Sergei Kuksin

The KdV equation under periodic boundary conditions and its perturbations

In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations.

KAM for the nonlinear beam equation

AbstractIn this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus

The effective equation method

u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0, \quad t \in {\mathbb{R}}, x \in {\mathbb{T}^d}, \quad (*)

Rigorous results in space-periodic two-dimensional turbulence

utt+Δ2u+mu+∂uG(x,u)=0,t∈R,x∈Td,(∗)where

Analyticity of solutions for quasilinear wave equations and other quasilinear systems

{G(x,u)=u^4+ O(u^5)}

KAM theory and the 3D Euler equation

G(x,u)=u4+O(u5). Namely, we show that, for generic m, many of the small amplitude invariant finite dimensional tori of the linear equation