scholarly journals New Exact Solutions of Some Nonlinear Systems of Partial Differential Equations Using the First Integral Method

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Shoukry Ibrahim Atia El-Ganaini
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Integral Method ◽  
Traveling Wave Solutions ◽  
First Integral ◽  
First Integrals ◽  
Partial Differential ◽  
First Integral Method ◽  
New Exact Solutions

The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE), (2 + 1)-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.

scholarly journals The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Shoukry Ibrahim Atia El-Ganaini
Keyword(s):  
Optical Fibers ◽  
Integral Method ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
First Integral ◽  
Higher Dimensions ◽  
First Integrals ◽  
First Integral Method ◽  
Nonlinear Schrödinger ◽  
Planar Systems

The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, the generalized nonlinear Schrodinger (GNLS) equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.

scholarly journals The First Integrals of a Second Order Ordinary Differential Equation and Application

2019 ◽  
pp. 1-15
Author(s):  
Yanxia Hu ◽  
Xiaofei Du
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
First Integrals ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Partial Differential ◽  
Wave Solutions ◽  
Symmetry Method

The first integrals of second order ordinary differential equations are considered. The necessary conditions of the existence of analytical first integrals for the equation are presented. Then, the first integrals of the equation are obtained using Lie symmetry method. The results of the first integrals are applied to certain classes of partial differential equations, the conditions of nonexistence of the traveling wave solutions of the partial differential equations are obtained, and traveling wave solutions of the equations under the certain parametric conditions are also obtained.

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