧_𝑚−1(𝑥,𝑡)=ℏ𝑅_𝑚𝑧^→_𝑚−1, (3.9) where𝑅_𝑚𝑧^→_𝑚−1=1()!𝜕𝑚−1^𝑚−1𝑁𝜙𝑥,𝑡;𝑝𝜕𝑝^𝑚−1_𝑝=0,𝜒_𝑚=0,𝑚≤1,1,𝑚>1. (3.10) It should be emphasized that 𝑧_𝑚(𝑥,𝑡) for 𝑚≽1 is governed by the linear equation (3.9) with the linear boundary conditions that come from original problem, which can be easily solved by symbolic computation software such as Maple.

First, we separate the complex high-frequency electric field 𝐸into real part and imaginary part; that is, 𝐸=𝐸₁+𝑖𝐸₂. Then we rewrite the system (1.1) in the following form:𝜕𝐸₁𝜕𝜕𝑡=−²𝐸₂𝜕𝑥²+2𝛽𝐸²₁+𝐸²₂𝐸₂−2𝑛𝐸₂,𝜕𝐸₂=𝜕𝜕𝑡²𝐸₁𝜕𝑥²−2𝛽𝐸²₁+𝐸²₂𝐸₁+2𝑛𝐸₁,𝜕²𝑛𝜕𝑡²=𝜕²𝑛𝜕𝑥²−𝜕²𝜕𝑥²𝐸²₁+𝐸²₂, (3.11) with the following initial conditions [27]:𝐸₁(𝑥,0)=𝑟tanh(𝑘𝑥)cos𝑘₁𝑥,𝐸₂(𝑥,0)=𝑟tanh(𝑘𝑥)sin𝑘₁𝑥,𝑛()𝑟𝑥,0=𝑠+²−4𝑘²₁+1tanh²(𝑘𝑥), (3.12) where 𝑟=𝑘²(4𝑘²₁−1)/(1+(4𝑘²₁−1)𝛽),𝑘,𝑠,𝛽, and 𝑘₁ are arbitrary constants.

