Physical Sciences and Mathematics Commons^™

Stability Of Discrete Solitons In The Presence Of Parametric Driving, H. Susanto, Q. E. Hoq, Panos Kevrekidis

Panos Kevrekidis

In this Brief Report, we consider parametrically driven bright solitons in the vicinity of the anticontinuum limit. We illustrate the mechanism through which these solitons become unstable due to the collision of the phase mode with the continuous spectrum, or eigenvalues bifurcating thereof. We show how this mechanism typically leads to complete destruction of the bright solitary wave.

Criticality For The Gehring Link Problem, Jason Cantarella, Joseph H.G. Fu, Robert Kusner, John M. Sullivan, Nancy C. Wrinkle

Robert Kusner

In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring’s problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality.

Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version of the Kuhn–Tucker theorem. We use this to prove that every critical link …

On A Notion Of Maps Between Orbifolds Ii. Homotopy And Cw-Complex, Weimin Chen Chen

Weimin Chen

This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants – the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem which asserts that a weak homotopy equivalence is a homotopy equivalence is extended to the orbifold category.

Remarks On The Combinatorial Intersection Cohomology Of Fans, Tom Braden

Tom Braden

No abstract provided.

Radiationless Travelling Waves In Saturable Nonlinear Schrödinger Lattices, T. R. O. Melvin, A. R. Champneys, Panos Kevrekidis, J. Cuevas

Panos Kevrekidis

The long-standing problem of moving discrete solitary waves in nonlinear Schrödinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. Genuinely localized traveling waves are computed as a function of the system parameters for the first time. The relevant solutions exist only for finite velocities.

On The Nondegeneracy Of Constant Mean Curvature Surfaces, Nick Korevaar, Robert Kusner, Jesse Ratzkin

Robert Kusner

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $f:\Sigma \to \R^3$ are nondegenerate; that is, the Jacobi operator Δf+|Af|2 has no L2 kernel. In fact, if Σ has genus zero and f(Σ) is contained in a half-space, then we find an explicit upper bound for the dimension of the L2 kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising …

Decoupling Of The General Scalar Field Mode And The Solution Space For Bianchi Type I And V Cosmologies Coupled To Perfect Fluid Sources, T. Christodoulakis, Th. Grammenos, Ch. Helias, Panos Kevrekidis, A. Spanou

Panos Kevrekidis

The scalar field degree of freedom in Einstein’s plus matter field equations is decoupled for Bianchi type I and V general cosmological models. The source, apart from the minimally coupled scalar field with arbitrary potential V(Φ), is provided by a perfect fluid obeying a general equation of state p = p(ρ). The resulting ODE is, by an appropriate choice of final time gauge affiliated to the scalar field, reduced to first order, and then the system is completely integrated for arbitrary choices of the potential and the equation of state.

Toric Modular Forms And Nonvanishing Of L-Functions, Lev A. Borisov, Paul E. Gunnells

Paul Gunnells

In a previous paper \cite{BorGunn}, we defined the space of toric forms $\TTT(l)$, and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f,1)≠0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

On 3+1 Dimensional Scalar Field Cosmologies, Panos Kevrekidis

Panos Kevrekidis

In this communication, we analyze the case of 3+1 dimensional scalar field cosmologies in the presence, as well as in the absence of spatial curvature, in isotropic, as well as in anisotropic settings. Our results extend those of Hawkins and Lidsey [Phys. Rev. D {\bf 66}, 023523 (2002)], by including the non-flat case. The Ermakov-Pinney methodology is developed in a general form, allowing through the converse results presented herein to use it as a tool for constructing new solutions to the original equations. As an example of this type a special blowup solution recently obtained in Christodoulakis {\it et al.} …