Existence of global solutions and global attractor for the third order Lugiato-Lefever equation on \(\mathbf T\)
The main concern of this paper is a Cauchy problem that consists of a third order Lugiato-Lefever equation and a classical initial condition. In fact, the equation is a generalized Lugiato-Lefever equation which has been attracting a great interest especially in the field of nonlinear optics. The authors show that this equation appears in different mathematical models, as such, for Kerr frequency comb generation in a whispering gallery mode resonator, octave-spanning Kerr frequency comb in a micro-ring resonator or cavity solitons in micro-ring resonator near zero group-dispersion. For the formulated Cauchy problem, the authors prove the global well-posedness in \(L^2\) and investigate the nonlinear smoothing effect. Based on the smoothing effect, the authors can obtain the existence of the global attractor in \(L^2\) for flows generated by the third order Lugiato-Lefever equation. In the last part it is proved that, without damping and forcing, the solution of the Cauchy problem satisfies the three conservations laws, namely, the mass, the momentum and the energy conservations laws.
Strichartz' estimate on one dimensional torus
scientific article; zbMATH DE number 6810596
scientific article