Initial-boundary value problem for integrable nonlinear evolution equations with $3\times 3$ Lax pairs on the interval
Initial-boundary value problem for integrable nonlinear evolution equations with $3\times 3$ Lax pairs on the interval
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发表时间：
2015-09
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通讯作者：
Jian Xu;E. Fan
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作者：
Jian Xu;E. Fan
本文给出了一种分析Lax对涉及3 × 3矩阵的可积方程在有限区间（0 × L，L为正常数）上的初边值问题的方法.可积非线性发展偏微分方程的边值问题可以用由Fokas提出并由他和他的合作者发展的统一方法来分析。在本文中，我们表明，该解决方案可以表示为一个$3\times 3$ Riemann-Hilbert问题的解决方案。相应的跳变矩阵以三个矩阵值谱函数s（k），S（k）和S_L（k）的形式给出，而这三个矩阵值谱函数又分别由初值，x=0的边界值和x=L的边界值定义.然而，这些谱函数不是独立的，它们满足整体关系。在这里，我们表明，在给定的初始和边界数据的未知边界值的表征明确描述的非线性演化PDE定义的间隔。此外，我们表明，在极限时的间隔的长度趋于infity，有关公式减少到类似的公式的情况下，制定的半线上的边值问题。
We present an approach for analyzing initial-boundary value problems which is formulated on the finite interval ($0\le x\le L$, where $L$ is a positive constant) for integrable equations whose Lax pairs involve $3\times 3$ matrices. Boundary value problems for integrable nonlinear evolution PDEs can be analyzed by the unified method introduced by Fokas and developed by him and his collaborators. In this paper, we show that the solution can be expressed in terms of the solution of a $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions $s(k)$,$S(k)$ and $S_L(k)$, which in turn are defined in terms of the initial values, boundary values at $x=0$ and boundary values at $x=L$, respectively. However, these spectral functions are not independent, they satisfy a global relation. Here, we show that the characterization of the unknown boundary values in terms of the given initial and boundary data is explicitly described for a nonlinear evolution PDE defined on the interval. Also, we show that in the limit when the length of the interval tends to infity, the relevant formulas reduce to the analogous formulas obtained for the case of boundary value problems formulated on the half-line.