The Lie Symmetry Method For Nonlinear Traveling Wave Equations

The real natural world exposes an infinite number of nonlinear phenomena.For a better understanding of our environment,nonlinear phenomena can be described by nonlinear differential equations.The latter is defined as a relation between one or more unknown functions and their derivatives.Studying nonlinear partial differential equations(NPDEs)has become an interesting topic for mathematicians and physicists,which requires them to investigate the unknown function either exactly or numerically to achieve the solution interpretation.And therefore the physical meaning.Differential equations have significant applications in many fields.Soliton is one of the most important nonlinear phenomena;it is a type of nonlinear wave that emerges unscathed after colliding with another soliton.The Lie symmetry method is the ideal method for unfamiliar equations.This technique is based on reducing NPDEs to a smaller dimension for the sake of obtaining a simpler equation to solve.This method has been successfully investigated for the solitary wave solution for integrable and nonintegrable equations.The main goal of this study is to investigate new solitary wave solutions by using the Lie symmetry method for different differential,intego-differential,integrable,and nonintegrable equations in(2+1),(3+ 1),and(4+ 1)dimensions.For better comprehension of the solitary wave propagations,the graphical interpretation and the comparison of it with real natural phenomena has been realized.The construction of explicit solutions that admit a multitude of types of solitary waves depends on the adjustment of the arbitrary constants and functions has been achieved.In the end,The development of new Schrodinger equation and exploring its physical meaning has been reached.These ideas has been analyzed in this dissertation.