标题： 一个在Cayley-Dickson代数上的新可积方程 显示英文标题

标题： A new integrable equation valued on a Cayley-Dickson algebra

摘要： 我们引入了一个在Cayley-Dickson（C-D）代数上取值的新可积方程。 在该代数退化为复数代数的特殊情况中，方程中的新相互作用项消失，方程变为已知的Korteweg-de Vries方程。 对于每个C-D代数，该方程都有一个无限序列的局部守恒量。 我们得到了该方程在Walhquist-Estabrook意义上的Bäcklund变换，并将其与广义Gardner方程相关联。 由此，无限序列的守恒量可以直接得出。 我们给出了其中前几个的显式表达式。 从Bäcklund变换中，我们得到了Lax对和单孤子与双孤子解，推广了四元数值KdV方程的已知解。 从Gardner方程中，我们得到了广义的修正KdV方程，该方程也具有无限序列的守恒量。 新的可积方程在C-D代数的自同构的一个子群下保持不变。 在八元数代数的特殊情况下，该方程在$SU(3)$下不变。 显示英文摘要

摘要： We introduce a new integrable equation valued on a Cayley-Dickson (C-D) algebra. In the particular case in which the algebra reduces to the complex one the new interacting term in the equation cancells and the equation becomes the known Korteweg-de Vries equation. For each C-D algebra the equation has an infinite sequence of local conserved quantities. We obtain a B\"{a}cklund transformation in the sense of Walhquist-Estabrook for the equation for any Cayley-Dickson algebra, and relate it to a generalized Gardner equation. From it, the infinite sequence of conserved quantities follows directly. We give the explicit expression for the first few of them. From the B\"{a}cklund transformation we get the Lax pair and the one-soliton and two-soliton solutions generalizing the known solutions for the quaternion valued KdV equation. From the Gardner equation we obtain the generalized modified KdV equation which also has an infinite sequence of conserved quantities. The new integrable equation is preserved under a subgroup of the automorphisms of the C-D algebra. In the particular case of the algebra of octonions, the equation is invariant under $SU(3)$.