752,882 research outputs found
A novel nonlinear evolution equation integrable by the inverse scattering method
A Backlund transformation for an evolution equation (ut+u ux)x+u=0 transformed into new coordinates is derived. An inverse scattering problem is formulated. The inverse scattering method has a third order eigenvalue problem. A procedure for finding the exact N-soliton solution of the Vakhnenko equation via the inverse scattering method is described
Symmetry, Structure and the Constitution of Objects
In this paper I focus on the impact on structuralism of the quantum treatment of objects in terms of symmetry groups and, in particular, on the question as to how we might eliminate, or better, reconceptualise such objects in structural terms. With regard to the former, both Cassirer and Eddington not only explicitly and famously tied their structuralism to the development of group theory but also drew on the quantum treatment in order to further their structuralist aims and here I sketch the relevant history with an eye on what lessons might be drawn. With regard to the latter, Ladyman has explicitly cited Castellani's work on the group-theoretical constitution of quantum objects and I indicate both how such an approach needs to be understood if it is to mesh with Ladyman's 'ontic' form of structural realism and how it might accommodate permutation symmetry through a consideration of Huggett's recent account
PT-symmetry broken by point-group symmetry
We discuss a PT-symmetric Hamiltonian with complex eigenvalues. It is based
on the dimensionless Schr\"{o}dinger equation for a particle in a square box
with the PT-symmetric potential . Perturbation theory clearly
shows that some of the eigenvalues are complex for sufficiently small values of
. Point-group symmetry proves useful to guess if some of the eigenvalues
may already be complex for all values of the coupling constant. We confirm
those conclusions by means of an accurate numerical calculation based on the
diagonalization method. On the other hand, the Schr\"odinger equation with the
potential exhibits real eigenvalues for sufficiently small
values of . Point group symmetry suggests that PT-symmetry may be broken
in the former case and unbroken in the latter one
Vowel Symmetry
Arrange the letters of the alphabet in columns of two
Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
We introduce the notion of algebraic higher symmetry, which generalizes
higher symmetry and is beyond higher group. We show that an algebraic higher
symmetry in a bosonic system in -dimensional space is characterized and
classified by a local fusion -category. We find another way to describe
algebraic higher symmetry by restricting to symmetric sub Hilbert space where
symmetry transformations all become trivial. In this case, algebraic higher
symmetry can be fully characterized by a non-invertible gravitational anomaly
(i.e. an topological order in one higher dimension). Thus we also refer to
non-invertible gravitational anomaly as categorical symmetry to stress its
connection to symmetry. This provides a holographic and entanglement view of
symmetries. For a system with a categorical symmetry, its gapped state must
spontaneously break part (not all) of the symmetry, and the state with the full
symmetry must be gapless. Using such a holographic point of view, we obtain (1)
the gauging of the algebraic higher symmetry; (2) the classification of
anomalies for an algebraic higher symmetry; (3) the equivalence between classes
of systems, with different (potentially anomalous) algebraic higher symmetries
or different sets of low energy excitations, as long as they have the same
categorical symmetry; (4) the classification of gapped liquid phases for
bosonic/fermionic systems with a categorical symmetry, as gapped boundaries of
a topological order in one higher dimension (that corresponds to the
categorical symmetry). This classification includes symmetry protected trivial
(SPT) orders and symmetry enriched topological (SET) orders with an algebraic
higher symmetry.Comment: 61 pages, 31 figure
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