Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equations
Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equations
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DOI：
10.1063/1.530509
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发表时间：
1994-05
影响因子：
1.3
通讯作者：
S. Lou
中科院分区：
文献类型：
--
作者：
S. Lou
利用Korteweg-de弗里斯（KdV）方程在平凡对称性和τ0对称性上的逆强对称性，得到了KdV方程的四组新的对称性.这些对称性通过KdV方程的Jost函数的多重积分来明确表达，并与已知对称性的两个层次一起构成无限维李代数。与一般观点相反，KdV和mKdV方程的时间无关对称群是非阿贝尔的，并且KdV和mKdV方程的无限维李代数是非同构的，尽管两个方程通过Miura变换相关联。从这些对称性出发，得到了四类积分微分KdV方程，它们可以用薛定谔逆散射变换方法求解。这些层次中的一些享有共同的强对称性和/或相同的局部守恒密度。
Using the inverse strong symmetry of the Korteweg–de Vries (KdV) equation on the trivial symmetry and τ0 symmetry, one gets four new sets of symmetries of the KdV equation. These symmetries are expressed explicitly by the multi‐integrations of the Jost function of the KdV equation and constitute an infinite dimensional Lie algebra together with two hierarchies of the known symmetries. Contrary to the general belief, the time‐independent symmetry groups of the KdV and mKdV equations are non‐Abelian and the infinite dimensional Lie algebras of the KdV and mKdV equations are nonisomorphic though two equations are related by the Miura transformation. Starting from these sets of symmetries, four hierarchies of the integrodifferential KdV equations, which can be solved by the Schrodinger inverse scattering transformation method, are obtained. Some of these hierarchies enjoy a common strong symmetry and/or same local conserved densities.