该项目基于谱分析、椭圆函数论、以及代数曲线，拟将研究可积非线性发展方程在周期背景下出现的怪波行为。对非线性发展方程的周期谱问题应用Lax对非线性化，借助于有限维可积Hamiltonian系统，探明周期位势所满足的Lax--Novikov方程，周期特征函数的解析表达式及其Floquet--Bloch谱带支点。结合Darboux变换，利用时空Lax系统的周期和非周期特征函数导出可积方程的新解析解。鉴于所得解的分析性质，明确其为周期波的模不稳定性所形成的怪波，并计算相关扩大因子。进而探求周期背景下的怪波在小振幅极限下的渐近行为，证实其恰为常数波背景下的怪波的自然推广。据此，研究内容将致力于发展一个系统、有效的方法寻求周期背景下怪波的解析表达式；研究结果将丰富周期和拟周期背景下的怪波的数学理论，提高人们对怪波的形成机制和动力学的理解，并可靠地预测其出现的几率，从而实现避免或减弱其对自然界的危害。

Based on the spectral analysis, the theory of elliptic functions, and the algebraic curves, the current proposal focuses on the study of rogue waves on the periodic background arising from integrable nonlinear evolution equations (NLEEs). Towards this object, the nonlinearization of Lax pair is generalized to the periodic spectral problems of NLEEs, from which the Lax--Novikov equation satisfied by the periodic potential is explored by means of finite-dimensional integrable Hamiltonian systems, as well as the analytical formulas to the periodic eigenfunctions and their branch points of Floquet--Bloch spectral bands. By using the Darboux transformation, some new analytical solutions to integrable NLEEs are deduced in view of periodic and non-periodic eigenfunctions of the space-time Lax systems. Followed by the asymptotic analysis, it turns out that the resultant solutions can be used to describe rogue waves on the periodic background due to the modulation instability of periodic waves, and then the associated magnification factor is figured out. Moreover, it follows from the asymptotic behavior under the small-amplitude limit that rogue waves on the periodic background are confirmed to be the natural generalizations of the classical rogue waves on the constant wave background. Summing up, a systematic and effective computational algorithm is developed to deduce analytical expressions for rogue waves on the periodic background; the proposed project would enrich the mathematical theory of rogue waves on both the periodic and the quasi-periodic backgrounds, and improve the understandings of mechanisms of formation and dynamics of rogue waves, and further reliably predict the occurrence rate of rogue waves, such that in a sense the destruction of rogue waves could be avoided or at least abated.