| It is well known that nonlinear phenomena such as physical phenomena in practical and social problems can be explained by nonlinear evolution equations.In this work,based on the inverse scattering theory and using the Riemann-Hilbert(RH)method as a tool,the Cauchy problems of several types for nonlinear evolution equations under different initial conditions are studied.The exact solution is obtained through analysis,and the propagation behaviors are simulated with images.Meanwhile,combined with the nonlinear steepest descent method,the long-time asymptotic behavior of the solution for the nonlinear evolution equation is discussed.In Chapter 1,the development history of the soliton theory and the method of solving the nonlinear evolution equation are briefly introduced.Furthermore,this work gives the research progress of the Riemann-Hilbert method and its application in the field of integrable systems.In Chapter 2,we consider the inverse scattering transform and multi-soliton solutions of the sextic nonlinear Schr ¨odinger(s NLS)equation.The Jost functions of spectral problem are derived directly,and the scattering data with = 0 are obtained accordingly to analyze the symmetry and other related properties of the Jost functions.Then we make use of translation transformation to get the relation between potential and kernel,and recover potential according to Gel’fand-Levitan-Marchenko integral equations.Furthermore,the time evolution of scattering data is considered,on the basis of that,the multi-soliton solutions are derived.In addition,some solutions of the equation are analyzed and revealed its dynamic behavior via graphical analysis.In Chapter 3,we employ the Riemann-Hilbert problem to study integrable threecomponent coupled Hirota(tc CH)equations.Thus,we investigate the spectral properties of the equations with a Lax pair of a matrix of order 4 × 4 and derive a RH problem,the solution of which is used in constructing -soliton solutions.While considering the spatiotemporal evolution of scattering data,the symmetry of the spectral problem is exploited.Graphical examples show new phenomena in soliton collision,including localized structures and dynamic behaviors of one-and two-soliton solutions.In Chapter 4,we solve the generalized modified Korteweg-de Vries(gm Kd V)equation by the Riemann-Hilbert mehod when the initial data of the gm Kd V equation belongs to Schwarz space.In the direct scattering transform,the analytical and asymptotic properties related to the Jost functions and the scattering matrix are given.On the basis of the above results,the appropriate RH problem is constructed.By solving the RH problem,we obtain the exact solution of the gm Kd V equation in the case of no reflection potential when the scattering data (6()has simple poles and higher-order poles.Furthermore,the three special solution under different zero points are given and the phenomenon of their spread is described respectively.In Chapter 5,we consider a matrix Riemann-Hilbert problem for the sextic nonlinear Schr ¨odinger equation with a non-zero boundary conditions at infinity.Before analyzing the spectrum problem,we introduce a Riemann surface and uniformization coordinate variable in order to avoid multi-value problems.Based on a new complex plane,the direct scattering problem perform a detailed analysis of the analytical,asymptotic and symmetry properties of the Jost functions and the scattering matrix.Then,a generalized RH problem is successfully established from the results of the direct scattering transform.In the inverse scattering problem,we discuss the discrete spectrum,residue condition,trace formula and theta condition under simple poles and double poles respectively,and further solve the solution of a generalized RH problem.Finally,we derive the solution of the equation for the cases of different poles without reflection potential.In addition,we analyze the localized structures and dynamic behaviors of the resulting soliton solutions by taking some appropriate values of the parameters appeared in the solutions.In Chapter 6,we investigate the long-time asymptotic behavior for the WadatiKonno-Ichikawa(WKI)equation with initial data belonging to Schwartz space at infinity by the nonlinear steepest descent method proposed by Deift and Zhou for the oscillatory Riemann-Hilbert problem.Based on the initial value condition,the original Riemann-Hilbert problem is constructed,which is related to the Wadati-KonnoIchikawa equation,so that the solution of the equation is transformed into the solution of the RH problem.After a series of deformation of the jump contour of the original RH problem,we finally obtain a solvable model RH problem.Furthermore,the long-time asymptotic solution of the WKI equation is represented by the solution of the parabolic cylindrical function.In chapter 7,we make a brief summary of the research content of the full text and prospects for future work. |
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