New Exact Traveling Wave Solutions And Modulation Instability Of Nonlinear Partial Differential Equation

Nonlinear partial differential equations(NLPDEs)are used in multiple study areas to define significant phenomena.Exact NLPDEs solutions play an important role in the research of physics,applied mathematics and engineering including solid state physics and wave propagation phenomena.Current studies are underway to find new techniques to extract traveling wave solution for NLPDEs.In this research work,the Simple Equation Method(SEM)has been used to obtain the exact solution of Generalized Korteweg–de Vries(g Kd V)and the Sawada Kotera nonlinear partial differential equation.These methods are used as the trial condition,since the SEM satisfies the first order Bernoulli differential equation or the first order Riccati differential equation.Using SEM,a balance equation is derived.By means of the balanced equations,the study obtains the exact solutions for the two equations.The exact solution attained from the proposed method imply that the approach is simple to apply and computationally feasible.Modified Simple Equation Method(MSEM)for solving fractional partial differential equations is also proposed.The traveling wave solutions of the space-time fractional HirotaSatsuma coupled Kd V equation and space-time fractional(2+1)dimensional long wave short wave resonance interaction equations are successfully obtained.The performance of this method is reliable,direct and effective.Again,by implementing the strong Generalized Elliptic Equation Rational Expansion(GEERE)technique,this thesis has effectively built fresh solitonic and reasonable regular solutions for the Davey-Stewartson Equation(DSE).Some freshly obtained solitary wave solutions are provided graphically to display their physical characteristics.The study concludes that the modulation instability analytical expression is gaining,which demonstrates that all the solutions are accurate and stable.The effectiveness and simplicity of the suggested Generalized Elliptic Equation Rational Expansion(GEERE)technique shows that it can be applied to distinct kinds of separatenonlinear models in different nonlinear science fields.Also,in this thesis,the exact traveling wave solutions for the(2+1)dimensional GardnerKP equation is studied.(2+1)dimensional Gardner-KP equation plays an important role in some complicated situations and makes it very difficult for us to solve the equation.The(2+1)dimensional Gardner-KP is reduced to the ordinary differential equation which is easy to solve by means of a proper transformation.Making full use of the Modified Extended Direct Algebraic(MEDA)method,the study discovers a wide range of explicit and exact traveling wave solutions.These solutions discovered in this thesis will help us understand the phenomena described in all the equations employed in this study.Comparing the study in this thesis and the work in literature,is found that more exact solutions with soliton and periodic structures and the rational function solution in this work are more general than the rational solution in existing literature.The technique used in this research work,which can be computerized and enable us to conduct complex and tedious algebraic computations on a desktop,can be expanded to many other NLPDEs.