4. Application the Homotopy Perturbation Method for the Generalized Zakharov Equations

To investigate the traveling wave solution of (3.11), we first construct a homotopy as follows:1−𝑝𝑣₁̇−𝐸̇_1,0+𝑝𝑣₁̇+𝑣₂^′′−2𝛽𝑣²₁+𝑣²₂𝑣₂+2𝑣₂𝑣₃=0,1−𝑝𝑣₂̇−𝐸̇_2,0+𝑝𝑣₂̇−𝑣₁^′′+2𝛽𝑣²₁+𝑣²₂𝑣₁−2𝑣₁𝑣₃=0,1−𝑝𝑣₃̈−𝑛₀̈+𝑝𝑣₃̈−𝑣₃^′′+𝑣²₁+𝑣²₂^′′=0, (4.1) where “′” denotes 𝜕/𝜕𝑥, and “⋅” denotes 𝜕/𝜕𝑡, and the initial approximations are as follows,𝑣_1,0(𝑥,𝑡)=𝐸_1,0(𝑥,𝑡)=𝐸₁(𝑥,0)=𝑟tanh(𝑘𝑥)cos𝑘₁𝑥,𝑣_2,0(𝑥,𝑡)=𝐸_2,0(𝑥,𝑡)=𝐸₂(𝑥,0)=𝑟tanh(𝑘𝑥)sin𝑘₁𝑥,𝑣_3,0(𝑥,𝑡)=𝑛₀()()𝑟𝑥,𝑡=𝑛𝑥,0=𝑠+²−4𝑘²₁+1tanh²(𝑘𝑥), (4.2) where 𝑟=𝑘²(4𝑘²₁−1)/(1+(4𝑘²₁−1)𝛽),𝑘,𝑠,𝛽, and 𝑘₁ are arbitrary constants, and𝑣₁=𝑣_1,0+𝑝𝑣_1,1+𝑝²𝑣_1,2+𝑝³𝑣_1,3𝑣+⋯,₂=𝑣_2,0+𝑝𝑣_2,1+𝑝²𝑣_2,2+𝑝³𝑣_2,3𝑣+⋯,₃=𝑣_3,0+𝑝𝑣_3,1+𝑝²𝑣_3,2+𝑝³𝑣_3,3+⋯, (4.3) where 𝑣_𝑖,𝑗(𝑥,𝑡),𝑖,𝑗=1,2,3,…, are functions yet to be determined. Substituting (4.3) into (3.11) and arranging the coefficients of “𝑝” powers, we have𝑣̇_1,1+𝑣̇_1,0+𝑣^′′_2,0−2𝛽𝑣²_1,0𝑣_2,0−2𝛽𝑣³_2,0+2𝑣_2,0𝑣_3,0𝑝+𝑣^′′_2,1−4𝛽𝑣_1,0𝑣_1,1𝑣_2,0−2𝛽𝑣²_1,0𝑣_2,1−6𝛽𝑣²_2,0𝑣_2,1+2𝑣_2,1𝑣_3,0+2𝑣_2,0𝑣_3,0+𝑣̇_1,2𝑝²+−2𝛽𝑣²_1,1𝑣_2,0−4𝛽𝑣_1,0𝑣_1,2𝑣_2,0−4𝛽𝑣_1,0𝑣_1,1𝑣_2,1−6𝛽𝑣_2,0𝑣²_2,1−2𝛽𝑣²_1,0𝑣_2,2−6𝛽𝑣²_2,0𝑣_2,2+2𝑣_2,2𝑣_3,0+2𝑣_2,1𝑣_3,1+2𝑣_2,0𝑣_3,0+𝑣̇_1,3+𝑣^′′_2,1𝑝³=0,𝑣̇_2,0+𝑣̇_2,1−𝑣^′′_1.0+2𝛽𝑣³_1,0+2𝛽𝑣_1,0𝑣²_2,0−2𝑣_1,0𝑣_3,0𝑝+6𝛽𝑣²_1,0𝑣_1,1+2𝛽𝑣_1,1𝑣²_2,0+4𝛽𝑣_1,0𝑣_2,0𝑣_2,1−2𝑣_1,1𝑣_3,0−2𝑣_1,0𝑣_3,1+𝑣̇_2,2−𝑣^′′_1,1𝑝²+6𝛽𝑣_1,0𝑣²_1,1+6𝛽𝑣²_1,0𝑣_1,2+2𝛽𝑣_1,2𝑣²_2,0+4𝛽𝑣_1,1𝑣_2,0𝑣_2,1+2𝛽𝑣_1,0𝑣²_2,1+4𝛽𝑣_1,0𝑣_2,0𝑣_2,2−2𝑣_1,2𝑣_3,0−2𝑣_1,1𝑣_3,1−2𝑣_1,0v_3,2+𝑣̇_2,3−𝑣^′′_1,2𝑝³=0,𝑣̈_3,0+𝑣̈_3,1+2𝑣^′2_1,2+2𝑣^′2_2,0+2𝑣_1,0𝑣^′′_1,0+2𝑣_2,0𝑣^′′_2,0−𝑣^′′_3,0𝑝+𝑣̈_3,2+4𝑣^′_1,0𝑣^′_1,1+4𝑣^′_2,0𝑣^′_2,1+2𝑣_1,1𝑣^′′_1,0+2𝑣_1,0𝑣^′′_1,1+2𝑣_2,1𝑣^′′_2,0+2𝑣_2,0𝑣^′′_2,1−𝑣^′′_3,1𝑝²+𝑣̈_3,3+2𝑣^′2_1,1+4𝑣^′_1,0𝑣^′_1,2+2𝑣^′2_2,1+4𝑣^′_2,0𝑣^′_2,2+2𝑣_1,2𝑣^′′_1,0+2𝑣_1,1𝑣^′′_1,1+2𝑣_1,0𝑣^′′_1,2+2𝑣_2,2𝑣^′′_2,0+2𝑣_2,1𝑣^′′_2,1+2𝑣_2,0𝑣^′′_2,2−𝑣^′′_3,2𝑝³=0. (4.4) To