论文标题
Riemann-Hilbert问题的六型非线性Schrödinger方程与非零边界条件
Riemann-Hilbert problem for the sextic nonlinear Schrödinger equation with non-zero boundary conditions
论文作者
论文摘要
我们考虑了无限度在无穷大的边界条件的六个非线性schrödinger方程的矩阵riemann-hilbert问题。在分析频谱问题之前，我们引入了Riemann表面和均匀化坐标变量，以避免多价问题。基于新的复杂平面，直接散射问题对JOST函数和散射矩阵的分析，渐近和对称性进行了详细分析。然后，从直接散射变换的结果中成功确定了广义的Riemann-Hilbert问题（RHP）。在逆散射问题中，我们分别在简单的极点和双极下讨论离散光谱，残基条件，痕量公式和theta条件，并进一步求解广义RHP的溶液。最后，我们得出方程式的解决方案对于不同的杆子的情况而没有反射潜力。此外，我们通过采用一些适当的参数值出现在解决方案中的参数值来分析所得孤子解的局部结构和动态行为。
We consider a matrix Riemann-Hilbert problem for the sextic nonlinear Schrödinger equation with a non-zero boundary conditions at infinity. Before analyzing the spectrum problem, we introduce a Riemann surface and uniformization coordinate variable in order to avoid multi-value problems. Based on a new complex plane, the direct scattering problem perform a detailed analysis of the analytical, asymptotic and symmetry properties of the Jost functions and the scattering matrix. Then, a generalized Riemann-Hilbert problem (RHP) is successfully established from the results of the direct scattering transform. In the inverse scattering problem, we discuss the discrete spectrum, residue condition, trace formula and theta condition under simple poles and double poles respectively, and further solve the solution of a generalized RHP. Finally, we derive the solution of the equation for the cases of different poles without reflection potential. In addition, we analyze the localized structures and dynamic behaviors of the resulting soliton solutions by taking some appropriate values of the parameters appeared in the solutions.
