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 (T) for the third order Lugiato-Lefever equation on T. Without damping and forcing terms, it has three conserved quantities, that is, the L 2 (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 (T). For the proof of the global attractor, we use the smoothing effect of cubic nonlinearity for the reduced equation.

The Lugiato–Lefever Equation with Nonlinear Damping Caused by Two Photon Absorption

“…Additionally, it was shown that all solutions of the initial value problem remain bounded in L 2 while the H 1 -norm is proved to grow at most like √ t as t → ∞. In the corresponding model with an additional third order dispersion effect well-posedness results and even the existence of a global attractor were proved in [21]. Convergence results for the numerical Strang-splitting scheme can be found in [11].…”

Quasi-invariance of Gaussian measures transported by the cubic NLS with third-order dispersion on T

“…where β is a real constant. It is known (see for instance [16]) that this equation is globally well posed in L 2 (T). More precisely, we consider the Gaussian measure µ α formally given by µ α (du) = C α e − 1 2 u 2 H α du, where C α is a normalization constant and study its evolution with respect to the flow of (1).…”

GLOBAL ATTRACTOR FOR WEAKLY DAMPED, FORCED mKdV EQUATION BELOW ENERGY SPACE

“…For this reason, the results of damped and forced KdV cannot be directly converted to those of damped and forced mKdV by the Miura transform unlike the case without damping and forcing terms. The study of global attractor is important as it characterizes the global behavior of all solutions see [13], [2], [11], [16], [14] [12], [19] and [24] for some of the equations. The asymptotic behavior of solutions below the energy space has not been known, though the global well-posedness below the energy space is already proved for the Cauchy problem of (1.1)-(1.2).…”