Solving The AB System With Two Different Boundary Conditions

The AB system is the basic integrable model to describe unstable baroclinic wave packets in geophysical fluids and propagation of mesoscale gravity flows in nonlinear optics.This thesis studies the AB system with two different boundary conditions.For constant boundary condition,we establish the Riemann-Hilbert problem of the AB system on the basis of the spectral analysis of Lax pair and the inverse scattering method.Then,the inverse problems are formulated and solved with the aid of the Riemann-Hilbert problem,from which the potentials can be reconstructed according to the asymptotic expansion of the sectional analytic function and the related symmetry relations.As an application,we obtain the multi-bright-dark soliton solutions to the AB system in the reflectionless case and discuss the dynamic behavior of elastic soliton collisions by choosing appropriate free parameters.For exponential boundary condition,Jost function solution with exponential boundary condition can be introduced based on the Lax pair of the system.Furthermore,the analytic and asymptotic behavior of Jost function are discussed.By solving the corresponding Riemann-Hilbert problem,-th order solution of the AB system is derived,from which one can obtain different type of breather solutions.