NUEVAS SOLUCIONES PARA LA ECUACIÓN KDV POR EL MÉTODO DE LA FUNCIÓN-EXP
NEW SOLUTIONS FOR THE KDV EQUATION BY THE EXP-FUNCTION METHOD
Alvaro H. Salas S.
Department of Mathematics, Universidad de Caldas, Universidad Nacional de Colombia - Manizales, Caldas.Correo electrónico: firstname.lastname@example.org
En este artículo se obtienen soluciones para la ecuación KdV. Estas soluciones son obtenidas a través del método de la function-Exp, con ayuda del computador.
ecuación diferencial no lineal, ecuación diferencial parcial no lineal, ecuación de evolución de tercer orden. Ecuación KdV, soluciones solitónicas,onda viajera, soliton, método de la function-Exp, ecuación diferencial parcial, ecuación de evolución no lineal.
In this paper we obtain some exact solutions for the KdV equation. These solutions are obtained via the Exp-function method with the aid of a computer.
Nonlinear differential equation, nonlinear partial differential equation, third order evolution equation, KdV equation, solitonic solution, traveling wave, soliton, Exp-function method, partial differential equation, nonlinear evolution equation.
Nonlinear evolution and wave equations are partial differential equations (PDEs) involving first or second-order derivatives with respect to time. Such equations have been intensively studied for the past decades [1, 2], and several new methodsto solve nonlinear PDEs, either numerically or analytically, are now available. When the dependent variable u in the PDE corresponds to a physical quantity (such as the surface height of a water wave, the magnitude of an electromagnetic wave,etc.), it is important to study the propagation or aggregation properties of u. This motivates the study of methods to analytically solve evolution or wave equations via symbolic methods. The goal is to find exact traveling wave solutions. If these solutions do not change their form during propagation, they are called solitary waves. Solitary waves that preservé their shape upon collision are called solitons . Solitary-waves and solitons arise due to a critical balance between dispersión and nonlinearity. Due-to the complexity of the mathematics involved in finding:exact solutions for these PDEs, the use of algorithmic techniques that can be implemented in the symbolic language of computer algebra systems becomes a necessity. Several computer algebra packages now exist to aid in the study of nonlinear PDEs [4, 5, 6]. For example, Painlev'e analysis offers an algorithm for testing whether or not a PDE is a good candidate to be completely integrable.In addition, the Painlev'e method allows one to construct solitary wave solutions in explicit form. A more powerful technique is Hirota's bilinear method  which allows one to find N-sóliton solutions of largé classes of completely integrable PDEs  ; The stor'y of the first observation of solitary waves*is worth telling. In 1834, while riding horseback beside the narrow Union canal near Edinburgh in Scotland, J. Scott Russell noticed that á bow wave, rolling away from a large barge, traveled ás a huge heap of water for quite a long distance before finálly dispersing into smaller ripples. In order to study this intriguing phenomenon, Russell did extensivé experiments in a large water tank. Further investigations of solitary waves were done by Airy, Stokes, Boussinesq, and Rayleigh -in an attempt tor understand the mechanism behind this remarkable phenomenon . The latter two scientists derived approximate models to describe solitary waves. In order to obtain his result, Boussinesq derived a one-dimensional'nonlinear wave equation which now bears his ñame. The issue was finally resolved (in 1895) bytwo Dutchmen, Korteweg and de Vries, when they derived a nonlinear evolution equation governing long, oñe-dimensional surface gravity waves (with small amplitude) propagating in shallow water:
(1)where is the surface elevation of the wave above the equilibrium level h,a is a small arbitrary constant related to the uniform motion of the liquid,g is the gravitational constant,T is the surface tension, and p is the density. The independent variables T and ζ are scaled versions of the time and space coordinates. Equation (1), which is called the Korteweg-de Vries (KdV) equation, can be brought into a non-dimensional form via the change of variables
Here subscripts denote partial derivatives,
After some algebra, one obtains
Despite this early derivación ofthe KdV equation,it was not until 1960 that any new applications of the equation were discovered . In 1960, while studying collision-free hydrodynamic waves,Gardner and Morikawa rediscovered the KdV equation . Amazingly, the KdV equation started to show up in a number of other physical contexts such as the study of stratified internal waves, ion-acoustic waves in plasma physics, lattice dynamics, and so on (further details can be found in Jeffrey and Kakutani , Scott et al. , Miura , Ablowitz and Segur , Lamb , Calogéro and Degasperis , Dodd et al. , and Novikov et al. ). Since the late 1960's, the study ofthe properties of solitons, and the search for solitonic equations and methods to solve them, has been an active and exciting area of research.
In this paper we give some new exact solutions of equation (4) by the exp-function method.
2. EXACT SOLUTIONS THE KDV EQUATION THE EXP-FUNCTION METHOD
Using the transformation
where λ, u are constants, Eq. (4) becomes
In view of the Exp-function method, we assume that the solution of Eq. (5) can be expressed in the form
where c , d, p and q are positive integers which are unknown to be determined later, an and bm are unknown constants.
In order to determine valúes of c and p , we balance the linear term of highest order in Eq. (6) with the highest order nonlinear term, and the linear term of lowest order in Eq. (6) with the lowest order nonlinear term, respectively.
By simple calculation, we have
where the ki are some constants. Balancing highest order of Exp-function in Eqs. (8) and (9), 7p + c = 2(3p + c) so that c =p. Similarly, to determine values of d and q, we ba lance the linear term of lowest order in Eq. (6)
where the ki are some constants. Balancing lowest order of Exp-function in Eqs. (10) and (11), we obtain 7q + d = 2(3q + d) so that d = q.
The considerations below say that any solution of the KdV equation (6) must have the form
We will consider two cases. In these cases we set b-p = 1, that is, the trial solution has the form
21 Case 1: p=c-=1 and d = q= 1
The trial solution Eq. (12) becomes
Substituting Eq. (13) into (6) and equating to zero the coefficients of all powers of exp (ζ) yields a set of algebraic equations. Solving it with the aid of a computer, we obtain the folloWing solutions:
For b0 = ± 2 the soliton sollitions correspondiiig to these values are :
For b0 ± 2 and real u
We obtain periodic solutions in the following cases :
2.2. Case 2: p = c = 2 and d = q = 2.
The trial solution Eq. (12) becomes
Substituting Eq. (16) into (6) and equating to zero the coefficients of all powers of exp() yields a set of algebraic equations. Solving it with the aid of a- computer, we obtain many solutions. For space reasons, we only give some of them.
In the case when b0 = + 2 we obtain periodic solutions. More exactly,
VIII. Finally, other interesting solitonic solutions are :
Figure 1 and Figure 2 illustrate graphically some solutions. The soliton solution U2 (x, t) and the periodic solution u14 (x, t)
Figure 1 : The solution U2 (X, t) for 711 = 2, al =0,1t1<1 and I X I < 2.
Figure .2: The function U14(x,t) for. q= 2 and :t, X E [-3,3].
In this paper, by using the exp-function method and the help of a computer, we obtained some exact solutions for the KdV equation (4). The method is direct and effective. We may apply this method to solve other partial and ordinary nonlinear differential equations. The Exp-function method is a promising and powerful new method for NLEEs arising in mathematical physics. Its applications are worth further studying.