Darboux Transformations And Exact Solutions For Nonlinear Evolution Equations Associated With Higher-order Matrix Spectral Problems

In this thesis,the classical Darboux transformations and multi-fold Darboux transformations of five nonlinear evolution equations associated with higher-order matrix spectral problems are constructed based on the internal structure analysis of these spectral problems.On the basis of that,the generalized Darboux transformations are obtained by using Taylor series and the limit technique.Then,the systematic algorithm for solving the exact solutions of these nonlinear evolution equations related to higher-order matrix spectral problems is presented.Utilizing the classical Darboux transformations,the multi-fold Darboux transformations and the generalized Darboux transformations,the localized wave solutions of these nonlinear evolution equations are constructed and their dynamics are given respectively,which can be applied conveniently.The following is a brief description of each chapter:In Chapter 2,starting from a 3 × 3 matrix spectral problem,a second-order three-wave interaction system and its Lax pair are derived.By analyzing the structure of the spectral problem,the classical Darboux transformation of the second-order three-wave interaction system is constructed,and the proof of the first-order classical Darboux transformation is given from the space and time parts of the Lax pair respectively.Based on the classical Darboux transformation,the generalized Darboux transformation is constructed.As an application,rogue wave solutions of the second-order three-wave interaction system are obtained,including dark/bright soliton + rogue wave,eye-shaped rogue wave and triangular rogue wave.Then,we analyse the dynamical behaviours of these solutions.In Chapter 3,we study a two-component generalized Sasa-Satsuma equation.Comparing with the problem in Chapter 2,there are three difficulties in this chapter:(1)The order of the spectral problem becomes 4;(2)In addition to the basic Hermitian symmetry,there is another symmetric relation of the spectral problem,which brings an additional constraint;(3)The two potential functions of the spectral problem are complex and real respectively.These properties lead to considerable difficulties for the construction and proof of the Darboux transformation.The classical Darboux transformation and the generalized Darboux transformation are constructed based on characteristic analysis and symmetric relations of the spectral problem,from which the traveling wave solution,the breather solution under the zero background and the rogue wave solution under the non-zero background are obtained,respectively.In Chapter 4,resorting to a 3 × 3 matrix spectral problem,a new generalized complex m Kd V equation and its Lax pair are proposed.Comparing with the two equations in Chapter 2 and Chapter 3,the Lax pair of the generalized complex m Kd V equation is two 3 × 3 matrices and has only one potential.Therefore,it is very complicated and difficult to construct the Darboux transformation of the generalized complex m Kd V equation.To overcome this difficulty,the compatible Darboux matrix is found by using two gauge transformations,and the first-order classical Darboux transformation and higher-order Darboux transformation of the generalized complex m Kd V equation are derived.As an application,the Darboux transformation formula is used to discuss the cases of = 1, = 2and = 3 respectively.Finally,the expressions of one-soliton,two-soliton,threesoliton,first-order breather,second-order breather solutions are obtained,and their images are shown.In Chapter 5,we investigate the four-component Fokas-Lenells equation associated with a 4 × 4 matrix spectral problem,which corresponds to the negative member of the generalized Kaup-Newell hierarchy.In contrast with the above three chapters,this chapter uses the idea of “package up” to rewrite the 4 × 4matrix spectral problem into the 2 × 2 block form,then the order of the spectral problem is reduced.However,in the calculation,the elements of the Darboux matrix all become matrices,which makes calculations and proofs extremely difficult.The construction process of the first-order classical Darboux transformation and the concrete proof steps are given in detail.By iterations,we get the -fold Darboux transformation for the four-component Fokas-Lenells equation and the corresponding proof is given by induction.Resorting to Taylor series and the limit technique,we obtain the generalized Darboux transformation.By utilizing the Darboux transformations and Mathematica software,various exact solutions for the four-component Fokas-Lenells equation are obtained,including soliton,breather and rogue wave solutions.In Chapter 6,a new vector hybrid m Kd V equation is derived based on a higher-order matrix spectral problem.The structure of the spectral problem associated with this equation is complex and lacks symmetry,which makes it extremely difficult to construct Darboux transformations.In order to get rid of the dependence on the symmetry of the spectral problem,we use the Riccati equation associated with the Lax pair to construct the Darboux matrix of the integrable equation.Then we derive the -order classical Darboux transformation of the vector hybrid m Kd V equation and obtain soliton solutions,breather solutions and so on.