Research On Riemann-hilbert Problems And Characteristics Of Analytical Solutions For Several Nonlinear Evolution Equations

It is well known that the nonlinear development equations can be used to explain some physical phenomena in real life and some applications in engineering.In this work,we mainly use Darboux transformation method,Hirota bilinear method to analyse the soliton solutions,breather wave solutions,rogue wave solutions and other nonlinear wave solutions for several nonlinear evolution equations.At the same time,we also discuss the application of Riemann-Hilbert method in the field of integrable systems,containing studying the multi-soliton solution of nonlinear Schr¨odinger equation and its long time asymptotic behavior.In the first chapter of this thesis,we mainly introduce the history and research situation of soliton theory,the relevant methods for solving nonlinear differential equations and the application of Riemann-Hilbert method in the initial value problem.In chapter 2,we consider the fully parity-time(PT)symmetric(2+1)dimensional nonlocal nonlinear Schr¨odinger equation,the generalized(2+1)dimensional NNV equation,and(3+1)dimensional BLMP equation.By developing the Hirota bilinear method,we derive the -soliton solution of these equations for the first time.Then the long-wave limit expansion of the -soliton solution is carried out,we construct its nonsingular rational solutions and semi-rational solutions.In addition,we use the relevant mathematical software to simulate and analyze the physical phenomena of the relevant solutions.In chapter 3,by extending Darboux transformation method,we initially study the high order rogue wave and breather wave solutions of the coupled nonlinear Schr¨odinger equations with alternate signs of nonlinearities.By adjusting the spectrum parameters,we can obtain the solution of the temporally periodic breather wave and the temporally periodic breather wave.The vector rogue wave solutions include the bright one-peak-two-valleys rogue wave and the bright rogue wave without valleys.Additionally,we successfully show different types of the distributions for the secondorder vector rogue waves.The existence condition for the rogue waves is discussed.For the coupled nonlinear Schr¨odinger equations with alternate signs of nonlinearities,we get an interesting rule that there are rogue wave solutions in the case of the baseband modulation instability(MI).At last,we also study the breather waves and rogue waves of a high order coupled nonlinear schr¨odinger equation by the generalized Darboux transformation.In chapter 4,the dn-periodic rogue wave of the Hirota equation is studied for the first time.We take the jacobian elliptic function dn as the seed solution.Interestingly,the seed solution presented modulation instability under the long-wave perturbation.Through nonlinearization of the Lax pair for Hirota equation,the corresponding periodic eigenfunctions are successfully obtained.Based on these periodic eigenfunctions,we further construct Lax pairs equation solution.Combining the Darboux transform of Hirota equation,we finally succeeded in obtaining the rogue wave solution under the periodic wave background for the equation.In chapter 5,the integrable three-component coupled nonlinear Schr¨odinger equations are researched.By developing the Riemann-Hilbert method,the direct scattering and inverse scattering problems of the three-component coupled nonlinear Schr¨odinger equation are analyzed for the first time,and the multi-soliton solutions of the equation are derived successfully.In addition,the dynamical behavior of these solitons is discussed by image simulation.Especially in the analysis of the collision behavior of two solitons,we find a new double soliton collision phenomenon,which is unique and not common in integrable systems.In chapter 6,we extend a 3×3 matrix value Riemann-Hilbert problem and successfully solve the initial value problem for the coupled cubic-quintic nonlinear Schr¨odinger equations.Using the unique solutions of the obtained 3×3 Riemann-Hilbert problem to represent the solutions for the coupled cubic-quintic nonlinear Schr¨odinger equations,we first derived the explicit long-time asymptotics for the coupled cubic-quintic nonlinear Schr¨odinger equations in the pure radiation case according to the approach of nonlinear steepest descent pioneered by Deift and Zhou.In chapter 7,we discuss the(2+1)dimensional BKP equation based on the bilinear method.For the first time,we construct the generalized lump solutions,lumpoff solutions and special rogue wave solutions for the equation,and point out that the rogue wave is predictable,which is a new and very interesting phenomenon.Then,with the help of the extended homoclinic test method,we studied the breather wave and rogue wave of the generalized(2+1)dimensional CDGKS equation.Finally,we successfully derive the bright soliton and dark soliton solutions of nonlinear schr¨odinger equation with high order odd and even terms by combining the method of undetermined coefficient and the method of symbolic computation.The propagation characteristics of these nonlinear waves are also simulated by modern scientific software.The last chapter summarizes the whole paper and prospects the future research work.