Research On Exact Solutions For Some Integrable Equations

This thesis concentrates on studying some nonlinear integrable equations in mathe-matical physics.By use of Riemann-Hilbert method and some ansatz algorithms,abundant exact solutions to the considered equations are constructed.First,in Chapter 1,the soliton theory and integrability as well as some classical approaches applied in this field are introduced briefly.Then,the main contents of this thesis are given.In Chapter 2,a fifth-order nonlinear Schrodinger equation related with 2×2 matrix spectral problem is first introduced.Under zero boundary condition at infinity,the spectral problem and time evolution equation are changed into the desired forms.Second,a matrix Riemann-Hilbert problem is constructed through analysis of the converted spectral prob-lem and some associated properties.Through reconstructing the potential and solving the Riemann-Hilbert problem under condition of no reflection,the expression of multi-soliton solutions of the fifth-order nonlinear Schrodinger equation is presented.Particularly,one-and two-soliton solutions are written out.In Chapter 3,the quintic nonlinear Schrodinger equation with 2×2 spectral problem is studied in the framework of Riemann-Hilbert problem.A new transformation is generated first through assuming that the potential decays to zero sufficiently fast at infinity,which helps to convert the given spectral problem and time evolution equation into the required forms.Second,a matrix Riemann-Hilbert problem is established on the real axis by anal-ysis of the transformed spectral problem and associated properties.By reconstructing the potential and solving the matrix Riemann-Hilbert problem in a special case(no reflection),the general expression of multi-soliton solutions of the quintic nonlinear Schrodinger equation are revealed.The one-and two-soliton solutions are also given.In a similar way as Chapters 2 and 3,in Chapter 4,the N-coupled Hirota equations related with(N+1)×(N+1)matrix spectral problem are studied under zero boundary condition at infinity.The spectral problem and time evolution equation are turned into the desired forms first on basis of decaying to zero sufficiently fast of potentials at infinity.Second,a matrix Riemann-Hilbert problem is established on the real axis by analysing the resulting spectral problem and some related properties.Then,the reconstruction formulae of potentials are revealed.Through solving the Riemann-Hilbert problem under constraint of no reflection,the general multi-soliton solutions of the N-coupled Hirota equations are generated.Finally,as a particular case,soliton solutions of the three-coupled Hirota equations are acquired.The Chapter 5 first focuses on the(3+1)-dimensional negative-order KdV equation and explores its a series of novel exact solutions by means of symbolic computation and the ansatz methods.Second,by employing symbolic computation and the generalized al-gorithms,the time-dependent Date-Jimbo-Kashiwara-Miwa equation in(2+1)-dimensions is investigated.Consequently,some novel solutions for this equation are obtained.Mean-while,some figures of the resulting solutions for both equations are made.The final chapter presents a summary of this thesis and the prospect of future research work.