Exact Solutions To The Davey-StewartsonⅠ Equation

In this PhD thesis,we investigate the D avey-Stewartsonl(DSI)equation by employing the Hirota bilinear method combined with the Kadomtsev-Petviashvili(KP)hierarchy reduction method.General soliton solutions with vanishing and nonvanishing boundary conditions,periodic solutions,rational and semi-rational solutions consisting of soliton,periodic solution and rational solution are constructed in terms of determi-nants whose matrix elements have simple algebraic expressions.In chapter 1,we review the research background and the current status of the Hi-rota’direct method and the KP hierarchy reduction method in detail.The main works of this thesis are summarised.In chapter 2,first we construct tau functions to the bilinear equations of the two-component KP hierarchy,then derive general soliton solutions to the DSI equation with vanishing boundary condition by reducing the obtained tau functions.Finally,we in-vestigate the soliton interactions in the DSI equation,whose profiles are similar to the wave patterns occurring on two flat beaches filmed by Ablowitz and Baldwin.In chapter 3,we give a general form of the tau function to the bilinear equations of one-component KP hierarchy.We start from one type of the tau function to derive the general soliton solutions to the DSI equation with vanishing boundary condition-s,which are dark solitons.We discuss the dynamical behaviours of the dark soliton solutions.In chapter 4,starting from another type of tau functions of the single component KP hierarchy,two types of periodic solutions to the DSI equation are given.The first type of periodic solution includes breather and line breather.By taking a long wave limit of the first type of periodic solutions,the rational solutions and semi-rational solutions of first type are generated.The rational solutions have two different dynamic behaviours:line rogue wave and lump.The semi-rational solutions describe the interaction between line rogue waves,lumps and breathers.The second type of periodic solutions contains breather,quasi-periodic solution consisting of breather and soliton,and a new type of periodic solution.The new periodic solution behaves as static anti-dark soliton in(x,y)plane and its amplitude is periodic along t.Taking a long wave limit of the second type of periodic solutions,it produces the second type of semi-rational solution which includes soliton solutions,periodic solutions and rational solutions.In chapter 5,by introducing two differential operators to act on the tau functions of the on e-component KP hierarchy,the second type of rational solutions to DSI e-quations are construct.Comparing to the first type of rational solutions,the second type of rational solutions is expressed by different types of determinants,and possesses more high-order rogue wave patterns.Under particular parameter constraint condition-s,general high-order rogue wave solutions to the(1+1)-dimension NLS equation are obtained as reductions of the second type of the rational solutions of the DSI equation.