We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis–Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash–Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure , the weak dispersive effects of the linear flow and the presence of strong resonant interactions require a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for special wave packet solutions and to perform a very careful algebraic analysis of the resonances. Our approach is quite general and can be applied also to other 1d integrable PDEs. We are confident for instance that the same strategy should work for the Camassa–Holm equation.

中文翻译：

---

Degasperis-Procesi 方程的可约 KAM Tori

我们开发了接近椭圆不动点的 KAM 理论，用于圆上色散 Degasperis-Procesi 方程的准线性哈密顿扰动。拟线性偏微分方程的 KAM 理论总体策略基于 Nash-Moser 非线性迭代、伪微分和范式技术。在目前的情况下，复杂的辛结构、线性流动的弱色散效应和强共振相互作用的存在需要一套新的想法。要点是利用未扰动方程的可积性，寻找特殊波包解并对共振进行非常仔细的代数分析。我们的方法非常通用，也可以应用于其他一维可积偏微分方程。