The Ginzburg-Landau Equation: Posed in a Quarter-Plane

We study the following Ginzburg-Landau equation (GL): ut = (ν + iα)uxx − (κ + iβ) |u|2u + γu, ν > 0, κ > 0, α ≠ 0. For a full-line problem with u(x, 0) = u0(x) ∈ H2(−∞, ∞), global existence-uniqueness is established. For a half-line problem with u(x, 0) = u0(x) ∈ H2[0, ∞), u(0, t) = Q(t) ∈ C2[0, ∞), u0(0) = Q(0), the following results are available: (1) local existence-uniqueness; (2) criteria for the existence of a small amplitude solution on any finite interval by means of small initial and boundary data; (3) global existence in the case |β| ≤ √3κ or αβ > 0.