The unified method: I. Nonlinearizable problems on the half-line
The unified method: I. Nonlinearizable problems on the half-line
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DOI：
10.1088/1751-8113/45/19/195201
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发表时间：
2012-05-18
影响因子：
2.1
通讯作者：
Lenells, J.
中科院分区：
文献类型：
--
作者：
Fokas, A. S.;Lenells, J.
半直线上可积非线性发展偏微分方程的边值问题可以用统一方法来分析，该方法已在文献中广泛使用。这种一般方法的实施，这类特殊的问题产生的解决方案的唯一解决方案的矩阵黎曼-希尔伯特问题制定在复杂的k-平面（傅立叶平面），其中有一个跳跃矩阵与明确的（x，t）-依赖涉及四个标量函数的k，称为谱函数。其中两个函数依赖于初始数据，而另外两个依赖于所有边界值。新方法的最困难的步骤是后两个谱函数的特性在给定的初始和边界数据，即消除未知的边界值。对于某些边界条件，称为线性化，这可以简单地使用代数操作来实现。在这里，我们提出了一个有效的表征谱函数在给定的初始和边界数据的一般情况下的非线性边界条件。这种表征是基于所谓的全球关系的分析，从全球关系通过一定的变换离开相关的线性化PDE不变的色散关系和计算的本征函数定义的相关谱函数的大k渐近方程的分析。
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving four scalar functions of k, called the spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant and on the computation of the large k asymptotics of the eigenfunctions defining the relevant spectral functions.