Initial–boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method☆
Initial–boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method☆
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DOI：
10.1016/j.jde.2016.09.033
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发表时间：
2017-01
影响因子：
2.4
通讯作者：
Shou‐Fu Tian
中科院分区：
文献类型：
--
作者：
Shou‐Fu Tian
可积非线性微分方程的边值问题可以用Fokas方法进行分析。本文利用这种方法研究了有限区间上具有3× 3 Lax对的一般耦合非线性薛定谔方程的初边值问题.该解可以写成3× 3黎曼-希尔伯特问题的解。相应的跳跃矩阵用三个矩阵值谱函数s（k），S（k）和SL（k）明确表示。通过这种整体关系，分析了相应的广义Dirichlet到Neumann映射.有趣的是，当区间的长度趋于无穷大时，相关公式可化为在极限中的半直线上的边值问题的类似公式。证明了刻画Dirichlet到Neumann映射的公式与通过Gelfand-Levitan-Marchenko表示得到的类似公式是一致的。
Boundary value problems for integrable nonlinear differential equations can be analyzed via the Fokas method. In this paper, this method is employed in order to study initial–boundary value problems of the general coupled nonlinear Schrödinger equation formulated on the finite interval with 3× 3 Lax pairs. The solution can be written in terms of the solution of a 3× 3 Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions s (k), S (k), and S L (k). The associated general Dirichlet to Neumann map is also analyzed via the global relation. It is interesting that the relevant formulas can be reduced to the analogous formulas derived for boundary value problems formulated on the half-line in the limit when the length of the interval tends to infinity. It is shown that the formulas characterizing the Dirichlet to Neumann map coincide with the analogous formulas obtained via a Gelfand–Levitan–Marchenko representation.