The complex Hamiltonian system in the Gerdjikov-Ivanov equation and its applications
The complex Hamiltonian system in the Gerdjikov-Ivanov equation and its applications
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DOI：
10.1007/s13324-022-00704-7
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发表时间：
2022-07
影响因子：
1.7
通讯作者：
Jinbing Chen;Yanpei Zhen
中科院分区：
文献类型：
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作者：
Jinbing Chen;Yanpei Zhen
将二阶非线性Schrödinger型Gerdjikov-Ivanov （GI）方程简化为两个具有与简单特征值相关的实值哈密顿量的复哈密顿系统，证明了它们在Liouville意义上是完全可积的。利用复哈密顿流的可交换性，复哈密顿系统的对合解得到GI方程的有限参数解，椭圆变量得到决定其动力学的Dubrovin型方程。由Abel-Jacobi变量可知，在黎曼曲面上，复哈密顿流的演化速度与无穷远处全纯微分的渐近展开系数有关。将黎曼-雅可比反演应用于线性化的复哈密顿流，利用黎曼定理和迹公式，得到了GI方程具有显式演化速度的黎曼函数表示的新的拟周期解。
The Gerdjikov–Ivanov (GI) equation of derivative nonlinear Schrödinger type is reduced to two complex Hamiltonian systems with real-valued Hamiltonians in relevance to simple eigenvalues, which are proved to be completely integrable in the Liouville sense. With the commutability of complex Hamiltonian flows, involutive solutions of the complex Hamiltonian systems result in finite parametric solutions to the GI equation, and elliptic variables give rise to the Dubrovin type equations determining their dynamics. It follows from Abel–Jacobi variables that on a Riemann surface the evolution velocities of complex Hamiltonian flows are connected with the asymptotic expansion coefficients of holomorphic differentials at infinities. The Riemann–Jacobi inversion is applied to the linearized complex Hamiltonian flows, from which a new quasi-periodic solution expressed by Riemann theta functions with explicit evolution velocities is obtained for the GI equation by means of the Riemann theorem and trace formulas.