On Algebro-Geometric Solutions of the Camassa-Holm Hierarchy
On Algebro-Geometric Solutions of the Camassa-Holm Hierarchy
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DOI：
10.1515/ans-2007-0303
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发表时间：
2007-08
影响因子：
1.8
通讯作者：
L. Zampogni
中科院分区：
文献类型：
--
作者：
L. Zampogni
摘要：我们找到了包含 Camassa-Holm 方程的新通勤层次结构的所有方程的代数几何类型的全局解。该层次结构的构建类似于经典的 K-dV 和 AKNS 层次结构。我们使用零曲率方法给出递推公式。解的时间演化是完全确定的，并且通过求解 Sturm-Liouville 方程 L(φ) = −φ″ + φ = λyφ 的反问题，获得了广义雅可比簇的非线性子簇 Y 上的运动。这是表达与 t 和 x 线性相关的解的自然设置，坐标位于此类子变量 φ 中包含的曲线平行四边形上。 φ 是通过将广义阿贝尔映射 I0 限制在 R 上无序 g 元组的空间 Symmg(R) 上而获得的，非线性平行四边形是通过动态 Sturm-Liouville 方程组的 Weyl m 函数 m±(x, t, λ) 的动极点 Pi(x, t) (i = 1, ..., g) 的受限广义阿贝尔映射的图像Lx(φ) = −φ″ + φ = λτx(y)φ，其中 τ 是平移流。事实证明，特定稳态初始条件 ug(x, t0) 的选择完全决定了层次结构中所有方程的解 u(x, t)，作为与 Sturm-Liouville 算子族 Lx 相对应的 Weyl m 函数的极点 Pi(x, t) 的函数，密度函数 y(x, t) = uxx(x, t)/2 − 2u(x, t)。对于每个 t∊ℝ，映射 x ↦ y(x, t) 位于相关 Sturm-Liouville 方程组 Lx(φ) 的等谱类中，并且完全通过为极点 Pi(x, t) 分配谱参数和初始条件来确定。
Abstract We find global solutions of algebro geometric type for all the equations of a new commuting hierarchy containing the Camassa-Holm equation. This hierarchy is built in analogy to the classical K-dV and AKNS hierarchies. We use a zero curvature method to give recursion formulas. The time evolution of the solutions is completely determined, and the motion on a nonlinear subvariety Υ of a generalized Jacobian variety is obtained by solving an inverse problem for the Sturm-Liouville equation L(φ) = −φ″ + φ = λyφ. This is the natural setting for the expression of the solutions which depend linearly with respect to t and x, with coordinates on a curvilinear parallelogram contained in such a subvariety φ. φ is obtained as the restriction of the generalized Abel map I0 to the space Symmg(R) of unordered g-tuples of points on R, and the nonlinear parallelogram is the image through the restricted generalized Abel map of the moving poles Pi(x, t) (i = 1, . . . , g) of the Weyl m-functions m±(x, t, λ) of the dynamical Sturm-Liouville family of equations Lx(φ) = −φ″ + φ = λτx(y)φ, where τ is the translation flow. It turns out that the choice of a particular stationary initial condition ug(x, t0) completely determines the solution u(x, t) of all the equations in the hierarchy, as functions of the poles Pi(x, t) of the Weyl m-functions corresponding to the family Lx of Sturm-Liouville operators, with density function y(x, t) = uxx(x, t)/2 − 2u(x, t). For every t∊ℝ, the maps x ↦ y(x, t) lie in an isospectral class of the associated family of Sturm-Liouville equations Lx(φ), and are completely determined by assigning spectral parameters and initial conditions for the poles Pi(x, t).