Investigations On Related Problems Of Exact Solutions Of Non-Linear Evolution Equation And Integrable System

The major contents in this paper include: the exact solutions of the non-linear evolution equation and the integrable system. In the second chapter, abundant new exact solutions of the (2+1)-dimensional Sine-Gordon equation are obtained by the generalized Jacobi elliptic function method. In the third chapter, firstly several (2+1)-dimensional isospectral problems are established based on the loopalgebra A₁. As its application, (2+1)-dimensional Dirac hierarchy is given.Secondly, a matrix loop algebra is constructed based on the loop algebra A₂ anda new isospectral problem is designed. Using Tu scheme, a Liouville integrable system is obtained, which possesses Hamiltonian structure and can be reduced to NLS-MKdV equation hierarchy. In the fourth chapter, a new matrix isospectral problem is constructed, discrete positive hierarchy and negative hierarchy are worked out respectively via different spectral parameter, which possess Hamiltonian structure and are integrable in Liouville sense. In the fifth chapter,firstly, vector loop algebra G_m is constructed and several isospectralproblems can be established from it. As its application, the integrable coupling of the multi-component Dirac hierarchy is worked out. It followed that the Hamiltonian structure of the above system is presented by taking advantage of the extending trace identity—quadratic-form identity. Secondly, the integrable coupling of the (2+1)-dimensional Dirac hierarchy is obtained by use of theexpanding loop algebra F of the loop algebra A₁. Finally, the extending loop algebra A_6m of the loop algebra A_3m is constructed by the method of semi-directsum of Lie algebra. Using A_6M the integrable coupling of the multi-componentNLS-MKdV equation hierarchy is given. In the sixth chapter, a proper Darboux matrix is constructed to the spectral problem proposed in chapter 4. By use of Darboux transformation method the single solution of the obtained soliton equation in chapter 4 is worked out and the figure of the single solution is presented via to Matlab. In the seventh chapter, the infinite conservation law of the obtained equation hierarchy in chapter 4 by use of direct method.