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Register a new userDiscussion on exp-function method and modified method of simplest equation
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Zlatinka I. Dimitrova
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We discuss the relation between the modified method of simplest equation and the exp-function method. First on the basis of our experience from the application of the method of simplest equation we generalize the exp-function ansatz. Then we apply the ansatz for obtaining exact solutions for members of a class of nonlinear PDEs which contains as particular cases several nonlinear PDEs that model the propagation of water waves.
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We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrodinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrodinger kind.
We discuss an extension of the modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations. The extension includes the possibility for use of: (i) more than one simplest equation; (ii) relationship that contains as particular cases the relationship used by Hirota cite{hirota} and the relationship used in the previous version of the methodology; (iii) transformation of the solution that contains as particular case the possibility of use of the Painleve expansion; (iv) more than one balance equation. The discussed version of the methodology allows: obtaining multi-soliton solutions of nonlinear partial differential equations if such solutions do exist and obtaining particular solutions of nonintegrable nonlinear partial differential equations. Examples for the application of the methodology are discussed.
We present a short review of the evolution of the methodology of the Method of simplest equation for obtaining exact particular solutions of nonlinear partial differential equations (NPDEs) and the recent extension of a version of this methodology called Modified method of simplest equation. This extension makes the methodology capable to lead to solutions of nonlinear partial differential equations that are more complicated than a single solitary wave.
We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of Weierstrass and Jacobi. We show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions. This system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution. Any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation. The methodology is illustrated by obtaining solitary wave solutions for the generalized Korteweg-deVries equation and by obtaining solutions of the higher order Korteweg-deVries equation.
We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order has the Painleve property if the only movable singularities connected to this equation are single poles. The importance of this property can be seen from the Ablowitz-Ramani-Segur conhecture that states that a nonlinear PDE is solvable by inverse scattering transformation only if each nonlinear ODE obtained by exact reduction of this PDE has the Painleve property. The Painleve property motivated motivated much research on obtaining exact solutions on nonlinear PDEs and leaded in particular to the method of simplest equation. A version of this method called modified method of simplest equation is discussed below.
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