Localized Wave Solutions For Nonlinear Evolution Equations On Hermitian Symmetric Spaces

Utilizing the classical and generalized Darboux transformations,this thesis focuses on four multi-component nonlinear evolution equations on Hermitian symmetric spaces.We obtain the localized wave solutions on a constant background,such as solitons,breathers and rogue waves.We analyse the dynamics for these solutions through Mathematica software.The four equations include the Hermitian symmetric space AIII AB system and three systems on the Hermitian symmetric space CI,which are AB system,the derivative nonlinear Schr(?)dinger equation and Fokas–Lenells equation.In Chapter 2,we construct the Darboux transformation of the Hermitian symmetric space AIII AB system.We obtain the eye-shaped,four-peaked and four-petalled rogue waves.We display the interactions between rogue waves and solitons or breathers.We show the fundamental,triangular and ring patterns for higher-order rogue waves.In addition,we analyse the dynamical behaviours of these solutions.In Chapter 3,we investigate the Hermitian symmetric space CI AB system.The two systems in Chapter 2 and 3 share one common feature,which can be reduced to AB system in canonical form and are related to Ablowitz–Kaup–Newell–Segur spectral problem.The system in the previous chapter is the vector generalization of AB system and associated with 3×3 spectral problems.However,the system in this chapter is the matrix generalization of AB system and related to 4 × 4 spectral problems with symmetric potential matrices.Comparing with the previous chapter,there are two difficulties in this chapter.One is how to reduce the high computational complexity due to increasing the orders of the matrix spectral problems.The other is how to construct Darboux transformation to ensure the symmetry of the new potential matrices.To this end,we rewrite this system in matrix form and “package up” its spectral problems into 2 ×2 block form.Then 2 × 2 matrix becomes the basic element and we reduce the computational complexity.In order to preserve the symmetry of the new potential matrices,we give a restrictive condition of the spectral functions to construct the generalized Darboux transformation.We get the rich localized waves,such as four-peaked rogue wave,two-peaked rogue wave,eye-shaped rogue wave,breathers,triangular and fan-shaped higher-order rogue waves.We display that four-peaked rogue wave can degenerate into two-peaked rogue wave,and discuss the fusion and fission processes for two “truncated” breathers.In Chapter 4,we study the Hermitian symmetric space CI derivative nonlinear Schr(?)dinger equation.Unlike Chapter 3,the equation is associated with Kaup–Newell spectral problem,whose spectral structure is more complex.It results in considering the paired spectral parameters to construct Darboux transformation.Meanwhile,this equation is related to 4 × 4 spectral problems with the symmetric potential matrix.Based on the two points,we get Darboux transformation for the equation and analyse the fusion and fission dynamics of the first-order rogue wave.Moreover,we display the fundamental and triangular second-order rogue waves,and analyse the modulation instability for this equation.In Chapter 5,we discuss the Hermitian symmetric space CI Fokas–Lenells equation associated with Kaup–Newell spectral problem.The difference is that the two equations in Chapter 4 and 5 correspond to the positive member and the negative member of Kaup–Newell hierarchy.We construct the Darboux transformations for this equation and solve various waves,such as multi-hump soliton,kink soliton,kink breather,W-shaped soliton,eye-shaped rogue wave and onepeaked rogue wave.We show the fission and fusion of the first-order rogue wave.For the second-order rogue wave,we exhibit the fundamental and triangular patterns.