Existence of global solutions and global attractor for the third order Lugiato-Lefever equation on T

We show the existence of global solution and the global attractor in L2(T) for the third order Lugiato-Lefever equation on T. Without damping and forcing terms, it has three conserved quantities, that is, the L2(T) norm, the momentum and the energy, but the leading term of the energy functional is not positive definite. So only the L2 norm conservation is useful for the third order Lugiato-Lefever equation unlike the KdV and the cubic NLS equations. Therefore, it seems important and natural to construct the global attractor in L2(T). For the proof of the global attractor, we use the smoothing effect of cubic nonlinearity for the reduced equation.