Gerdjikov-Ivanov方程的Rogue波和呼吸子解
The Rogue Wave and breather solution of the Gerdjikov-Ivanov equation
摘要：
$q$和$r$的Gerdjikov-Ivanov（GI）系统由具有$2 \times 2$矩阵系数的二次多项式谱问题定义。该系统的n重达布变换矩阵的每个元素都用特征函数的$(n+1)\times (n+1)$行列式与$n\times n$行列式之比表示，这意味着由已知解$q$和$r$生成的$q^{[n]}$和$r^{[n]}$的行列式表示.通过根据约化条件$q^{[n]}=-(r^{[n]})^*$选择一些特殊的特征值和特征函数，$q^{[n]}$的行列式表示提供了GI方程的一些新的解.作为例子，呼吸者的解决方案和流氓波的GI明确给出了从一个周期性的“种子”与一个恒定的振幅的两倍DT。
英文摘要：
The Gerdjikov-Ivanov (GI) system of $q$ and $r$ is defined by a quadratic polynomial spectral problem with $2 \times 2$ matrix coefficients. Each element of the matrix of n-fold Darboux transformation of this system is expressed by a ratio of $(n+1)\times (n+1)$ determinant and $n\times n$ determinant of eigenfunctions, which implies the determinant representation of $q^{[n]}$ and $r^{[n]}$ generated from known solution $q$ and $r$. By choosing some special eigenvalues and eigenfunctions according to the reduction conditions $q^{[n]}=-(r^{[n]})^*$, the determinant representation of $q^{[n]}$ provides some new solutions of the GI equation. As examples, the breather solutions and rogue wave of the GI is given explicitly by two-fold DT from a periodic "seed" with a constant amplitude.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
