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تسجيل مستخدم جديدModified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: past and present
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Nikolay K. Vitanov
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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.
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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 different
ial 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.
The method of simplest equation is applied for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. The used simplest equ
ation is $f_xi^2 = n^2(f^2 -f^{(2n+2)/n})$. The developed methodology is illustrated on two examples of classes of nonlinear partial differential equations that contain: (i) only monomials of odd grade with respect to participating derivatives; (ii) only monomials of even grade with respect to participating derivatives. The obtained solitary wave solution for the case (i) contains as particular cases the solitary wave solutions of Korteweg-deVries equation and of a version of the modified Korteweg-deVries equation.
We discuss the last version as well as applications of a method for obtaining exact solutions of nonlinear partial differential equations. As this version is based on more than one simple equation we call it Simple Equations Method (SEsM). SEsM conta
ins as particular case the Modified Method of Simplest Equation (MMSE) for the case when we use one simple equation and the solution is searched as power series of the solution of the simple equation. SEsM contains as particular cases many other methodologies for obtaining exact solutions of non-linear partial differential equations. We demonstrate that SEsM can lead to multisoliton solutions of integrable nonlinear partial differential equations and in addition we demonstrate that SEsM keeps the property of the Modified Method of Simplest Equation to lead to exact solutions of nonitegrable nonlinear partial differential equations.
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 a version the methodology for obtaining exact solutions of nonlinear partial differential equations based on the possibility for use of: (i) more than one simplest equation; (ii) relationship that contains as particular cases the relations
hip 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: (i) obtaining multi-soliton solutions of nonlinear partial differential equations if such solutions do exist; (ii) obtaining particular solutions of nonintegrable nonlinear partial differential equations. Several examples for the application of the methodology are discussed. Special attention is devoted to the use of the simplest equation $f_xi =n[f^{(n-1)/n} - f^{(n+1)/n}]$ where $n$ is a positive real number. This simplest equation allows us to obtain exact solutions of nonlinear partial differential equations containing fractional powers.
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