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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 340824, 25 pages
Solving Nonlinear Partial Differential Equations by the sn-ns Method
FIZMAKO Research Group and Department of Mathematics, University of Caldas/National University of Colombia, Campus la Nubia, Manizales, Colombia
Received 2 January 2012; Accepted 30 January 2012
Academic Editor: Muhammad Aslam Noor
Copyright © 2012 Alvaro H. Salas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present the application of the sn-ns method to solve nonlinear partial differential equations. We show that the well-known tanh-coth method is a particular case of the sn-ns method.
The search of explicit solutions to nonlinear partial differential equations (NLPDEs) by using computational methods is one of the principal objectives in nonlinear science problems. Some powerful methods have been extensively used in the past decade to handle nonlinear PDEs. Some of them are the tanh-method , the tanh-coth method , the exp-function method , the projective Riccati equation method , and the Jacobi elliptic functions method. Practically, there is no unified method that could be used to handle all types of nonlinear problems.
The main purpose of this work consists in solving nonlinear polynomial PDE starting from the idea of the projective Riccati equations method. We derive exact solutions to the following equations: Duffing equation, cubic nonlinear Schrodinger equation, Klein-Gordon-Zakharov equations, quadratic Duffing equation, KdV equation, Gardner equation, Boussinesq equation, symmetric regular long wave equation, generalized shallow water wave equation, Klein-Gordon equation with quadratic nonlinearity, Fitzhugh-Nagumo-Huxley equation, and double sine-Gordon equation.
2. The Main Idea
In the search of the traveling wave solutions to nonlinear partial differential equation of the form the first step consists in considering the wave transformation for a suitable function , where is a constant. Usually, (the identity function).
To find solutions to (2.3), we suppose that can be expressed as where is a rational function in the new variables , which satisfy the system with being some constant to be determined and a rational function in the variable . We show that the system (2.5) may be solved exactly in certain cases. In fact, taking where and , system (2.5) reduces to From (2.7) we get Equation (2.8) is of elliptic type. Choosing and adequately, we may obtain distinct methods to solve nonlinear PDEs. More exactly, suppose we have solved (2.8). Then, in view of (2.5) and (2.6) functions and may be computed by formulae To solve (2.3) we try one of the following ansatz: We substitute any of these ansatz into (2.3) and we obtain a polynomial equation either in the variables and or . We equate the coefficients of () to zero, and we obtain a system of polynomial equations in the variables Solving this system with the aid of a symbolic computational package such as Mathematica 8 or Maple 15, we obtain the desired solutions. Sometimes, we replace with and then the corresponding system is regarded w.r.t. the variables and , where const.
We also may solve coupled systems of nonlinear equations. Indeed, suppose that we have a coupled system of two equations in the form We first apply the wave transformation for a suitable pair of functions and in order to obtain a coupled system of two ODEs in the form where and are polynomials w.r.t. the variables .
We seek solutions to system (2.14) in the forms The integers and are determined by the balancing method.
We substitute any of ansatz (2.15), (2.16), or (2.17) into (2.14), and we obtain a system of two polynomial equations either w.r.t. the variables and or w.r.t. the variable . We equate the coefficients of (resp., the coefficients of ) () to zero, and we obtain a system of polynomial equations w.r.t the variables Solving this system with the aid of a symbolic computational package such as Mathematica or Maple, we obtain the desired solutions. Sometimes, with the aim to add an additional parameter , we replace with and then the corresponding system is regarded w.r.t. the variables and , where .
The same technique is applied for solving systems of three or more equations.
3. The sn-ns Method and Its Derivation
Let , and , where This choice gives us (2.8) in the form We may express the general solution of this equation in terms of the Jacobi elliptic functions ns or nd as follows: where Solution (3.3) is valid for , , and .
On the other hand, function ns is real valued for any real , , , and . We may verify that function where and are given by (3.4), is a solution to equation Thus, we always may find a solution to (3.2) when and for any .
Now, let us assume that . It may be verified that a solution to (3.2) is where Observe that for any function nd is real valued for any real values of , , and . We conclude that (3.2) has exact solutions for any and .
Projective equations are Taking in (3.3) we see that solutions to system (3.9) are This motivates us to seek solutions to (2.3) in the form Usually, and then has the forms In the case when , this gives the tanh-coth method since and .
Another possible ansatz suggested from (2.10) is If , this ansatz reads We may consider similar ansatz by replacing sn by dn and ns by nd, respectively. We will call this the dn-nd method. Thus, two possible ansatz for this method are An other useful ansatz to handle other equations of the form (2.3) is We will call this the cn-nc method.
