KAM for autonomous quasi-linear perturbations of mKdV (Q295302)

This work focuses on the the stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear (i.e., strongly nonlinear) autonomous Hamiltonian differentiable perturbations of the KdV equation NEWLINE\[NEWLINEu_{t}+u_{xxx}-6uu_{x}+\mathcal{N}_{4}(x,u,u_{x}, u_{xx}, u_{xxx})=0, NEWLINE\]NEWLINE under periodic boundary conditions \(x\in\mathbb{T}\), where NEWLINE\[NEWLINE\mathcal{N}_{4}(x,u,u_{x}, u_{xx}, u_{xxx}):=-\partial_{x}[(\partial_{u}f)(x,u,u_{x})-\partial_{x}((\partial_{u_{x}}f)(x,u,u_{x}))]. NEWLINE\]NEWLINE This is the first result that extends KAM theory to quasi-linear autonomous and parameter-independent PDEs. The main methods are a weak Birkhoff normal form algorithm and a convergent Nash-Moser scheme.