On Transformations of the Rabelo Equations

We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The other two are transformed into a linear equation and the Liouville equation, and in this way their general solutions are obtained.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Global well-posedness of the short-pulse and sine-Gordon equations in energy space

We prove global well-posedness of the short-pulse equation with small initial data in Sobolev space $H^2$. Our analysis relies on local well-posedness results of Sch\"afer & Wayne, the correspondence of the short-pulse equation to the sine-Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also prove local and global well-posedness of the sine-Gordon equation in an appropriate function space.Comment: 17 pages, revised versio

A new integrable generalization of the Korteweg - de Vries equation

A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg - de Vries equation with a source. A Lax representation and a Backlund self-transformation are found of the new equation, and its travelling wave solutions and generalized symmetries are studied.Comment: 13 pages, 2 figure

Integrated source of path-entangled photon pairs with efficient pump self-rejection

This research was supported by the EU Flagship on Quantum Technologies, project PhoG (820365). D.M., A.S. and A.M. also acknowledge support from the National Academy of Sciences of Belarus program “Convergence”. The authors gratefully acknowledge the support from the Scottish Universities Physics Alliance (SUPA) and the Engineering and Physical Sciences Research Council (EPSRC).We present a theoretical proposal for an integrated four-wave mixing source of narrow-band path-entangled photon pairs with efficient spatial pump self-rejection. The scheme is based on correlated loss in a system of waveguides in Kerr nonlinear media. We calculate that this setup gives the possibility for upwards of 100 dB pump rejection, without additional filtering. The effect is reached by driving the symmetric collective mode that is strongly attenuated by an engineered dissipation, while photon pairs are born in the antisymmetric mode. A similar set-up can additionally be realized for the generation of two-photon NOON states, also with pump self-rejection. We discuss the implementation of the scheme by means of the coherent diffusive photonics, and demostrate its feasibility in both glass (such as fused silica-glass and IG2) and planar semiconductor waveguide structures in indium phosphide (InP) and in silicon.Publisher PDFPeer reviewe

Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations

We prove the most general theorem about spectral stability of multi-site breathers in the discrete Klein-Gordon equation with a small coupling constant. In the anti-continuum limit, multi-site breathers represent excited oscillations at different sites of the lattice separated by a number of "holes" (sites at rest). The theorem describes how the stability or instability of a multi-site breather depends on the phase difference and distance between the excited oscillators. Previously, only multi-site breathers with adjacent excited sites were considered within the first-order perturbation theory. We show that the stability of multi-site breathers with one-site holes change for large-amplitude oscillations in soft nonlinear potentials. We also discover and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure

Coherent diffusive photon gun for generating nonclassical states

Funding: EU Flagship on Quantum Technologies, project PhoG (820365). D.M., A.S., and A.M. also acknowledge support from the National Academy of Sciences of Belarus program “Convergence,” and the BRRFI project F18U-006.We suggest and discuss a concept of a deterministic integrated source of nonclassical light based on the coherent diffusive photonics, a coherent light flow in a system of dissipatively coupled waveguides. We show how this practical quantum device can be realized with a system of single-mode waveguides laser inscribed in nonlinear glass. We describe a hierarchy of models, from the complete multimode model of the waveguide network to the single mode coupled to a bath, analyze the conditions for validity of the simplest single-mode model and demonstrate feasibility of the generation of bright sub-Poissonian light states merely from a coherent input. Notably, the generation of nonclassical states occurs at the initial stages of the dynamics, and can be accounted for in the linear model that allows us to circumvent the prohibiting computational complexity of the exact full quantum representation.Publisher PDFPeer reviewe