Damped nonlinear Ginzburg–Landau equation with saturation. Part I. Existence of solutions on general domains

Résumé

We study the complex Ginzburg–Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schr¨odinger equation and dissipative parabolic dynamics through a complex timederivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains.

Mots-clés

Damped Ginzburg–Landau equation,; Saturated nonlinearity,; Finite time extinction,; Maximal monotone operators,; Existence and regularity of weak solutions;