Darboux Transformations in Integrable Systems
Darboux Transformations in Integrable Systems
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DOI：
10.1007/1-4020-3088-6
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
C. Gu;He-sheng Hu;Zixiang Zhou
中科院分区：
文献类型：
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作者：
C. Gu;He-sheng Hu;Zixiang Zhou
孤波理论是非线性科学的一个重要分支。一方面，它描述了实际中出现的各种稳定运动，如孤立水波、光中孤立信号等; bre等，并且在科学和技术（如光信号通信）中具有许多应用。另一方面，它提供了许多E？非线性偏微分方程显式解的有效方法埃塞尔方程因此，它引起了物理学家和数学家的极大关注。Nonlinearpartialdi？基本方程出现在许多科学中？cpr-lems。得到明确的解决方案通常是一个di？邪教任务只有在某些特殊情况下，解才能被显式地写出。然而，对于许多孤立子方程，人们已经找到了许多方法来获得显式解。最著名的是逆散射法、B] acklund变换等。逆散射方法是基于普通散射的光谱理论。埃塞尔方程多个孤立子方程的柯西问题可转化为求解一个线性积分方程组。当积分方程的核是退化的时，可以导出显式解。B ac] klund变换通过求解一个完全可积的部分微分方程组，从已知解给出一个新解。埃塞尔方程一些复杂的”非线性叠加公式”取代了线性科学中的叠加原理。
GU Chaohao The soliton theory is an important branch of nonlinear science. On one hand, it describes various kinds of stable motions appearing in-ture, such as solitary water wave, solitary signals in optical? bre etc., and has many applications in science and technology (like optical signal communication). On the other hand, it gives many e? ective methods ofgetting explicit solutions of nonlinear partial di? erential equations. Therefore, it has attracted much attention from physicists as well as mathematicians. Nonlinearpartialdi? erentialequationsappearinmanyscienti? cpr-lems. Getting explicit solutions is usually a di? cult task. Only in c-tain special cases can the solutions be written down explicitly. However, for many soliton equations, people have found quite a few methods to get explicit solutions. The most famous ones are the inverse scattering method, B] acklund transformation etc.. The inverse scattering method is based on the spectral theory of ordinary di? erential equations. The Cauchyproblemofmanysolitonequationscanbetransformedtosolving a system of linear integral equations. Explicit solutions can be derived when the kernel of the integral equation is degenerate. The B] ac] klund transformation gives a new solution from a known solution by solving a system of completely integrable partial di? erential equations. Some complicated" nonlinear superposition formula" arise to substitute the superposition principlein linear science.