We also may try the following ansatz: For example, this ansatz may be successfully applied to the cubic-quintic Duffing equation, which is defined by
In this section we solve various important models related to nonlinear science by the methods described in previous sections.
4.1. Duffing Equation
Let us consider the equation where and are nonzero constants. This equation is very important since some relevant physical models described by a nonlinear PDEs may be studied once this equation is solved. Two of them are related to cubic nonlinear the Schrodinger equation and the Klein-Gordon-Zakharov equations.
To find solutions we multiply (4.1) by and we integrate it w.r.t. . The resulting equation is where is the constant of integration. This equation has form (3.2) and we already know that there exists an exact solution to it for any and . Instead, we may apply directly the sn-ns method (resp., the dn-nd method) to it. Balancing gives . We seek solutions in the form Inserting (4.3) into (4.1), we obtain a polynomial equation w.r.t. the variable . Equating to zero the coefficients of yields the following algebraic system: Solving system (4.4) gives solutions as follows:
(ii) ,,, ,
(iv) , ,,,
Letting , we obtain trigonometric and hyperbolic solutions:
4.2. Cubic Nonlinear Schrodinger Equation
This equation reads where is a complex-valued function of two real variables and and is a nonzero real parameter and . The physical model of the cubic nonlinear Schrodinger equation (shortly, NLS equation) (4.11) and its generalized variants occur in various areas of physics such as nonlinear optics, water waves, plasma physics, quantum mechanics, superconductivity, and the Bose-Einstein condensate theory. It also has applications in optics since it models many nonlinearity effects in a fiber, including but not limited to self-phase modulation, four-wave mixing, second harmonic generation, stimulated Raman scattering, and so forth. For water waves, the NLS equation (4.11) describes the evolution of the envelope of modulated nonlinear wave groups. All these physical phenomena can be better understood with the help of exact solutions for a given . When , the equation is said to be attractive. If we say that it is repulsive. Recently, Ma and Chen  obtained some solutions to (4.11).
We seek solutions to (4.11) in the form where and are some real constants to be determined. Inserting (4.12) into (4.11) and simplifying, we obtain This last equation has the form (4.1) with Thus, making use of solutions (4.5)–(4.10) for the choices given by (4.14) we obtain exact solutions to the Schrodinger equation (4.11) in the form (4.12).
4.3. Quadratic Duffing Equation
Let us consider the following second-order and second-degree nonlinear ODE: where , , and are constants and . Solutions to this equation may be used to study some important physical models whose associated PDEs may be solved after making the traveling wave transformation (2.2). As we will show in next subsections some examples of nonlinear partial differential equations where this equation arises are the following.(i)KdV equation: .(ii)Gardner equation (also called combined KdV-mKdV equation): , ,(iii) Boussinesq equation: ,(iv) symmetric regular long wave equation: ,(v)generalized shallow water wave equation: ,(vi)Klein-Gordon equation with quadratic nonlinearity: .
Balancing gives . We seek solutions to (4.15) in the form (3.13), that is, Inserting this ansatz into (4.15) gives the following algebraic system: Solving this system, we obtain the following solutions:(i),,,,,, (ii), , , , , , (iii) , , , , ,, (iv), ,, , , , (v) , , ,, , , (vi) , , , , ,,
Letting , we obtain trigonometric and hyperbolic solutions: Now, let us seek solutions in the ansatz form (3.16), that is, Inserting this ansatz into (4.15) and solving the resulting algebraic system yields the following solutions to (4.15):(i) , , , , , , (ii), ,,,,,
We may obtain other solutions by making use of the dn-nd method.
4.4. KdV Equation
This is the equation If , , this equation takes the form Integrating this equation w.r.t. yields where is the constant of integration. Equation (4.30) has the form (4.15) with Exact solutions to KdV equation may be derived from (4.18)–(4.25) and (4.31).
The KdV equation may also be solved by the Weierstrass elliptic functions method.
4.5. Gardner Equation
This equation reads This equation reduces to mKdV equation if we apply the similarity transformations to the Gardner equation (4.32). We obtain the mKdV equation where .
4.6. Boussinesq Equation
This equation reads After the traveling wave transformation , and integrating twice w.r.t , we obtain the following ODE: where and are the constants of integration. Setting , we obtain an equation of the form (4.15) with As we can see, exact solutions to the Boussinesq equation are calculated from (4.18)–(4.25) and (4.38).
