114 research outputs found
Darboux integrability of trapezoidal and families of lattice equations I: First integrals
In this paper we prove that the trapezoidal and the families
of quad-equations are Darboux integrable systems. This result sheds light on
the fact that such equations are linearizable as it was proved using the
Algebraic Entropy test [G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic
entropy, symmetries and linearization for quad equations consistent on the
cube, \emph{J. Nonlinear Math. Phys.}, 23(4):507543, 2016]. We conclude with
some suggestions on how first integrals can be used to obtain general
solutions.Comment: 34 page
Integrability Test for Discrete Equations via Generalized Symmetries
In this article we present some integrability conditions for partial
difference equations obtained using the formal symmetries approach. We apply
them to find integrable partial difference equations contained in a class of
equations obtained by the multiple scale analysis of the general multilinear
dispersive difference equation defined on the square.Comment: Proceedings of the Symposium in Memoriam Marcos Moshinsk
Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations
In this paper we construct the autonomous quad-equations which admit as
symmetries the five-point differential-difference equations belonging to known
lists found by Garifullin, Yamilov and Levi. The obtained equations are
classified up to autonomous point transformations and some simple
non-autonomous transformations. We discuss our results in the framework of the
known literature. There are among them a few new examples of both sine-Gordon
and Liouville type equations.Comment: 27 page
Classification of five-point differential-difference equations
Using the generalized symmetry method, we carry out, up to autonomous point
transformations, the classification of integrable equations of a subclass of
the autonomous five-point differential-difference equations. This subclass
includes such well-known examples as the Itoh-Narita-Bogoyavlensky and the
discrete Sawada-Kotera equations. The resulting list contains 17 equations some
of which seem to be new. We have found non-point transformations relating most
of the resulting equations among themselves and their generalized symmetries.Comment: 29 page
Inverse Design of Perfectly Transmitting Eigenchannels in Scattering Media
Light-matter interactions inside turbid medium can be controlled by tailoring
the spatial distribution of energy density throughout the system. Wavefront
shaping allows selective coupling of incident light to different transmission
eigenchannels, producing dramatically different spatial intensity profiles. In
contrast to the density of transmission eigenvalues that is dictated by the
universal bimodal distribution, the spatial structures of the eigenchannels are
not universal and depend on the confinement geometry of the system. Here, we
develop and verify a model for the transmission eigenchannel with the
corresponding eigenvalue close to unity. By projecting the original problem of
two-dimensional diffusion in a homogeneous scattering medium onto a
one-dimensional inhomogeneous diffusion, we obtain an analytical expression
relating the intensity profile to the shape of the confining waveguide.
Inverting this relationship enables the inverse design of the waveguide shape
to achieve the desired energy distribution for the perfectly transmitting
eigenchannel. Our approach also allows to predict the intensity profile of such
channel in a disordered slab with open boundaries, pointing to the possibility
of controllable delivery of light to different depths with local illumination.Comment: 9 pages, 6 figure
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