## Abstract

As we all know, perturbation theory is closely related to methods used in the numerical analysis fields. In this chapter, we focus on introducing two homotopy asymptotic methods and their applications. In order to search for analytical approximate solutions of two types of typical nonlinear partial differential equations by using the famous homotopy analysis method (HAM) and the homotopy perturbation method (HPM), we consider these two systems including the generalized perturbed Kortewerg-de Vries-Burgers equation and the generalized perturbed nonlinear Schrödinger equation (GPNLS). The approximate solution with arbitrary degree of accuracy for these two equations is researched, and the efficiency, accuracy and convergence of the approximate solution are also discussed.

### Keywords

- homotopy analysis method
- homotopy perturbation method
- generalized KdV-Burgers equation
- generalized perturbed nonlinear Schrödinger equation
- approximate solutions
- Fourier transformation

## 1. Introduction

In the past decades, due to the numerous applications of nonlinear partial differential equations (NPDEs) in the areas of nonlinear science [1, 2], many important phenomena can be described successfully using the NPDEs models, such as engineering and physics, dielectric polarization, fluid dynamics, optical fibers and quantitative finance and so on [3–5]. Searching for analytical exact solutions of these NPDEs plays an important and a significant role in all aspects of this subject. Many authors presented various powerful methods to deal with this problem, such as inverse scattering transformation method, Hirota bilinear method, homogeneous balance method, Bäcklund transformation, Darboux transformation, the generalized Jacobi elliptic function expansion method, the mapping deformation method and so on [6–10]. But once people noticed the complexity of nonlinear terms of NPDEs, they could not find the exact analytic solutions for many of them, especially with disturbed terms. Researchers had to develop some approximate and numerical methods for nonlinear theory; a great deal of efforts has been proposed for these problems, such as the multiple-scale method, the variational iteration method, the indirect matching method, the renormalization method, the Adomian decomposition method (ADM), the generalized differential transform method and so forth [11–13], among them the perturbation method [14], including the regular perturbation method, the singular perturbation method and the homotopy perturbation method (HPM) and so on.

Perturbation theory is widely used in numerical analysis as we all know. The earliest perturbation theory was built to deal with the unsolvable mathematical problems in the calculation of the motions of planets in the solar system [15]. The gradually increasing accuracy of astronomical observations led to incremental demands in the accuracy of solutions to Newton’s gravitational equations, which extended and generalized the methods of perturbation theory. In the nineteenth century, Charles-Eugène Delaunay discovered the problem of small denominators which appeared in the

The homotopy analysis method (HAM) was firstly proposed in 1992 by Liao [19], which yields a rapid convergence in most of the situations [20]. It also showed a high accuracy to solutions of the nonlinear differential systems. After this, many types of nonlinear problems were solved with HAM by others, such as nonlinear Schrödinger equation, fractional KdV-Burgers-Kuramoto equation, a generalized Hirota-Satsuma coupled KdV equation, discrete KdV equation and so on [21–24]. With this basic idea of HAM (as

In this chapter, we extend the applications of HAM and HPM with the aid of Fourier transformation to solve the generalized perturbed KdV-Burgers equation with power-law nonlinearity and a class of disturbed nonlinear Schrödinger equations in nonlinear optics. Many useful results are researched.

### 1.1. The homotopy analysis method (HAM)

Let us consider the following nonlinear equation

where

With the basic idea of the traditional homotopy method, we construct the following zero-order deformation equation

where

Thus, as

where

If the auxiliary linear operator, the initial guess, the auxiliary parameter and the auxiliary function are so properly chosen such that they are smooth enough, the Taylor’s series (4) with respect to

which must be one of the solutions of the original nonlinear equation, as proved by Liao. As

Eq. (7) is used mostly in the HPM, whereas the solution is obtained directly, without using Taylor’s series. As

which is used in the HAM when it is not introduced in the set of base functions. According to definition (5), the governing equation can be deduced from Eq. (2). Define the vector

Differentiating Eq. (2)

where

And

It should be emphasized that

### 1.2. The homotopy perturbation method

To illustrate the basic concept of the homotopy perturbation method, consider the following nonlinear system of differential equations with boundary conditions

where

We construct the following homotopy mapping

where

The approximate solution can be obtained by setting

If we let

To study the convergence of the method, let us state the following theorem.