4.7. Symmetric Regular Long Wave Equation
This equation is defined by Let , . Applying this transformation and integrating twice w.r.t. , we obtain the ODE where and are the constants of integration. Setting , we obtain an equation of the form (4.15) with Thus, exact solutions to symmetric regular long wave equation are easily found from (4.18)–(4.25) taking into account (4.41).
4.8. Generalized Shallow Water Wave Equation
This equation is given by Let Substituting (4.43) into (4.42) and integrating once w.r.t. , we obtain where is the constant of integration. Equation (4.15) has the form (4.15) with It is evident that we may find exact solutions to generalized shallow water wave equation (4.42) from (4.18)–(4.25) for the choices given by (4.45).
4.9. Klein-Gordon Equation with Quadratic Nonlinearity
The general Klein-Gordon equation has the form In the case when we obtain the so-called Klein-Gordon equation with quadratic nonlinearity: Let , . After this traveling wave transformation, (4.47) reduces to which is an equation of the form (4.15) with Again, exact solutions to (4.47) are obtained from (4.18)–(4.25) taking into account (4.49).
4.10. Fitzhugh-Nagumo-Huxley Equation
This equation reads This equation is an important model in the study of neuron axon . Let The corresponding ODE is Application of the sn-ns method gives only trivial solutions since we get . Instead, we may use other methods. If we apply the tanh-coth method or the exp method, we obtain nontrivial solutions. Indeed, balancing gives . Following the tanh-coth method, we try the ansatz Inserting (4.53) into (4.50) and solving the corresponding algebraic system gives the following solutions to (4.50):
4.11. Double Sine-Gordon Equation
Our last example deals with the double sine-Gordon equation. This equation in a normalized form reads This equation is an important model in the study of the DNA molecule .
The application of the tanh-coth method gives only the trivial solution . If we apply the sn-ns method, we obtain nontrivial solutions. Indeed, let Inserting ansatz (4.56) into (4.55) gives the ODE Let Substituting (4.58) into (4.57) and solving the corresponding algebraic system gives the following solutions to (4.55):
(iii) , , ,,
5. Comparison with Other Methods to Solve Nonlinear PDEs
There are some other powerful and systematical approaches for solving nonlinear partial differential equations, such as the expansion along the integrable ODE [9, 10], the transformed rational function method , and the multiple expfunction method . Even about linear DEs, there is some recent study on solution representations  and the linear superposition principle has been applied to bilinear equations .
5.1. Multiple Exp Function Method
Let us consider the following equation: Equation (5.1) is also called the Sawada-Kotera equation . In a recent work , the authors obtained multisoliton solutions to (5.1) by using Hirota’s bilinear approach.
Introducing the potential , defined by(5.1) may be written in the form Integrating (5.3) once w.r.t and taking the constant of integration equal to zero, the following partial differential equation is obtained: We will call (5.4) the potential Caudrey-Dodd-Gibbon equation associated with (5.1).
In view of the multiple exp method one-soliton solutions to (5.4) have the form where , , and are some constants to be determined. Inserting ansatz (5.5) into (5.4) and simplifying, we obtain the following polynomial equation in the variable : Equating the coefficients of different powers of to zero gives an algebraic system. Solving it with either Mathematica 8 or Maple 15, we obtain Thus, the following is a one-soliton solution (or one wave solution in Ma’s terminology) to (5.4): In view of (5.2) one-soliton solution to Caudrey-Dodd-Gibbon equation (5.1) is Observe that solution (5.9) is the same solution obtained in  by using Hirota’s bilinear method. We conclude that Hirota’s method and multiple exp method give the same result for one-soliton solutions. Let us examine two-soliton solutions. In view of Ma’s method the two-soliton solutions to (5.4) are of the form Let . Inserting ansatz (5.10) into (5.4) and simplifying, we obtain the following polynomial equation w.r.t the variables and : Equating the coefficients of andto zero gives a system of algebraic equations. Solving it, we obtain the following nontrivial solution: We see that solution (5.12) coincides with solutions obtained in . The same is valid if we set We conclude that Ma’s method does not give any new solutions compared with Hirota’s method.
5.2. The Transformed Rational Function Method
Given a nonlinear ODE in the unknown , or . we search for traveling wave solutions determined by where and are two natural numbers and are normally constants but could be functions of the independent variables and is a solution to equation with A, B, C, D, and E being some constants. Observe that equations are particular cases of (5.14). This method was applied in  to solve the - dimensional Jimbo-Miwa equation which converts into the nonlinear ODE after the traveling wave transformation If we integrate (5.14) once w.r.t , then the following equation follows: