Research On Inverse Scattering Transformation,Riemann-Hilbert Problem And Analytical Solutions Of Some Nonlinear Evolution Equations

It is well-known that the nonlinear differential equations can be used to explain some physical phenomena in real life and some applications in biology and chemistry.In this work,on the basis of the Riemann-Hilbert method,initial value problems of several types of nonlinear integrable equations are investigated.For different types of initial value conditions,the exact solutions of nonlinear evolution equations are successfully constructed by referring to the extended Riemann-Hilbert method,and some new and interesting phenomena are obtained.Besides,a hierarchy of nonlocal nonlinear evolution equations is studied via extending the(?)-dressing method.In Chapter 1,the development process of soliton theory and the relevant methods for solving nonlinear differential equations are introduced.Meanwhile,the history of the Riemann-Hilbert method and its research status in integrable systems are also briefly described.In Chapter 2,different methods are developed to study the exact solutions of nonlinear differential equations.We first construct the exact solutions of(3+1)-dimensional nonlinear evolution equation via popularizing the Hirota bilinear method and discover some interesting phenomena,including the predictable special rogue wave.Next,the Riemann-Hilbert method is extended to study the higher order nonlinear Schr ¨odingerMaxwell-Bloch(HNLS-MB)equations with high-order matrix spectrum problem.Under the condition that the initial value belongs to the Schwartz space,the multi-soliton solutions of the HNLS-MB equations are given for the first time.Also,the structures and the propagation characteristics of the solutions of the HNLS-MB equations are analyzed.In Chapter 3,under the condition that the initial value belongs to the Schwartz space,the initial value problem of the higher-order dispersive nonlinear Schr ¨odinger equation and variable-coefficient nonlinear Schr ¨odinger equation are studied via extending the Riemann-Hilbert method.For both cases of single and double poles,the constructed Riemann-Hilbert problems corresponding to the two equations are solved,and the formulae of -soliton solutions are displayed.Additionally,for the variablecoefficient nonlinear Schr ¨odinger equation,via adjusting the coefficients of the equation,some new and interesting phenomena are analyzed graphically.In Chapter 4,the Riemann-Hilbert method is extended to construct the RiemannHilbert problem with arbitrary-order poles corresponding to the variable-coefficient fifth-order nonlinear Schr ¨odinger equation under the condition that the initial value belongs to the Schwartz space.Furthermore,the explicit expressions of multi-soliton solutions of the equation are derived when the reflection coefficient possesses a highorder pole and multiple high-order poles.Besides,under multiple high-order pole conditions,some new and interesting phenomena of the solutions are displayed.In Chapter 5,on the basis of Riemann-Hilbert method,the the higher-order dispersive nonlinear Schr ¨odinger equation and variable-coefficient nonlinear Schr ¨odinger equation are investigated with the nonzero boundary conditions at the infinity,and the multi-soliton solutions are successfully derived,including breather solution.In the process of analysis,an appropriate Riemann surface and uniformization coordinate variable need to be introduced to deal with the double-valued functions.Then,on the complex plane of the modified spectral parameters,the Riemann-Hilbert problems corresponding to the two equations are constructed.Furthermore,for both cases of single and double poles,the explicit formulae of -soliton solutions are derived.Meanwhile,via evaluating the impact of the variable coefficient,some new and interesting phenomena of these solutions are analyzed graphically,and the influence law of the variable coefficient is obtained.In Chapter 6,the (?)-dressing method is employed to investigate a hierarchy of nonlocal nonlinear evolution equations and their soliton solutions.Starting from the matrix(?) problem,a hierarchy of nonlocal nonlinear evolution equations associated with 2 × 2 matrix problem,which contains the nonlocal extended modified Kortewegde Vries(em Kd V)equation,is derived via introducing appropriate recursive operator.Furthermore,via selecting a proper spectral transformation matrix,the -soliton solutions of the nonlocal em Kd V equation are constructed.In chapter 7,we make a brief summary of the research content of this work and prospect for further work.