On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions
On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions
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DOI：
10.1088/0266-5611/2/3/005
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发表时间：
1986-08
期刊：
影响因子：
2.1
通讯作者：
M. Boiti;J. Leon;M. Manna;F. Pempinelli
中科院分区：
文献类型：
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作者：
M. Boiti;J. Leon;M. Manna;F. Pempinelli
Korteweg-de Vries方程在2+1维上的推广与谱问题（x2- y2-p(x,y)） (x,y;k)=0有关。它可以包含任意x+y或x-y和时间的函数。柯西问题与初始数据在无穷远处迅速衰减有关，通过将光谱变换技术扩展到两个空间维度来线性化。用初始数据明确地定义了谱数据，并将反问题表述为非局部黎曼-希尔伯特边值问题。演化方程中x+y和x-y的任意函数的存在意味着光谱数据的时间演化是线性的，但不是局域的。离散谱数据是禁止的，因此，局部孤子解是不允许的。
A generalisation in 2+1 dimensions of the Korteweg-de Vries equation is related to the spectral problem ( delta x2- delta y2-p(x,y)) phi (x,y;k)=0. It can contain arbitrary functions of x+y or x-y and time. The Cauchy problem, associated with initial data decaying sufficiently rapidly at infinity, is linearised by an extension of the spectral transform technique to two spatial dimensions. The spectral data are explicitly defined in terms of the initial data and the inverse problem is formulated as a non-local Riemann-Hilbert boundary-value problem. The presence of arbitrary functions of x+y and x-y in the evolution equation implies that the time evolution of the spectral data is linear but non-local. Discrete spectral data are forbidden and, consequently, localised soliton solutions are not allowed.