A partial differential equation is a mathematical equation derived from the models of physical application and engineering fields.It comprises two or more independent variables,an unknown function,and partial derivatives of the unknown function concerning the independent variables.Most physical phenomena like heat flow and wave propagation,plasma physics,quantum mechanics,and many other models in science fields are well described by the partial differential equation.During the past decade,the classical partial differential equations such as Laplace equation,Helmholtz equation,and Poisson equation were created with broad utility in mechanical engineering and theoretical physics.Later on,most of them have been derived due to the progress of modern science and technology to describe the physical phenomena in real-world problems;for example,the KleinGordon equation,the Wave equation,the Schr¨odinger equation,the Heat equation,and the Maxwell’s equation.These are all linear partial differential equations.Besides,nonlinear partial differential equations are extensively used as models to express many important natural phenomena,which arise from many fields of mathematics and other branches of science such as physics as well as finance,mechanics,and material science;For example,the Hamilton-Jacobi equation from classical mechanics,Nonlinear Schr¨odinger equation from quantum mechanics,MongeAmp`ere equation from differential geometry,Inviscid Burger’s equation from fluid mechanics,and Kadomtsev-Petviashvili equation from nonlinear wave motion and so on.Exact solutions of nonlinear partial differential equations represent a major part in nonlinear science and engineering,which have been studied extensively by many scientists and researchers.In the studies of solitary wave theory,a variety of methods have been used such as the Hirota method,nonlocal symmetry method,the Test Function method,the B¨acklund transformation,the inverse scattering transformation,the Darboux transformation,source generation procedure,the Wronskian technique,and so on.More and more methods have been introduced to construct various exact solutions in different fields of sciences,which help us to understand the physical mechanism of it.This thesis is concerned with solutions to nonlinear partial differential equations.In particular,we examine three specific equations;a new generalized（2+1）-dimensional model,a 2+1 dimensional Ablowitz-Kaup-Newell-Segur equation,and a（3+1）-dimensional nonlinear evolution equation.The main objective is to give more emphasis on the Hirota bilinear method to find solitary wave solutions with an emphasis on the Kadomtsev-Petviashvili hierarchy reduction method,which leads to getting rational and more interaction solutions for understanding the dynamical behavior of nonlinear evolution equations.The thesis contains six-chapter arranged as followsChapter 1: In this chapter,we provide a brief introduction to the nonlinear partial differential equation,exact solutions,and their role in practical life,as well as the objective and methodology used in this study.Chapter 2: In this chapter,we give a summary of some methods that are used to solve nonlinear partial differential equations such as,the inverse scattering method,the Darboux transformation,the B¨acklund transformation,and the Hirota bilinear method.With its history,current status of science,and achievement.Chapter 3: In this chapter,we introduce the new generalized（2+1）-dimensional model,derived from the Kadomtsev-Petviashvili equation regarded as a system of nonlinear evolution partial differential equations.We construct N-soliton solutions of the integrable system with the use of the Hirota bilinear method and the KadomtsevPetviashvili hierarchy reduction method.Mentioning the case of s =-1 in the linear differential operators,L1 and L2,and a specific set of parameters,we obtain two dark solitons,mixed solutions comprising of soliton-type and periodic waves solution.We derive one and two rogue wave solutions expressed in terms of rational functions on the basis of the specific definition of the matrix elements.The fundamental rogue waves are shown to be line rogue waves,which is different from the feature of the traveling line solitons of the soliton equations.Chapter 4: In this chapter,the 2+1 dimensional Ablowitz-Kaup-NewellSegur equation is presented,which derived from the potential Boiti-Leon-MannaPempinelli equation.We construct the rational solutions consisting of rogue wave and lump soliton solutions via the bilinear method and ansatz technique,where we discuss the condition of guaranteeing the lump solutions’ positivity and analyticity.The set of an exponential function with the quadratic one representing rationalexponential solutions is described,where the interaction consisting of one lump and one soliton with fission and fusion phenomena.The interaction of line rogue wave and soliton solution,which is inelastic,is the second form of interaction.The homoclinic breather-wave solution is obtained by the use of the extended homoclinic test approach.The attributes of these different solutions are represented and graphically illustrated.Chapter 5: In this chapter,we create a new（3+1）-dimensional nonlinear evolution equation,via the generalized bilinear operators based on prime number p =3.The one,two lump,and breather-type periodic soliton solutions are derived by maple symbolic calculation,where the state of lump solution’ positivity and analyticity are considered.By combining multiexponential or trigonometric sine and cosine functions with a quadratic function,we construct the interaction solutions between the lump and multi-kink soliton,the interaction between the lump and breather-type periodic soliton.Furthermore,using the ansatz method,new interaction solutions are obtained between a lump,periodic wave,and one,two,even three-kink solitons.Finally,the features of these different solutions are exhibited and graphically illustrated.The last chapter summarizes the main results of this study and future study directions.