非零边界条件下六次非线性薛定谔方程的Riemann-Hilbert问题
Riemann-Hilbert problem for the sextic nonlinear Schrödinger equation with non-zero boundary conditions
摘要：
考虑无穷远处具有非零边界条件的六次非线性薛定谔方程的矩阵Riemann-Hilbert问题.在分析谱问题之前，为了避免多值问题，我们引入了黎曼曲面和一致化坐标变量。基于一个新的复平面，对直接散射问题中Jost函数和散射矩阵的解析性、渐近性和对称性进行了详细的分析。然后，从直接散射变换的结果成功地建立了一个广义Riemann-Hilbert问题（RHP）。在逆散射问题中，我们分别讨论了单极点和双极点下的离散谱、留数条件、迹公式和θ条件，并进一步求解了广义RHP的解。最后，我们推导了无反射势时不同极点情况下的方程解。此外，通过选取适当的参数值，我们分析了孤子解的局域结构和动力学行为。
英文摘要：
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.
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