obtain the unknown 𝑣_𝑖,𝑗(𝑥,𝑡),𝑖,𝑗=1,2,3, we must construct and solve the following system which includes nine equations with nine unknowns, considering the initial conditions of 𝑣_𝑖,𝑗(𝑥,0)=0,𝑖,𝑗=1,2,3, 𝑣̇_1,1+𝑣̇_1,0+𝑣^′′_2,0−2𝛽𝑣²_1,0𝑣_2,0−2𝛽𝑣³_2,0+2𝑣_2,0𝑣_3,0=0, (4.5)𝑣^′′_2,1−4𝛽𝑣_1,0𝑣_1,1𝑣_2,0−2𝛽𝑣²_1,0𝑣_2,1−6𝛽𝑣²_2,0𝑣_2,1+2𝑣_2,1𝑣_3,0+2𝑣_2,0𝑣_3,0+𝑣̇_1,2=0, (4.6)−2𝛽𝑣²_1,1𝑣_2,0−4𝛽𝑣_1,0𝑣_1,2𝑣_2,0−4𝛽𝑣_1,0𝑣_1,1𝑣_2,1−6𝛽𝑣_2,0𝑣²_2,1−2𝛽𝑣²_1,0𝑣_2,2−6𝛽𝑣²_2,0𝑣_2,2+2𝑣_2,2𝑣_3,0+2𝑣_2,1𝑣_3,1+2𝑣_2,0𝑣_3,0+𝑣̇_1,3+𝑣^′′_2,1=0, (4.7)𝑣̇_2,0+𝑣̇_2,1−𝑣^′′_1.0+2𝛽𝑣³_1,0+2𝛽𝑣_1,0𝑣²_2,0−2𝑣_1,0𝑣_3,0=0, (4.8)6𝛽𝑣²_1,0𝑣_1,1+2𝛽𝑣_1,1𝑣²_2,0+4𝛽𝑣_1,0𝑣_2,0𝑣_2,1−2𝑣_1,1𝑣_3,0−2𝑣_1,0𝑣_3,1+𝑣̇_2,2−𝑣^′′_1,1=0, (4.9)6𝛽𝑣_1,0𝑣²_1,1+6𝛽𝑣²_1,0𝑣_1,2+2𝛽𝑣_1,2𝑣²_2,0+4𝛽𝑣_1,1𝑣_2,0𝑣_2,1+2𝛽𝑣_1,0𝑣²_2,1+4𝛽𝑣_1,0𝑣_2,0𝑣_2,2−2𝑣_1,2𝑣_3,0−2𝑣_1,1𝑣_3,1−2𝑣_1,0𝑣_2,3+𝑣̇_3,2−𝑣^′′_1,2=0, (4.10)𝑣̈_3,0+𝑣̈_3,1+2𝑣^′2_1,2+2𝑣^′2_2,0+2𝑣_1,0𝑣^′′_1,0+2𝑣_2,0𝑣^′′_2,0−𝑣^′′_3,0=0, (4.11)𝑣̈_3,2+4𝑣^′_1,0𝑣^′_1,14𝑣^′_2,0𝑣^′_2,1+2𝑣_1,1𝑣^′′_1,0+2𝑣_1,0𝑣^′′_1,1+2𝑣_2,1𝑣^′′_2,0+2𝑣_2,0𝑣^′′_2,1−𝑣^′′_3,1=0, (4.12)𝑣̈_3,3+2𝑣^′2_1,1+4𝑣^′_1,0𝑣^′_1,2+2𝑣^′2_2,1+4𝑣^′_2,0𝑣^′_2,2+2𝑣_1,2𝑣^′′_1,0+2𝑣_1,1𝑣^′′_1,1+2𝑣_1,0𝑣^′′_1,2+2𝑣_2,2𝑣^′′_2,0+2𝑣_2,1𝑣^′′_2,1+2𝑣_2,0𝑣^′′_2,2−𝑣^′′_3,2=0. (4.13) From (4.3), if the three approximations are sufficient, we will obtain𝐸₁(𝑥,𝑡)=lim_𝑝→1𝑣₁(𝑥,𝑡)=³_𝑘=0𝑣_1,𝑘(),𝐸𝑥,𝑡₂(𝑥,𝑡)=lim_𝑝→1𝑣₂(𝑥,𝑡)=³_𝑘=0𝑣_2,𝑘(),𝑛()𝑥,𝑡𝑥,𝑡=lim_𝑝→1𝑣₃(𝑥,𝑡)=³_𝑘=0𝑣_3,𝑘(𝑥,𝑡). (4.14)

