Traveling Wave Solutions And Stability Analysis Of Nonlinear Partial Differential Equations

Many real-world phenomena are modeled by nonlinear partial differential equations(NLPDEs).In this thesis,we study the nonlinear traveling wave solutions of Benjamin-Ono equations,complex hyperbolic Schrodinger equation,Nizhnik-Novikov-Vesselov(NNV)equation,Caudrey-Dodd-Gibbon(CDG)equation,Jaulent-Miodek(UM)equation,and the Maccari system by extending traditional theoretical methods.These NLPDEs have sev-eral applications in various fields,which include fluid dynamics,fiber,and geometrical optics,water wave dynamics,and oceanic sciences.The extended simple equation method(ESEM)and the generalized elliptic equation rational expansion method(GEEREM)are applied to extract some new solutions for these NLPDEs.We construct soliton and solitary wave solutions of first and second order Benjamin-Ono equations via ESEM,which has critical applications in fluid dynamics.Further,the con-ditions for the formation of kink,bright and dark solitons are also given on parameters for the Benjamin-Ono equations.The moments of obtained solutions are graphically pre-sented to demonstrate the physical phenomena of the models.The stability analysis of the two models is discussed using modulation instability to establish the sensitivity of the sep-tic nonlinearity in the models.The stability analysis confirms that all obtained solutions are analytical and stable in a given condition.We also apply ESEM to study the dispersive traveling wave solutions of(2+1)-dimensional Nizhnik-Novikov-Vesselov(NNV),Caudrey-Dodd-Gibbon(CDG)and Jaulent-Miodek(JM)hierarchy nonlinear equations.A set of exact,periodic,and soliton solutions is ob-tained for these models confirming the effectiveness of the proposed method.These models are essential for several application areas,especially in the field of mathematical physics.Significant figures are used to illustrate the physical properties of some obtained results.A comparison between the obtained solutions and established results in the literature is also given.Stability analysis on obtained solutions revealed that all the steady-state solutions of the models are stable against wave number perturbations.We obtained new complex solutions for the complex hyperbolic Schrodinger equation are constructed using GEEREM.The acquired new solutions are complex rational,trigonomet-ric,and hyperbolic solution expressed by kink solitons,bright solitons,singular solitons,and periodic solutions.The moments of obtained solutions are graphically illustrated to show the physical properties of this model.The conditions for the stability of obtained re-sults are also discussed using modulation instability analysis.Fractional traveling wave so-lutions for complex(2+ 1)-dimensional Maccari dynamical system and Schrodinger equa-tion are obtained using ESEM.These solutions are in the form of exact,periodic,and soliton solutions in fractional forms.These achieved fractional results have a variety of applica-tions in fiber optics,quantum mechanics,oceanic and optical sciences.The extracted solutions and computational work in this thesis verify that the proposed methods are simple,reduces computational complexity,provides exact solutions in gen-eral type,and can be utilized effectively in various applications in mathematical physics.The obtained results in this study provide a theoretical basis for practical applications of these NLPDE models in wave dynamics control problems.