Invariant tori of full dimension for second KdV equations with the external parameters (Q1692094)

This work focuses on the persistence of full dimension invariant tori (corresponding to almost periodic solutions) of a fifth-order KdV equations: \[ u_{t}=\partial^{5}_{x}u+(M_{\sigma}u+u^{3}), \] under zero mean-value periodic boundary conditions \[ u(t, x+2\pi)=u(t,x),\quad \int^{2\pi}_{0}u(t,x)\mathrm d\,x=0, \] where \(M_{\sigma}\) is a real Fourier multiplier. The authors prove that the equations admits a Whitney smooth family of small amplitude, real analytic almost periodic solutions with all frequencies. The proof is based on a conserved quantity \(\int^{2\pi}_{0}u^{2}\mathrm d\,x\), Töplitz-Lipschitz property of the perturbation and an abstract infinite-dimensional KAM theorem. This is the first attempt to construct almost periodic solutions for the unbounded perturbation case.