To calculate the terms of the homotopy series (4.14) for 𝐸₁(𝑥,𝑡),𝐸₂(𝑥,𝑡), and 𝑛(𝑥,𝑡), we substitute the initial conditions (4.2) into the system (4.4), and using Mathematica software, from (4.5), we obtain𝑣_1,1(𝑥,𝑡)=𝑟𝑡2𝑘secℎ²(𝑘𝑥)−𝑘₁cos𝑘₁𝑥+𝑘sin𝑘₁𝑥tanh(𝑘𝑥)+sin𝑘₁𝑥×tanh(𝑘𝑥)𝑘²₁−2𝑠+2𝑟²1−𝛽+4𝑘²₁𝛽tanh²(𝑘𝑥)4𝑘²₁;−1 (4.15) substituting (4.15) into (4.8), we obtain𝑣_2,1(𝑥,𝑡)=𝑟𝑡−2𝑘secℎ²(𝑘𝑥)𝑘₁sin𝑘₁𝑥+𝑘cos𝑘₁𝑥tanh(𝑘𝑥)+cos𝑘₁𝑥×tanh(𝑘𝑥)−𝑘²₁+2𝑠+2𝑟²−1+𝛽−4𝑘²₁𝛽tanh²(𝑘𝑥)4𝑘²₁;−1 (4.16) substituting (4.15), (4.16) into (4.11),we obtain𝑣_3,1(𝑥,𝑡)=4𝑘²₁𝑘²𝑟²𝑡²4𝑘²₁(−1(−2+Cosh2𝑘𝑥))Secℎ⁴(𝑘𝑥). (4.17) in This Manner the Other Components 𝑣_1,2(𝑥,𝑡),𝑣_2,2(𝑥,𝑡),𝑣_3,2(𝑥,𝑡),𝑣_1,3(𝑥,𝑡),𝑣_2,3(𝑥,𝑡), and 𝑣_3,3(𝑥,𝑡) Can be obtains from (4.6), (4.9), (4.12), (4.7), (4.10), and (4.13), Respectively, and Substituting Into (4.14) to Obtain 𝐸₁(𝑥,𝑡),𝐸₂(𝑥,𝑡), and 𝑛(𝑥,𝑡).

