Interaction of nonlinear waves

彩虹之美只是光谱分解物理现象的壮观表现，即太阳光分解成最简单的成分或基本颜色的现象。具有足够任意来源的光束可以分解成组件，反之亦然，这些组件可以组合在一起以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和来表示的，每个函数都以一定的系数进入和。可以被分解成简单的组成部分，然后以这种方式从这些组成部分重建的现象称为线性现象。然而，在自然界中，并非所有现象都是线性的，大多数现象实际上是非线性的。因此，问题出现了，是否有可能将描述非线性现象的函数分解成最简单的分量，以及是否有可能将这些简单的分量组合起来构建这样的函数。这种现象的数学描述要比线性现象复杂得多。在建议中，我们考虑由数学家称为可积系统描述的非线性现象。这样的系统会产生孤子。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。这些描述最简单波的分量可以组合在一起，得到描述各种各样现象的函数。

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.