Abstract
Three different methods are applied to construct new types of solutions of nonlinear evolution equations. First, the Csch method is used to carry out the solutions; then the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions. The effectiveness of these methods is demonstrated by applications to the RKL model, the generalized derivative NLS equation. The solitary wave solutions and trigonometric function solutions are obtained. The obtained solutions are very useful in the nonlinear pulse propagation through optical fibers.
1. Introduction
Partial differential equations describe various nonlinear phenomena in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, biology, and mathematical finance. Partial differential equations which arise in real-world physical problems are often too complicated to be solved exactly. It is of significant importance to solve nonlinear partial differential equations (NLPDEs) from both theoretical and practical points of view. The analysis of some physical phenomena is investigated by the exact solutions of nonlinear evolution equations (NLEEs) [1–9].
In this paper, the third-order generalized NLS equation is studied, which is proposed by Radhakrishnan, Kundu, and Lakshmanan (RKL) [10]. The normalized RKL model can be written as Equation (1) describes the propagation of femtosecond optical pulses, represents normalized complex slowly varying amplitude of the pulse envelope, and , and are real constants. Some solitary wave solutions and combined Jacobian elliptic function solution were constructed by different methods [3, 4].
The Csch method is used to carry out the solutions. Then, the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions of this equation.
2. Traveling Wave Solution
Consider the nonlinear partial differential equation in the formwhere is a traveling wave solution of nonlinear partial differential equation (2). We use the transformations,where . This enables us to use the following changes:Using (4) to transfer the nonlinear partial differential equation (2) to nonlinear ordinary differential equation,The ordinary differential equation (5) is then integrated as long as all terms contain derivatives, where we neglect the integration constants.
3. The Generalized NLS Equation (RKL)
In this section, the generalized third-order NLS equation (RKL) (1) is chosen to illustrate the effectiveness of three methods.
The solution of (1) may be supposed asSubstituting (6) into (1) and by defining the derivatives,then decomposing (1) into real and imaginary parts yields a pair of relations which represented nonlinear ordinary differential equations. The real part iswhile the imaginary part isIntegrating (9) once and setting the integration constant to zero, we obtainEquations (8) and (10) will be equivalent, provided thatfrom which we get the parametric constraintsmultiplying both sides of (10) by and integrating with respect to with zero constant, we getassume thatThen
4. Methodology
In this section we will apply three different methods to solve (15). These methods are Csch method, Extended Tanh-Coth method, and the modified simple equation method (MSEM).
4.1. Csch Function Method
The solution of many nonlinear equations can be expressed in the form [11]and their derivativewhere , , and are parameters to be determined and and are the wave number and the wave speed, respectively. We substitute (16)-(17) into the reduced equation (15); we getBalance the terms of the Csch functions to find We next collect all terms in (18) with the same power in and set their coefficients to zero to get a system of algebraic equations among the unknowns , , and and solve the subsequent systemSolving the system of equations in (20), we getthenthereforeFigure 1 represents the solitary wave in (23) for , , , , and then .
4.2. Tanh-Coth Method
The key step is to introduce the ansatz, the new independent variable [12, 13] that leads to the change of variables:AssumeEquation (15) can be written asThe next step is that the solution of (29) is expressed in the formwhere the parameter can be found by balancing the highest-order linear term with the nonlinear terms in (29).
We balance with , to obtain ; then . The Tanh-Coth method admits the use of the finite expansion for are to be determined. Substituting (31) into (25) and then into (29) will yield a set of algebraic equations because all coefficients of have to vanishEquation (32) can be written asEquating expressions at to zero, we have the following system of equations:Solving the system of equations (34), we getSubstitute for from (14), and thenthereforeThenfor , , , and then Figure 2 represents the solitary wave in (39).
4.3. The Modified Simple Equation Method
We look for solutions of (29) in the form [14]Then (29) can be written asThen (41) can be written asEquating expressions in (42) at , and to zero, we have the following system of equations:Solving the system of equations in (43),
Family 1
Family 2
Family 3
Family 4
Family 5
Family 6
Family 7
Family 8
Family 9
Family 10
Family 11
Family 12
5. Conclusion
In this paper, series of new traveling wave solutions have been obtained. The Csch method and the Extended Tanh-Coth method and modified simple equation method are used to carry out the integration of the generalized NLS equation, which is RKL. These methods can be also applied to solve other types of the generalized nonlinear evolution equations with complex coefficients. The solitary waves in Figures 1 and 2 obtained by the Csch and Tanh-Coth methods, respectively, are identical in form and behavior. The obtained solutions are very useful and may be important to explain some physical phenomena and find applications in the nonlinear pulse propagation through optical fibers.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research has been supported by Al-Rafidain University College.