与Hermite对称空间相关的多分量可积方程的局域波

本项目拟研究与Hermite对称空间相关的多分量可积方程在常数背景下和周期背景下的局域波及其动力学行为。首先，考虑Hermite对称空间的代数结构，拓展基于Riccati方程构造Darboux变换的方法，求解关于Hermite对称空间的(1+1)维可积方程在常数背景下的怪波、呼吸子等局域波。然后，将上述内容推广到(2+1)维情形，研究KP方程和DS方程关于Hermite对称空间Sp(n)/U(n)的广义版本在常数背景下的局域波。结合Darboux变换方法和非线性化方法，研究NLS方程和mKdV方程关于Hermite对称空间SU(p+q)/S(U(p)×U(q))的二分量版本在周期背景下的局域波。最后，根据(2+1)维方程的可积分解，求得KP方程和mKP方程在周期背景下的局域波，并将该方法推广到这两方程关于Hermite对称空间SU(p+q)/S(U(p)×U(q))的二分量版本。

In this project, we investigate localized waves on constant or periodic backgrounds and their dynamics for multi-component integrable equations associated with Hermitian symmetric spaces. Firstly, considering the algebraic structures of Hermitian symmetric spaces, we shall develop the way to construct Darboux transformation by virtue of Riccati equations. Consequently, the localized waves on constant backgrounds of (1+1)-dimensional integrable equations associated with Hermitian symmetric spaces are obtained, such as rogue waves and breathers. As a (2+1)-dimensional extension of the above, we get localized waves on constant backgrounds for the generalizations of KP and DS equations associated with Hermitian symmetric spaces Sp(n)/U(n). Combining Darboux transformation with the nonlinearization method, we study localized waves on periodic backgrounds for the two-component coupled versions of NLS and mKdV equations associated with Hermitian symmetric space SU(p+q)/S(U(p)×U(q)). Finally, we utilize the integrable decompositions of (2+1)-dimensional equations to derive localized waves on periodic backgrounds of KP and mKP equations. We extend this method to analyze their two-component versions associated with Hermitian symmetric space SU(p+q)/S(U(p)×U(q)).