×

Defocusing nonlinear Schrödinger equations. (English) Zbl 1458.35002

Cambridge Tracts in Mathematics 217. Cambridge: Cambridge University Press (ISBN 978-1-108-47208-1/hbk; 978-1-108-59051-8/ebook). xii, 242 p. (2019).
The book offers a consistent mathematically rigorous treatment of well-posedness and existence problems for the nonlinear Schrödinger equation for complex function \(u(t,x)\) with a power-type nonlinear term, the term in front of which corresponds to the repulsive self-interaction: \[ i\partial_t u+\Delta u - |u|^{2\nu} u=0, \] where \(\Delta\) is the Laplacian in the \(d\)-dimensional space. The book presents a systematic collection of proofs of theorems which demonstrate the existence of solutions to this equation with initial values. Chiefly, this is done for the case of sufficiently small initial data, and the proof states the existence of solutions of the scattering type, i.e., small-amplitude (for this reason, quasi-linear) spreading configurations initiated by the input. Special care is paid to the cases of the “mass-critical” nonlinearity, with \(\nu=2/d\) in the above equation, for all \(d\geq1\), and the “energy-critical” nonlinearity, with \(\nu=2/(d-2)\), for \(d\geq3\) (in the case of the opposite, self-focusing sign of the nonlinearity, the mass-critical nonlinearity term is the one which gives rise to the critical collapse.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35K55 Nonlinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI Link