5. Application the Homotopy Analysis Method for the Generalized Zakharov Equations

In order to apply the homotopy analysis method, we choose the linear operator𝐿𝜙_𝑖𝑥,𝑡;𝑝=𝜕𝜙_𝑖𝐿𝜙𝜕𝑡,𝑖=1,2,₃=𝜕𝑥,𝑡;𝑝²𝜙₃𝜕𝑡², (5.1) with the property 𝐿[𝑐]=0 where 𝑐 is constant; from (3.11), we define a system of nonlinear operators as𝑁₁𝜙₁𝑥,𝑡;𝑝,𝜙₂𝑥,𝑡;𝑝,𝜙₃𝑥,𝑡;𝑝=𝜕𝜙₁𝑥,𝑡;𝑝+𝜕𝜕𝑡²𝜙₂𝑥,𝑡;𝑝𝜕𝑥²−2𝛽𝜙²₁𝑥,𝑡;𝑝+𝜙²₂𝑥,𝑡;𝑝𝜙₂𝑥,𝑡;𝑝+2𝜙₂𝑥,𝑡;𝑝𝜙₃,𝑁𝑥,𝑡;𝑝₂𝜙₁𝑥,𝑡;𝑝,𝜙₂𝑥,𝑡;𝑝,𝜙₃𝑥,𝑡;𝑝=𝜕𝜙₂𝑥,𝑡;𝑝−𝜕𝜕𝑡²𝜙₁𝑥,𝑡;𝑝𝜕𝑥²+2𝛽𝜙²₁𝑥,𝑡;𝑝+𝜙²₂𝑥,𝑡;𝑝𝜙₁𝑥,𝑡;𝑝−2𝜙₁𝑥,𝑡;𝑝𝜙₃,𝑁𝑥,𝑡;𝑝₃𝜙₁𝑥,𝑡;𝑝,𝜙₂𝑥,𝑡;𝑝,𝜙₃=𝜕𝑥,𝑡;𝑝²𝜙₃𝑥,𝑡;𝑝𝜕𝑡²−𝜕²𝜙₃𝑥,𝑡;𝑝𝜕𝑥²+𝜕²𝜙²₁𝑥,𝑡;𝑝+𝜙²₂𝑥,𝑡;𝑝𝜕𝑥². (5.2)

By using the above definition, we construct the zero-order deformation equations:1−𝑝𝐿𝜙_𝑖𝑥,𝑡;𝑝−𝑧_𝑖,0(𝑥,𝑡)=𝑝ℏ𝑁_𝑖𝜙₁𝑥,𝑡;𝑝,𝜙₂𝑥,𝑡;𝑝,𝜙₃𝑥,𝑡;𝑝,𝑖=1,2,3. (5.3) When 𝑝=0,𝜙₁(𝑥,𝑡;0)=𝑧_1,0(𝑥,𝑡)=𝐸₁(𝑥,0)=𝑟tanh(𝑘𝑥)cos𝑘₁𝑥,𝜙₂(𝑥,𝑡;0)=𝑧_2,0(𝑥,𝑡)=𝐸₂(𝑥,0)=𝑟tanh(𝑘𝑥)sin𝑘₁𝑥,𝜙₃(𝑥,𝑡;0)=𝑧_3,0()()𝑟𝑥,𝑡=𝑛𝑥,0=𝑠+²−4𝑘²₁+1tanh²(𝑘𝑥). (5.4) When 𝑝=1,𝜙₁(𝑥,𝑡;1)=𝐸₁(𝑥,𝑡),𝜙₂(𝑥,𝑡;1)=𝐸₂(𝑥,𝑡),𝜙₃(𝑥,𝑡;1)=𝑛(𝑥,𝑡). (5.5) Therefore, as the embedding parameter 𝑝 increases from 0 to 1, 𝜙_𝑖(𝑥,𝑡;𝑝) varies from initial guess 𝑧_𝑖,0(𝑥,𝑡) to the solutions 𝐸₁(𝑥,𝑡),𝐸₂(𝑥,𝑡), and 𝑛(𝑥,𝑡), for 𝑖=1,2,3, respectively.

