Interaction of nonlinear waves

彩虹的美丽只是光谱分解物理现象的壮观表现，光谱分解是阳光分裂成最简单的成分或基本颜色的现象。一束来源足够任意的光束可以分解成成分，反之亦然，这些成分可以组合在一起，获得任何色调或颜色。在数学上，基本颜色的组合是通过描述基本颜色的函数的和来表示的，每个函数以一定的系数输入和。可以分解成简单分量，然后以这种方式从这些分量重建的现象称为线性现象。然而，在自然界中，并不是所有的现象都是线性的，大多数现象实际上都是非线性的。因此，问题就出现了，是否可以将描述非线性现象的函数分解成最简单的分量，以及是否可以将简单的分量组合起来构造这样的函数。对此的数学描述将比线性现象的数学描述要复杂得多。在该提案中，我们考虑了由数学家所称的可积系统描述的非线性现象。在这种系统中，孤子的产生。在没有孤子的情况下，描述孤子的函数可以分解成类似于光的基本颜色的非线性分量。这些描述最简单的波的分量可以组合在一起，得到描述各种现象的函数。

The beauty of the rainbow is just a spectacular manifestation of the physical phenomenon of spectral decomposition, the phenomenon of the sunlight spliting up into simplest components or basic colors.  A beam of light of sufficiently arbitrary origin may be decomposed into components and vice versa the components may be combined together to obtain any hue or color. Mathematically the combinations of basic colors are expressed by means of sums of functions describing basic colors with each function entering the sum with a certain coefficient. Phenomena that can be broken up into simple components and then reconstructed from these components in such a manner are called linear phenomena.       In nature, however, not all phenomena are linear, most phenomena are actually nonlinear.   So the question arises whether it is possible to break up functions describing nonlinear phenomena into simplest components and whether it is possible to combine the simple components to construct such functions. The mathematical description of this would be significantly more complicated than that of  linear phenomena.    In the proposal we consider the nonlinear phenomena described by what mathematicians call integrable systems.  Such systems give rise to solitons. In the absence of solitons the functions describing them may be decomposed into nonlinear components analogous to basic colors of light.  These components describing the simplest waves may be combined together to obtain functions describing a large variety of phenomena.