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

Abstract

We show the existence of global solution and the global attractor in $L^2(\mathbf{T})$ for the third order Lugiato–Lefever equation on $\mathbf{T}$. Without damping and forcing terms, it has three conserved quantities, that is, the $L^2(\mathbf{T})$ norm, the momentum and the energy, but the leading term of the energy functional is not positive definite. So only the $L^2$ 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 $L^2(\mathbf{T})$. For the proof of the global attractor, we use the smoothing effect of cubic nonlinearity for the reduced equation.