Expanding 𝜙_𝑖(𝑥,𝑡;𝑝) in the Taylor series with respect to 𝑝 for 𝑖=1,2,3, one has𝜙_𝑖𝑥,𝑡;𝑝=𝑧_𝑖,0(𝑥,𝑡)+^∞_𝑚=1𝑧_𝑖,𝑚(𝑥,𝑡)𝑝^𝑚, (5.6) where𝑧_𝑖,𝑚()=1𝑥,𝑡𝜕𝑚!^𝑚𝜙_𝑖(𝑥,𝑡;𝑝)𝜕𝑝^𝑚_𝑝=0. (5.7) If the auxiliary linear operator, the initial guess, and the auxiliary parameters ℏ_𝑖 are so properly chosen, the above series, converge at 𝑝=1, has𝐸₁(𝑥,𝑡)=𝑧_1,0(𝑥,𝑡)+^∞_𝑚=1𝑧_1,𝑚(),𝐸𝑥,𝑡₂(𝑥,𝑡)=𝑧_2,0(𝑥,𝑡)+^∞_𝑚=1𝑧_2,𝑚(),𝑛()𝑥,𝑡𝑥,𝑡=𝑧_3,0(𝑥,𝑡)+^∞_𝑚=1𝑧_3,𝑚(𝑥,𝑡), (5.8) which must be one of solutions of the original nonlinear equation, as proved by Liao [18]. Define the vectors𝑧^→_𝑖,𝑛=𝑧_𝑖,0(𝑥,𝑡),𝑧_𝑖,1(𝑥,𝑡),…,𝑧_𝑖,𝑛(𝑥,𝑡),𝑖=1,2,3. (5.9)

We have the 𝑚th-order deformatiom equations:𝐿𝑧_𝑖,𝑚(𝑥,𝑡)−𝜒_𝑚𝑧_{𝑖,𝑚−1}(𝑥,𝑡)=ℏ_𝑖𝑅_𝑖,𝑚𝑧^→_1,𝑚−1,𝑧^→_2,𝑚−1,𝑧^→_3,𝑚−1,𝑖=1,2,3, (5.10) where𝑅_1,𝑚𝑧^→_1,𝑚−1,𝑧^→_2,𝑚−1,𝑧^→_3,𝑚−1=𝜕𝑧_1,𝑚−1+𝜕𝜕𝑡²𝑧_2,𝑚−1𝜕𝑥²−2𝛽^𝑚−1𝑗_𝑗=0𝑘=0𝑧_1,𝑘𝑧_{1,𝑗−𝑘}𝑧_{2,𝑚−1−𝑗}+^𝑚−1𝑗_𝑗=0𝑘=0𝑧_2,𝑘𝑧_{2,𝑗−𝑘}𝑧_{2,𝑚−1−𝑗}+2^𝑚−1_𝑗=0𝑧_2,𝑗𝑧_{3,𝑚−1−𝑗},𝑅_2,𝑚𝑧^→_1,𝑚−1,𝑧^→_2,𝑚−1,𝑧^→_3,𝑚−1=𝜕𝑧_2,𝑚−1−𝜕𝜕𝑡²𝑧_1,𝑚−1𝜕𝑥²+2𝛽^𝑚−1𝑗_𝑗=0𝑘=0𝑧_1,𝑘𝑧_{1,𝑗−𝑘}𝑧_{1,𝑚−1−𝑗}+^𝑚−1𝑗_𝑗=0𝑘=0𝑧_1,𝑘𝑧_{2,𝑗−𝑘}𝑧_{2,𝑚−1−𝑗}−2^𝑚−1_𝑗=0𝑧_1,𝑗𝑧_{3,𝑚−1−𝑗},𝑅_3,𝑚𝑧^→_1,𝑚−1,𝑧^→_2,𝑚−1,𝑧^→_3,𝑚−1=𝜕²𝑧_3,𝑚−1𝜕𝑡²−𝜕²𝑧_3,𝑚−1𝜕𝑥²+𝜕²𝜕𝑥^2𝑚−1_𝑗=0𝑧_1,𝑗𝑧_{1,𝑚−1−𝑗}+^𝑚−1_𝑗=0𝑧_2,𝑗𝑧_{2,𝑚−1−𝑗}, (5.11) where 𝑧₁,𝑧₂, and 𝑧₃ are functions of 𝑥 and 𝑡. Now, the solutions of the 𝑚th-order deformation equation (5.10) for 𝑚≽1 become𝑧_𝑖,𝑚(𝑥,𝑡)=𝜒_𝑚𝑧_{𝑖,𝑚−1}(𝑥,𝑡)+ℏ_𝑖𝐿⁻¹𝑅_𝑖,𝑚𝑧^→_1,𝑚−1,𝑧^→_2,𝑚−1,𝑧^→_3,𝑚−1,𝑖=1,2, (5.12) where 𝐿⁻¹=∫^𝑡₀(⋅)𝑑𝑡 and𝑧_3,𝑚(𝑥,𝑡)=𝜒_𝑚𝑧_3,𝑚−1(𝑥,𝑡)+ℏ₃𝐿⁻¹𝑅_3,𝑚𝑧^→_1,𝑚−1,𝑧^→_2,𝑚−1,𝑧^→_3,𝑚−1,𝑖=1,2, (5.13) where 𝐿⁻¹=∬^𝑡₀(⋅)𝑑𝑡𝑑𝑡. For simplicity, we suppose ℏ₁=ℏ₂=ℏ₃=ℏ.

