New Integrable Models With N-peakons And Algebro-geometric Constructions To The Solution Of Soliton Equations

The thesis can be mainly divided into two parts. First, with the help of Lenardrecursion equations and the zero-curvature equation, we derive three different soliton hi-erarchies, which are associated with one4×4and two3×3matrix spectral problemsrespectively. Moreover, generalized Hamiltonian structure and infinite conservation lawsof the hierarchy and soliton equation are established; On the other hand, We present theexplicit solutions of soliton equations above. In chapter two, we derive peaked solutionof some CH type equations. And based on the theory of trigonal curve and the knowledgeof algebraic geometry, we construct the algebro-geometric solutions of two hierarchies ofsoliton equations associated with two different3×3matrix spectral problems in chapterfour and five, respectively.In chapter two, by introducing the negative flow, we derive three CH type equations,two of which admit N-peakons.Algebro-geometric solutions of soliton equations reveal inherent structure mecha-nism of solutions, and describe the quasi-periodic behavior of nonlinear phenomenon.Chapter three mainly concentrates on Riemann surface and Theta function, and the con-cepts, lemmas and theorems do a good favor to understand the trigonal curve. In chapterfour and five, we propose a systemic method to construct the trigonal curve, and thenintroduce the appropriate Baker-Akhiezer function, meromorphic function and ellipticvariables on the three-sheeted Riemann surface, from which soliton equations are decom-posed into the system of solvable Dubrovin-type ordinary differential equations. Further-more, in accordance with the properties of the zeros and singularities of the meromor-phic function and Baker-Akhiezer function, we get their Riemann theta function repre-sentations by means of the second and third Abel differentials, Riemann theorem andRiemann-Roch theorem. Combining the Riemann theta function representations of the meromorphic function and the Baker-Akhiezer function with their asymptotic properties,we finally obtain the algebro-geometric solutions of soliton equations.