| The objectives of this work are threefold. Firstly, a fixed point the-orem is used to investigate a nonlinear generalized asset pricing model. Theanalytic properties of solutions for the model are established. Secondly, severaltypes of solutions for a generalized Boussinesq water equation are derived byusing the reduction of order of di?erential equations. Thirdly, a mathemati-cal technique based on auxiliary equation is developed to investigate physicalstructures for two nonlinear dispersive equations.Chapter 1 provides the derivation of the nonlinear integral equation for ageneralized asset pricing model, which yields an analytic price-dividend func-tion of one state variable. Under some assumptions and using the fixed pointtheorem, the existence and uniqueness of the price-dividend function, which isthe solution of the integral equation, are investigated. The analytic propertiesof the solution are in detail discussed in the complex plane.In chapter 2, the nonlinear variants of the generalized Boussinesq waterequation with positive and negative exponents are studied. The analytic ex-pressions of the compactons, solitons, solitary patterns and periodic solutionsfor the equations are obtained by using a technique based on the reduction of order of di?erential equations. It is shown that the nonlinear variants, or non-linear variants together with the wave numbers, directly lead to the qualitativechange in the physical structures of the solutions.In chapter 3, the auxiliary di?erential equation approach and the symboliccomputation system Maple are used to investigate two nonlinear dispersiveequations with variable coe?cients. Under certain circumstances, the exactsolutions to the equation are constructed in the form of semi-travelling wavesolutions. It is shown that the variable coe?cients of the derivative termsof the equation determine the physical structures of the semi-travelling wavesolutions.
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