We consider the solutions of (3.11) with the initial conditions (5.4); we obtain𝑧_1,1(𝑥,𝑡)=𝑟𝑡ℏ2𝑘secℎ²(𝑘𝑥)𝑘₁cos𝑘₁𝑥−𝑘sin𝑘₁𝑥tanh(𝑘𝑥)+sin𝑘₁𝑥×tanh(𝑘𝑥)−𝑘²₁+2𝑠+2𝑟²−1+𝛽−4𝑘²₁𝛽tanh²(𝑘𝑥)4𝑘²₁,𝑧−1_2,1(𝑥,𝑡)=𝑟𝑡ℏ2𝑘secℎ²(𝑘𝑥)𝑘₁sin𝑘₁𝑥+𝑘cos𝑘₁𝑥tanh(𝑘𝑥)+cos𝑘₁𝑥×tanh(𝑘𝑥)𝑘²₁−2𝑠+2𝑟²1−𝛽+4𝑘²₁𝛽tanh²(𝑘𝑥)4𝑘²₁,𝑧−1_3,1(𝑥,𝑡)=−4𝑘²₁𝑘²𝑟²𝑡²ℏ4𝑘²₁(−1(−2+cosh2𝑘𝑥))secℎ⁴(𝑘𝑥). (5.14)(5.15)(5.16)

Obviously, for ℏ=−1 the obtained solutions are the same homotopy perturbation method in (4.15)–(4.17); we continue to evaluate eight terms of HAM.

Using a Taylor series, then the closed-form solutions yield as follows:𝐸(𝑥,𝑡)=𝑟tanh(𝑘𝑥−𝑤𝑡)exp𝑖𝑘₁,𝑛()𝑟𝑥−Ω𝑡𝑥,𝑡=𝑠+²−4𝑘²₁+1tanh²(𝑘𝑥−𝑤𝑡), (5.17) where 𝑤=2𝑘₁𝑘,Ω=−2𝑠+𝑘²₁+2𝑘²,𝑟=𝑘²(4𝑘²₁−1)/(1+(4𝑘²₁−1)𝛽),𝑘,𝑠,𝛽, and 𝑘₁ are arbitrary constants.

6. Comparing the HPM Results with the HAM Results and the ADM Results and the Exact Solutions

To demonstrate the convergence of the HPM, the results of the numerical example are presented, and only few terms are required to obtain accurate solutions. Tables 1 and 4 show the absolute errors between the analytical solutions and the HPM solutions of the GZE with initial conditions (3.12) for 𝐸(𝑥,𝑡),𝑛(𝑥,𝑡) are very small with the present choice of 𝑡 and 𝑥; Tables 2, 3, 5, and 6 help us to compare the HPM results with the ADM results, and the HAM results when ℏ=−1 through the absolute errors. Both the analytical solutions, the HPM result, the ADM result and the HAM result for 𝐸(𝑥,𝑡) and 𝑛(𝑥,𝑡) are plotted in Figures 1 and 2. The diagrams of the results obtained for ℏ=−1.1,ℏ=−1, and ℏ=−0.9 in comparison with the ADM solutions and the exact solutions for 𝐸(𝑥,𝑡) and 𝑛(𝑥,𝑡) are shown in Figures 3 and 4, respectively.

