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

  • Tomoyuki Miyaji

    Meiji Institute for Adv. Stud. Math. Sci., Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan
  • Yoshio Tsutsumi

    Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

Abstract

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

Cite this article

Tomoyuki Miyaji, Yoshio Tsutsumi, Existence of global solutions and global attractor for the third order Lugiato–Lefever equation on T. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 7, pp. 1707–1725

DOI 10.1016/J.ANIHPC.2016.12.004