Dynamics And Multiple Soliton Solutions Of Nonlinear Dispersive Wave Equations Regarding Fluid Mechanics And Heat Transfer

Investigation of nonlinear dispersive waves is one of the important ob-servable phenomena in nature.Waves propagate through a material medium(solid,liquid,or gas)at a wave speed depends on the elastic and inertial properties of that medium.The study is related to the fluid dynamics and convective heat transfer as well.Part of this thesis deals with the propagation of surface gravity waves on a flat water-air inter-face under the gravity force and surface tension effects.Water waves are the most fascinating and varied parameter of the area of wave motions.Mathematical as well as physical problems deal with water waves and their breaking on beaches.Another part focuses on the propagation of surface waves in a horizontal fluid layer opened to air under gravity effect and vertical temperature gradient.Class of nonlinear PDEs can be reduced to a more tractable single non-linear equation via the lowest order of the perturbed reductive technique.Three-dimensional nonlinear and dispersive waves of shallow-water(SW)model throughout a finite depth of fluid under the effect of surface tension and gravity force was investigated in an attempt to derive the Davey-Stewartson(DS)system for the modulation of 2-D harmonic waves.The linear analysis of the model was studied to construct the dispersion prop-erties.Likewise,conservation laws of DS equations were constructed and discussed in details.We applied the Painleve analysis not only to investi-gate the integrability of DS equations but also to construct the Backlund transformation via the truncation Painleve expansion.By employing the Backlund transformation,Hamiltonian approach and the improved(G'/G)-expansion method to DS equations,new traveling solitary and kink wave solutions were obtained.While using the simplest equation method,we established exact trav-eling wave solutions and a general form of multiple-soliton solution of DS-model.Results obtained that the more increase in the Ursell param-eter,the more decrease in waves' amplitudes.While the wave profile has a similar trend with time.It is also revealed that the consistency of the results with conservation of potential energy increases with increas-ing Ursell parameter.In addition,in the Hamiltonian approach,it was found that the amplitude of the waves increases with raising the energy constant.Moreover,the phase plane method was applied to the result-ing nonlinear first-order differential equations of DS model to reveal its stability.The surface waves of a horizontal fluid layer opened to air under the gravity field and vertical temperature gradient effects were investigated.The governing equations which describe the problem were presented and converted to a nonlinear evolution equation which is the perturbed Korteweg-de Vries(pKdV)equation.This equation was studied to exam-ine the evolution of long surface waves in a convective fluid.Dispersion relation and its concepts for pKdV equation were constructed and dis-cussed.The Painleve analysis was applied to check the integrability of pKdV equation and establish the Backlund transformation form of that equation.In addition,new traveling wave solutions and multiple-soliton solutions of pKdV equation were found using the Backlund transforma-tion,simplest equation method with Bernoulli,Riccati and using Burg-ers equations as simplest equations,and the new form of factorization method.The final part of the thesis is concerned with studying the coupled cubic-quintic complex Ginzburg-Landau(cc-qcGL)equations.These equa-tions can be used to describe the nonlinear evolution of slowly varying envelopes of periodic spatial-temporal patterns in a convective binary fluid.Dispersion relation of cc-qcGL equations and its properties were constructed.The Painleve analysis was used not only to check the inte-grability of cc-qcGL equations but also to establish the Backlund trans-formation form.In addition,new traveling wave solutions and a general form of multiple-soliton solutions of cc-qcGL equations were obtained via the Backlund transformation and simplest equation methods that were used in the latter two models.The solutions of all our models were investigated by using various analytical methods and illustrated in sev-eral 3-D and 2-D graphics showing the shock and solitary waves nature in the flow.