Fourier Parameterization of Attractors for Dissipative Equations in One Space Dimension

For dissipative equations of the form $$u_t + ( - 1)^m u^{(2m)} + f(x,u,u_x ,...,u^{(2m - 1)} ) = 0$$ we show that the global attractor is a Lipschitz graph over a finite dimensional Fourier eigenspace. In particular, the statement applies to the Burgers equation u _t ?u _xx +uu _x =f and the modified Burgers equation u _t ?u _xx +uu _x ?u=0.