Exactly Analytic Solutions Of Some Discrete And Continuous Soliton Equations

This thesis investigates three fully discrete integrable models and a (2+1)-dimensional derivative Schwarzian KdV equation. It consists chiefly of three parts:Firstly, the H1model and the special H3model in the Adler-Bobenko-Suris list, i.e. the lattice potential KdV equation and the lattice potential MKdV equation are discussed. New Lax pairs of them are given, by which integrable symplectic maps are constructed through a non-linearization procedure. Resorting to these maps and the permutability of the discrete phase flows sharing the same Liouville platform, finite genus solutions of the H1model as well as the special H3model are calculated. Besides, a special solution expressed with theta function of the lattice KdV equation with Nijhoff’s discretization is also arrived.Secondly, the Veselov’s discrete Neumann system is derived through non-linearization of a discrete spectral problem. Based on the commutation relation between the Lax matrix and the Darboux matrix with finite genus potentials, a special solution is obtained with the help of the Baker-Akhiezer-Kriechever function.Finally, the zero-curvature expressions of two (1+1)-dimensional derivative Schwarzian KdV equations are constructed. According to there compatibility, a (2+1)-dimensional derivative Schwarzian KdV equation is got.Through non-linearization of the Lax pairs and straightening out of the Hamilton phase flows, the special solutions with finite pa-rameters and the Abel-Jacobi solutions are calculated.