A Study Of Some Nonlinear Water Wave Problems Using Homotopy Analysis Method

Nonlinear water wave theory is a hot research topic in the field of naval architecture and ocean engineering hydrodynamics.Although the history of nonlinear water wave study is long,a lot of problems in nonlinear water waves field remain open today,due to their difficulties.As the population of world increase,the resource of land can not meet the needs of people.Now the ocean is becoming the resource base and the second living space for human being.Exploitation and utilization of sea resources depend on development of ocean engineering.The scale of sea exploitation and utilization is becoming much more complex and bigger,so that exploitation of oil and gas in port,seashore is employed in deep water.The platforms and ships are required to endure strong nonlinear water wave loads in practical cases.The energy of these strong nonlinear water waves is concentrated, of great damage.The study of nonlinear waves has wide applications in ocean engineering.Among the methods for solving nonlinear problems are analytic method,numerical method and experiment.Nowadays as the computing technology has highly developed, numerical simulation is main technique for solving nonlinear problems.Numerical simulation can solve nonlinear problems with complex computational fields,while analytic method can be merely applied to nonlinear problems with simple computational fields.However,discontinuity often occur in numerical simulation,so it will cost lots of efforts and time for computation.On the other hand,in case of singularity and multiple solutions,numerical simulation will be invalid.Above all,analytic method has the following advantages:Analytic solutions can be used as a test of numerical solutions;analytic solution can reveal the properties of different variables,and it is convenient for parameter optimization.By means of perturbation techniques,a lot of important properties and interesting phenomena of non-linear problems have been revealed.Perturbation techniques are based on the existence of a small/large parameter or variable.Obviously,the existence of perturbation quantities is a cornerstone of perturbation teclmiques.Perturbation method is valid for weak nonlinear problems.For strong nonlinear problems,we can not get the convergent solutions.In recent years,Homotopy analysis method has been proposed,and it can control convergence region and adjust convergence rate.In this thesis,we apply the homotopy analysis method to solving nonlinear water waves.We proposed a new method for differential-difference equations based on Homotopy analysis method.We called it differential-difference equation-Homotopy analysis method(DDE-HAM).We study the initial guess in homotopy analysis method.And apply it the nonlinear shallow water,that is,one-dimensional,two-dimensional,threedimensional shallow water,we apply it to solve nonlinear Schr(o|¨)dinger equation.Such a kind of explicit,analytic solutions are useful for analyzing periodic wave groups and envelope solitary gravity waves.The work of this thesis shows the wide applicability of Homotopy analysis method in science and engineeing.Homotopy analysis method has the following advantages, such as it is valid even if a given non-linear problem does not contain any small/large parameters at all,it itself can provide us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary.The work shows that Homotopy analysis method is more efficient and effective analytic methods in solving nonlinear problems.Especially,some of the results indicates that HAM is a useful tool for those nonlinear problems with analytic solutions as a test of the results by numerical methods or experimental methods.All these examples given in this thesis might be helpful to keep us an open mind for solving nonlinear problems in ocean engineering.