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The diversity of steady state solutions of the complex Ginzburg-Landau equation. (English) Zbl 0972.35540

Summary: The structure of the phase space of stationary and quasi-stationary (i.e., uniformly translating) solutions of 1D CGLE is investigated by methods of the qualitative theory of ordinary differential equations. The Nozaki-Bekki holes are seen as heteroclinic connections which are made structurally stable by an involution symmetry in phase space. The existence of a countable set of double-loop heteroclinic trajectories is proved, which corresponds to complex shock-hole-shock structures both motionless and moving with constant velocity \(v_{0}\) along the \(x\)-axis.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
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