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We study homeomorphisms of a Cantor set with$$k$$($$k<+\infty$$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where$$k\geq 2$$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition$$(X,\unicode[STIX]{x1D70E})$$is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least$$k$$vertices at each level, and the image of the index map must consist of infinitesimals.
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This content will become publicly available on January 1, 2027
Numerical inverse scattering transform for the defocusing nonlinear Schrödinger equation with box-type initial conditions on a nonzero background
Abstract We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude$$q_o \gt 0$$as$$x\to \pm \infty$$by implementing numerically the corresponding inverse scattering transform (IST). The Riemann–Hilbert problem associated with the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift–Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). The numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the$$(x,t)$$-plane depending on whether$$|x/(2t)| \lt q_o$$or$$|x/(2t)| \gt q_o$$, as$$t \to \infty$$.
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- PAR ID:
- 10692787
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Nonlinear Waves
- Volume:
- 2
- ISSN:
- 3033-4268
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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