Rational And Semi-rational Solutions In Two Types Of The Davey-Stewartson Equation

Integrable Davey-Stewartson(DS)equation is an important mathematical model in fluid mechanic,nonlinear optics and plasma physics,etc.According to water wave surface tension,it contains two types:DS? equation and DS? equation.In this thesis,first,we construct N-fold Darboux transformation of DS? equation,which is expressed in the form of determinant.Next,we prove that the Taylor expansion coefficients of an eigenfunction also satisfy the Lax pair equation of the DS equation for the same"seed" solution.These expansion coefficients are thus used to construct the Darboux transformation and the following results are obtained:1.For DS? equation with zero background,semi-rational solutions involving ratio-nal solution and soliton are obtained.As |t|??,these solutions can degenerate to the line soliton solution of the DS? equation.2.For DS? equation with plane wave background,three types of lump solution-s(bright lump,dark lump and four-petal lump),line rogue wave and their combinations are constructed.Among them,higher-order lump solutions describe typical interaction-s consisting of head-on collision with a ?/2 normal scattering angle.In addition,unlike two-line rogue wave whose maximum amplitude does not exceed 4|a|,in this thesis,the limit of the largest amplitude of first-order coexisting line rogue wave can arrive at 9|a|.Further,higher-order line rogue wave possesses richer parabola-shaped rogue wave configuration at certain time.3.For DS? equation with zero background,lump,rogue wave and their combina-tions are given.Among them,two dimensional(2D)rogue waves are obtained by the even-fold Darboux transformation.The obtained rogue waves,which are localized in both space and time,can be viewed as a 2D analogue of the Peregrine soliton and are thus natural candidates to describe oceanic rogue wave phenomena.4.For DS? equation with plane wave background,using Darboux transformation,lumps,line rogue waves and their combinations are constructed.Compared to lumps in DS? equation,this scattering process presents a novel feature,i.e.,the scattering angle is in(0,?).