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An Excursion from Enumerative Geometry to Solving Systems of Polynomial Equations with Macaulay 2
To solve the polynomial equations mathematics geometry cross theory
2014/12/29
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics ha...
Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions
The coupling instability nonlinear schrodinger equation matter waves lens type conversion
2014/12/29
The modulational instability of the nonlinear Schrödinger (NLS) equation is examined in the case with a quadratic external potential. This study is motivated by recent experimental results in the...
Modulational instabilities and domain walls in coupled discrete nonlinear Schrödinger equations
Discrete nonlinear schrodinger equation linear coupling plane wave coupling
2014/12/29
We consider a system of two discrete nonlinear Schrödinger equations, coupled by nonlinear and linear terms. For various physically relevant cases, we derive a modulational instability criterion ...
Solitary wave interactions in dispersive equations using Manton’s approach
Invariant to calculate solitary wave dispersion equation solitary wave vries equation and nonlinear schrodinger equation
2014/12/29
We generalize the approach first proposed by Manton [Nucl. Phys. B 150, 397 (1979)] to compute solitary wave interactions in translationally invariant, dispersive equations that support such localized...
Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein–Gordon equations
Discrete schrodinger propagator klein Gordon equation form the analysis of decay estimate
2014/12/29
We show decay estimates for the propagator of the discrete Schrödinger and Klein–Gordon equations in the form {{\| {U(t)f} \|}_{{l^\infty}}} \leq C (1+|t|)^{-d/3}{{\| {f} \|}_{{l^1}}} . This impl...
Discrete nonlinear Schrödinger equations free of the Peierls–Nabarro potential
Discrete nonlinear schrodinger equation nonlinear discrete fixed discrete soliton
2014/12/25
We derive a class of discrete nonlinear Schrödinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is ...
On a class of spatial discretizations of equations of the nonlinear Schrödinger type
Discrete nonlinear schrodinger equation nonlinear discrete model and discrete
2014/12/25
We demonstrate the systematic derivation of a class of discretizations of nonlinear Schrödinger (NLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a ...
Emergence of unstable modes in an expanding domain for energy-conserving wave equations
Simulated energy wave equation the nonlinear schrodinger equation the coupling
2014/12/25
Motivated by recent work on instabilities in expanding domains in reaction–diffusion settings, we propose an analog of such mechanisms in energy-conserving wave equations. In particular, we consider a...
SYMMETRY-BREAKING BIFURCATION IN NONLINEAR SCHRODINGER/GROSS-PITAEVSKII EQUATIONS
Nonlinear schrodinger mathematical modeling nonlinear optics macroscopic quantum phenomena the number of particles the mixed pattern
2014/12/25
We consider a class of nonlinear Schrödinger/Gross–Pitaeveskii (NLS-GP) equations, i.e., NLS with a linear potential. NLS-GP plays an important role in the mathematical modeling of nonlinear opti...
Soliton dynamics in linearly coupled discrete nonlinear Schrodinger equations
Soliton dynamics system coupling discrete nonlinear schrodinger equation
2014/12/24
We study soliton dynamics in a system of two linearly coupled discrete nonlinear Schrödinger equations, which describe the dynamics of a two-component Bose gas, coupled by an electromagnetic fiel...
Short pulse equations and localized structures in frequency band gaps of nonlinear metamaterials
Short pulse propagation nonlinear metamaterials nonlinear dielectric short pulse linear electromagnetic waves magnetic permeability
2014/12/24
We consider short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. Two short-pulse equations (SPEs) are derived for the high- a...
Deterministic Equations for Stochastic Spatial Evolutionary Games
Coarse-grained dynamics monte carlo simulation grid the stochastic dynamics systematic lattice spin flip dynamics
2014/12/24
In this paper we investigate the approximation properties of the coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice stochastic dynamics. We provide both analytical and num...
Deterministic Equations for Stochastic Spatial Evolutionary Games
Coarse-grained dynamics monte carlo simulation grid the stochastic dynamics systematic lattice spin flip dynamics
2014/12/24
In this paper we investigate the approximation properties of the coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice stochastic dynamics. We provide both analytical and num...
Penalized importance sampling for parameter estimation in stochastic differential equations
Chronic wasting disease Euler-Maruyama scheme Maximum likelihood estimation Partially observed discrete sparse data Penalized importance sampling Stochastic di
2013/6/14
We consider the problem of estimating parameters of stochastic differential equations with discrete-time observations that are either completely or partially observed. The transition density between t...
Estimation in Systems of Ordinary Differential Equations Linear in the Parameters
local polynomials Lotka-Volterra nonparamet-ric regression ordinary differential equation plug-in estimators
2013/6/14
Many phenomena in biology, chemistry, physics, and engineering are modeled by a system of possibly nonlinear ordinary differential equations that are linear in their unknown constants. Current methods...