**Theorem** (Sufficient Condition of Convergence).

Suppose that

Then, according to Banach’s fixed point theorem,

and suppose that

**Proof**. (i) By inductive approach, for

(ii) Because of

## 2. Application to the generalized perturbed KdV-Burgers equation

Consider the following generalized perturbed KdV-Burgers equation

where

This equation with

Fitzhugh-Nagumo equation [32]:

Burgers-Huxley equation [33]

Burgers-Fisher equation [34]

It’s significant for us to handle Eq. (22).

### 2.1. The generalized KdV-Burgers equation

If we let

Eq. (26) is solved on the infinite line

where

Eq. (26) is reduced to the following form:

where the derivatives are performed with respect to the coordinate

### 2.2. The approximate solutions by using HAM

To solve Eq. (22) by means of HAM, we choose the initial approximation

where

According to Eq. (1), we define the nonlinear operator

It is reasonable to express the solution

with the property

From Eqs. (10, 11 and 32), we have

where

and

Now, the solution of the mth-order deformation in Eq. (10) with initial condition

Thus, from Eqs. (31, 35 and 38), we can successively obtain

We obtain the mth-order approximate solution and exact solution of Eq. (22) as follows

if we choose

From Eqs. (39–44), we can obtain the corresponding approximate solution of Eq. (22).

### 2.3. Example

In the following, three examples are presented to illustrate the effectiveness of the HAM. We first plot the so-called

Now, we consider the small perturbation term

**Example 1**. Consider the CKdV equation with small disturbed term

with the initial exact solution

From Section 2.2, we have

The

**Example 2**. Consider the KdV-Burgers equation with small disturbed term

with the initial exact solution

From Section 2.2, we have

The

**Example 3**. Consider the Burgers-Fisher equation

with the exact solution and the initial exact solution

From Section 2.2, we have

The

## 3. Application to the generalized perturbed NLS equation

In this section, we will use the HPM and Fourier’s transformation to search for the solution of the generalized perturbed nonlinear Schrödinger equation (GPNLS)

If we let

where disturbed term

We make the transformation

With the following consistency conditions,

where

If we let

By using the general mapping deformation method [10, 40], we can obtain the following solutions of the corresponding undisturbed Eq. (70) when

In order to obtain the solution of Eq. (70), we introduce the following homotopic mapping

where

Obviously, from mapping Eq. (72),

### 3.1. Approximate solution

In order to obtain the solution of Eq. (70), set

If we let

where

From Eq. (75) we have

If we select

where

We obtain the first- and second-order approximate solutions

With the same process, we can also obtain the N-order approximate solution

where

### 3.2. Comparison of accuracy

In order to explain the accuracy of the expressions of the approximate solution represented by Eq. (86), we consider the small perturbation term

where

From the discussion of Section 3.1, we obtain the second-order approximate Jacobi-like elliptic function solution of Eq. (88) as follows

Set

where

Therefore, from the above result, we know that the approximate solution,

Set

## 4. Conclusions

We research the generalized perturbed KdV-Burgers equation and GPNLS equation by using the HAM and HPM; these two powerful straightforward methods are much more simple and efficient than some other asymptotic methods such as perturbation method and Adomian decomposition method and so on. The Jacobi elliptic function and solitary wave approximate solution with arbitrary degree of accuracy for the disturbed equation are researched, which shows that these two methods have wide applications in science and engineering and also can be used in the soliton equation with complex variables, but it is still worth to research whether or not these two methods can be used in the system with high dimension and high order.

## Acknowledgments

The work is supported by the Scientific Research Foundation of Nanjing Institute of Technology (Grant No. ZKJ201513,2016YB22).