Study on the behavior of solutions in the zero dispersion limit of the nonlinear wave equations with the integral kernel

1.可精确解的本征值问题在逆散射方法下，求解了非局部非线性薛定谔(NLS)方程的线性本征值问题。与精确可解动力系统的关系从复平面3动力系统的观点出发，研究了非局域NLS方程的孤子极点和周期解的运动。调制理论分析了非局部NLS方程的周期解的调制问题。推导了波参数的调制方程，并进行了详细的求解。利用逆散射变换方法建立了非局部NLS方程的初值问题。导出了约斯特函数的方程组。孤子解的新公式本杰明-Ono方程的多孤子解是从一个非线性代数方程组导出的。这一结果是逆散射法方案从未预料到的新发现。

1. Exactly solvable eigenvalue problemThe linear eigenvalue problem is solved for the nonlocal nolonear Schrodinger (NLS) equation in the context of the inverse scattering method when the potential is the multisolison solution2. Relation to exactly solvable dynamical systemsThe motion of poles of the soliton and periodic solutions of the nonlocal NLS equation is investigated from the viewpoint of the dynamical system in the complex plane3. Modulation theoryThe modulation problem is analyzed for the periodic solution of the nonlocal NLS equation. The modulation equations for the wave parameters are derived and they are solved explicitly4. Formulation of the initial value problemThe initial value problem of the nonlocal NLS equation is formulated by means of the inverse scattering transform method. A system of equations are derived for the Jost functions5. New formula for the soliton solutionThe multisoliton solution of the Benjamin-Ono equation is derived from a system of nonlinear algebraic equations. This result is a new finding which is never anticipated from the scheme of the inverse scattering method