Rogue Wave Solution, Soliton Solution And Integrability Of The Nonlinear Models

Based on symbolic computation and by using the generalized Darboux transforma-tion (DT), classical DT, binary DT and symmetry theory, we study the rogue wave so-lution, soliton solution and integrability of some nonlinear models in nonlinear mathe-matical physics. A software package based on the symbolic computation software Maple platform of DT and rogue wave solution automatic deduction is developed. The main work and innovative points contain:Chapter 1:The background and research status of the rogue wave, soliton and inte-grable system, DT and symbolic computation are introduced, the main research results of this thesis are also illustrated.Chapter 2:The classical DT and soliton solutions of the 2+1 dimensional nonlinear models are studied. The classical DT for the 2+1 dimensional CDGKS equation and 2+1 dimensional mKdV equation are constructed, a general expression of N-soliton solution for the two equations is given, and the dynamic behaviors of the bright, dark and kink solitons are analyzed.Chapter 3:The generalized DT and rogue wave solutions for the nonlinear models which are related to 2×2 spectral problems are studied. The generalized DT for the AB system and Kundu-Eckhaus (KE) equation are constructed, and a unified Nth-order rogue wave solution for the two equations is derived. For AB system, the’four-peaky’shaped rogue wave is discovered. For KE equation, through the numerical computation, it is found that the higher-order nonlinear and Raman scattering terms only affect the spacial distributions of the rogue waves, and have no effects on their temporal distributions and amplitudes.Chapter 4:The generalized DT and rogue wave solutions for the nonlinear models which are related to 3 x 3 spectral problems are discussed. The generalized DT for the coupled Hirota equations, Manakov system and three wave resonant (TWR) equations are constructed, and a unified Nth-order rogue wave solution for the three different types of equations is presented. For coupled Hirota equations, three different kinds of nonlin-ear waves which include vector higher-order rogue waves, interactions between higher-order rogue waves and multi-dark-bright solitons, interactions between higher-order rogue waves and multi-breathers are discovered. For Manakov system, the interactional prop-erties between higher-order rogue waves and other nonlinear waves are discussed. For TWR equations, the classification of the composite higher-order rogue waves is investi-gated, and based on the numerical simulation method, the stability of higher-order rogue wave solutions are analyzed.Chapter 5:The integrability and soliton solutions of the variant-coefficients and dis-crete nonlinear models are discussed. The Lax pair and conjugate Lax pair for the 2+1 dimensional variant-coefficients Gardner equation are given, the binary DT is constructed, the N-soliton solution is obtained, and the dynamic behaviors of the bright and dark, reso-nant bright and dark solitons are analyzed. The Lie symmetry, general symmetry, conser-vation laws and soliton solutions for the four-fields Blaszak-Marciniak lattice equations are studied, the integrability of the models are proved from the view of symmetry.Chapter 6:The automatic deduction software package is developed. On the Maple platform, the DT and rogue wave solution software package DTRWSI is developed, and its practicality and effectiveness are verifyed through a lot of examples.Chapter 7:The summary of the whole thesis is given, and the prospect of the future work is presented.