The generalized Dirichlet‐to‐Neumann map for certain nonlinear evolution PDEs
The generalized Dirichlet‐to‐Neumann map for certain nonlinear evolution PDEs
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DOI：
10.1002/cpa.20076
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发表时间：
2005-05
影响因子：
3
通讯作者：
A. S. Fokas
中科院分区：
文献类型：
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作者：
A. S. Fokas
设q（x,t）满足一个最高空间导数为n阶的非线性可积演化PDE。如果规定了初始条件，以及x = 0处的n个边界条件，那么这种PDE的半线上的初始边值问题至少是线性适定的，其中n偶n等于n/2， n奇n等于n - 1/2或n+1/2，取决于最高导数的符号。例如，对于非线性Schrödinger （NLS）和sin‐Gordon (sG), N = 1，而对于修改的Korteweg‐deVries (mKdV), N = 1或N = 2取决于三阶导数的符号。构造广义Dirichlet - to - Neumann映射意味着根据给定的初始条件和边界条件确定在x = 0处不作为边界条件规定的边界值。给出了用两个非线性ode系统的解来构造该映射的一般方法。这个公式意味着，对于聚焦NLS、sG和两个聚焦版本的mKdV，这个地图在时间上是全局的。这似乎是文献中第一次明确描述非线性偏微分方程的这种表征。这里还表明，对于边界条件的特定选择，上述映射可以线性化。©2005 Wiley期刊公司
Let q(x,t) satisfy a nonlinear integrable evolution PDE whose highest spatial derivative is of order n. An initial boundary value problem on the half‐line for such a PDE is at least linearly well‐posed if one prescribes initial conditions, as well as N boundary conditions at x = 0, where for n even N equals n/2 and for n odd, depending on the sign of the highest derivative, N equals either n−1/2 or n+1/2. For example, for the nonlinear Schrödinger (NLS) and the sine‐Gordon (sG), N = 1, while for the modified Korteweg‐deVries (mKdV) N = 1 or N = 2 depending on the sign of the third derivative. Constructing the generalized Dirichlet‐to‐Neumann map means determining those boundary values at x = 0 that are not prescribed as boundary conditions in terms of the given initial and boundary conditions. A general methodology is presented that constructs this map in terms of the solution of a system of two nonlinear ODEs. This formulation implies that for the focusing NLS, for the sG, and for the two focusing versions of the mKdV, this map is global in time. It appears that this is the first time in the literature that such a characterization for nonlinear PDEs is explicitly described. It is also shown here that for particular choices of the boundary conditions the above map can be linearized. © 2005 Wiley Periodicals, Inc.