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Invariant manifolds and fibrations for perturbed nonlinear Schrödinger equations. (English) Zbl 0897.35070

Applied Mathematical Sciences. 128. New York, NY: Springer. viii, 170 p. (1997).
The authors consider the perturbed nonlinear Schrödinger equation (PNLS) \[ iq_t=q_{xx} +2| q|^2 q-2\omega^2 q+i\varepsilon (-\alpha q+\Gamma +\widehat D^2q) \] with even and periodic boundary conditions, where \(\varepsilon\) is small and \(\widehat D^2\) is a regularized (i.e. bounded) Laplace operator. The authors prove the existence of infinite-dimensional local center-stable, center-unstable and center manifolds near the family \(q(x)= \omega e^{i\theta}\) of homogeneous fixed points. The first two manifolds are of codimension one, while the center manifold is of codimension two. In addition, the existence of smooth stable and unstable foliations of the center-stable and center-unstable manifolds by one-dimensional fibers is shown.
In the first chapter, a comprehensive list of references regarding invariant manifolds for infinite-dimensional dynamical systems is given. The PNLS is introduced in the second chapter. It is shown that the linearized flow near the family of fixed points has linear invariant manifolds of the kind described above. The next two chapters contain the main part of the book. Here, the authors prove that the aforementioned manifolds persist for the full nonlinear PNSL and are foliated by stable and unstable fibers. The technique used to prove the results is the graph transform. The authors’ analysis is based upon the fact that the PNLS admits a flow and not only a semiflow.
The book is generally well-written. The proofs of the main results are very detailed. Applications of the results are not given; instead the reader is referred to the literature where the presented results have been used.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B32 Bifurcations in context of PDEs
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