某些非线性方程的双解:孤子和混沌及其意义

Double solutions of some nonlinear equations:soliton and chaos and their meanings

在非线性方程中,孤子和混沌的基本特征是完全不同的.但各种具有孤子解的非线性方程都可以得到混沌.而只有某些具有混沌解的非线性方程有孤子解.两种解的条件是不同的,某些参数是某个常数时得到孤子,而这些参数在一定区域变化时出现分岔-混沌,也许它联系于混沌的控制.双解可能对应于量子理论中的波-粒二象性,联系于非线性波动力学的双重解.某些非线性方程具有孤子和混沌双解,在数学、物理和粒子理论中存在若干新的意义.

The fundamental characteristics of soliton and chaos in nonlinear equation are completely different.But all nonlinear equations with a soliton solution may derive chaos.While only some equations with a chaos solution have a soliton solution.The conditions of the two solutions are different.When some parameters are certain constants,the soliton is derived;while these parameters vary in a certain region,the bifurcation-chaos appears.It connects a chaotic control probably.The double solutions correspond possibly to the wave-particle duality in quantum theory,and connect the double solution theory of the nonlinear wave mechanics.Some nonlinear equations possess soliton and chaos,whose new meanings are discussed briefly in mathematics,physics and particle theory.