Table 1 

The HPM results for 𝐸(𝑥,𝑡) for the first three approximation in comparison with the analytical solution with initial conditions (3.11) when 𝑘=0.05,𝑘₁=1,𝑠=0.33,𝛽=1.

Table 2 

The ADM results for 𝐸(𝑥,𝑡) for the first six approximations in comparison with the analytical solution with initial conditions (3.11) when 𝑘=0.05,𝑘₁=1,𝑠=0.33,𝛽=1.

Table 3 

The HAM results for 𝐸(𝑥,𝑡) for the 8th-order approximate in comparison with the analytical solution with initial conditions (3.11) when ℏ=−1,𝑘=0.05,𝑘₁=1,𝑠=0.33,𝛽=1.

Table 4 

The HPM results for 𝑛(𝑥,𝑡) for the first three approximations in comparison with the analytical solution with initial conditions (3.11) when 𝑘=0.05,𝑘₁=1,𝑠=0.33,𝛽=1.

Table 5 

The ADM results for 𝑛(𝑥,𝑡) for the first six approximations in comparison with the analytical solution with initial conditions (3.11) when 𝑘=0.05,𝑘₁=1,𝑠=0.33,𝛽=1.

Table 6 

The HAM results for 𝑛(𝑥,𝑡) for the 8th-order approximate in comparison with the analytical solution with initial conditions (3.11) when ℏ=−1,𝑘=0.05,𝑘₁=1,𝑠=0.33,𝛽=1.

(a) The exact solution for |𝐸|2

(b) The HPM solution for |∑3𝑘=0𝐸2𝑘|

(c) The ADM solution for |∑6𝑘=0𝐸2𝑖|

(d) The HAM solution for |𝐸|2 for the 8th order

Figure 1 

Comparison between the exact solution, the HPM solution, the ADM solution, and the HAM solution for 𝐸(𝑥,𝑡) with initial conditions (3.11) when 𝑘=0.05,𝑘1=1,𝑠=0.33,𝛽=1,ℏ=−1.

(a) The exact solution for 𝑛(𝑥,𝑡)

(b) The HPM solution for 𝑛(𝑥,𝑡)

(c) The ADM solution for 𝑛(𝑥,𝑡)

(d) The HAM solution for 𝑛(𝑥,𝑡) for the 8th order

Figure 2 

Comparison between the exact solution, the HPM solution, the ADM solution, and the HAM solution for 𝑛(𝑥,𝑡) with initial conditions (3.11) when 𝑘=0.05, 𝑘1=1, 𝑠=0.33, 𝛽=1, ℏ=−1.

(a)

(b)

(c)

(d)

Figure 3 

The results obtained by HAM for various ℏ by the 8th-order approximate solutions for 𝐸(𝑥,𝑡), in comparison with ADM solutions and the exact solutions with initial conditions (3.11) when 𝑘=0.05, 𝑘1=1, 𝑠=0.33, 𝛽=1, (a)𝑥=0.25, (b)=0.5, (c)=0.75, (d)=1.

(a)

(b)

(c)

(d)

Figure 4 

The results obtained by HAM for various ℏ by 8th-order approximate solutions for 𝑛(𝑥,𝑡), in comparison with ADM solutions and the exact solutions with initial conditions (3.11) when 𝑘=0.05, 𝑘1=1, 𝑠=0.33, 𝛽=1, (a)𝑥=0.25, (b)=0.5, (c)=0.75, (d)=1.

Acknowledgments

This paper was founded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (25/130/1431). The authors, therefore, acknowledge with thanks the DSR technical and financial support.