Initial-boundary value problems for the coupled nonlinear Schrodinger equation on the half-line
Initial-boundary value problems for the coupled nonlinear Schrodinger equation on the half-line
复制标题
半线上耦合非线性薛定谔方程的初边值问题
DOI：
10.1111/sapm.12088
复制
发表时间：
2015
影响因子：
2.7
通讯作者：
Junyi Zhu
中科院分区：
文献类型：
--
作者：
Xianguo Geng;Huan Liu;Junyi Zhu
通过 Fokas 方法研究了半线上耦合非线性薛定谔方程的初始边值问题。结果表明，该解可以用在复平面上表述的矩阵黎曼-希尔伯特问题的唯一解来表示，其跳跃矩阵是根据矩阵谱函数定义的，并且分别取决于初始数据和所有边界值。如果存在满足全局关系的谱函数，则可以证明上述黎曼-希尔伯特问题定义的函数求解耦合非线性薛定谔方程并且与规定的初始值和边界值一致。该方法实现中最具挑战性的问题是表征谱函数中出现的未知边界值。对于一类特定的边界条件，即所谓的可线性化边界条件，可以通过使用全局关系的代数操作来计算给定边界条件的谱函数。对于边界条件的一般情况，可以通过采用摄动展开来获得未知边界值的有效表征。
Initial‐boundary value problems for the coupled nonlinear Schrödinger equation on the half‐line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complexk‐plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so‐called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.