# Poster Session and Reception

Tuesday, November 1, 2016 - 4:15pm - 6:00pm

Lind 400

**The Time-dependent Schrödinger Equation with Piecewise Constant Potentials**

Natalie Sheils (University of Minnesota, Twin Cities)

The linear Schrödinger equation with piecewise constant potential in one spatial dimension is a well-studied textbook problem. It is one of only a few solvable models in quantum mechanics and shares many qualitative features with physically important models. In examples such as particle in a box and tunneling, attention is restricted to the time-independent Schrödinger equation. This paper combines the Fokas method and recent insights for interface problems to present fully explicit solutions for the time-dependent problem.**Symmetry Breaking Bifurcation for Bound States in Nonlinear Schrodinger Equation**

Heeyeon Kim (University of Illinois at Urbana-Champaign)

We consider symmetry breaking bifurcation in NLS equation with attractive (focusing) nonlinearity. For sufficiently large separations of double well potentials, the second bifurcation of symmetric ground states occurs and a new branch past the bifurcation point is asymmetric. Moreover, the symmetric branch becomes orbitally unstable past the bifurcation point while the stability of the asymmetric branch depends on the power of nonlinearity.**Coherence and Nonclassicality in Noncommutative Spaces with Generalised Uncertainty Principle**

Sanjib Dey (University of Montreal)

We construct several quantum optical models; such as, coherent states, squeezed states, Schrodinger cat states and photon-added coherent states based on a noncommutative space arising from the generalised uncertainty principle. In spite of arising from mathematical backgrounds, we indicate the physical reality of our models with the help of the standard techniques of PT-symmetry. We analyse the classical-like as well as various nonclassical properties of our models by computing the existing mechanisms of quadrature and photon number squeezing, entanglement, etc, however, they are more challenging from the computational point of view.**Asymptotics and Computations of Band-Edge Solitons in PT-symmetric NLS Equations with Periodic Potentials**

Jessica Taylor (University of California, Merced)

The bifurcation of nonlinear bound states from the spectral edges of the NLS equation with periodic parity-time (PT)- symmetric potentials is studied asymptotically and computationally. These modes undergo a transition near the breakdown point of the PT symmetry. This is joint work with Boaz Ilan.**Quantum Hydrodynamic Theory for Nonlinear Plasmonics**

Cristian Ciraci (Istituto Italiano di Tecnologia)

Metallic structures supporting gap-plasmon modes have the ability to squeeze the light in nanometer size volumes producing incredibly strong local fields. Advances in nano-fabrication techniques have made possible to achieve distances between to metallic elements (i.e. between two nanoparticles, or between nanoparticles and a metal film) of only few nanometers. At such distances nonlocal or quantum effects become non-negligible, producing modifications of the macroscopic optical response of the system. At the same time, metals are compelling candidates for nonlinear optics, as they possess nonlinear susceptibilities that are orders of magnitude larger than dielectric materials. The origin of nonlinearity in metals arises from both volume and surface contributions. Nonlinear surface contributions are strictly related to the response of the electrons within few angstroms from the metal boundaries. In this sub-nanometer realm, electron-electron interactions become non-negligible, and quantum and nonlocal effects must be taken into account. While the nonlinear hydrodynamic model (within the limit of the Tomas-Fermi theory) has been successfully used to describe second-harmonic generation in metallic systems, its basic assumption of a constant equilibrium electron density, prevents its applicability to more extreme condition in which a precise description of the electron density profile is required.

We present a generalization of the nonlinear hydrodynamic theory that takes into account a more accurate description of the free-electron gas internal energy by adding to the Thomas-Fermi energy functional, the nonlocal (i.e. depending on the gradient of the charge density) von Weizsäcker energy functional. We show that the hydrodynamic equation can be directly obtained from a single particle Kohn-Sham equation that includes the contribution of an external vector potential. This derivation allows to straightforwardly incorporate in the hydrodynamic equation an exchange-correlation viscoelastic term, so that broadening of collective excitation can be taken into account, as well as a correction to the plasmon dispersion. The result is an accurate self-consistent and computationally efficient hydrodynamic description of the free electron gas.**On the Hénon Equation with a Neumann Boundary Condition: Asymptotic Profile of Ground States**

Sangdon Jin (Korea Advanced Institute of Science and Technology (KAIST))

We consider the Hénon equation with the homogeneous Neumann boundary condition. We are concerned on the existence of a ground state solution and its asymptotic behavior as the parameter approaches to a threshold for an existence and a nonexistence of a ground state solution.**Exact Optical Solitons in Optical Fibers by Fan Sub-equation Method**

Muhammad Younis (University of the Punjab)

This paper studies the optical solitons in optical fibers. The dynamics of the improved perturbed nonlinear Schrodinger equation, which describes the propagation of optical solitons through optical fibers in nonlinear optics, is investigated analytically by the extended Fan sub-equation method. As a result, explicit soliton solutions and trigonometric function travelling wave solutions are constructed. The important fact of this method is to take the full advantage of clear relationship among general elliptic equation involving five parameters and other existing sub-equations involving three parameters. So it is preferable to use this method to solve improved perturbed nonlinear Schrodinger equation because this method gives us all the solutions obtained previously by the application of at least four methods (the method of using Riccati equation, or auxiliary ordinary differential equation method, or first kind elliptic equation or the generalized Riccati equation as mapping equation) in a unified manner.**Spatiotemporal Optical Vortices**

Nihal Jhajj (University of Maryland)

Spatiotemporal optical vortices (STOVs) are phase vortices carried by pulsed beams with phase circulation in both spatial and temporal dimensions. We present the first experimental detection of a STOV, where STOVs are seen to be self-generated by nonlinearly propagating beams. A simple model is developed which argues that STOV generation is a universal feature of the optical collapse arrest of a pulsed beam. STOVs exhibit qualitatively different dynamics than spatial optical vortices (heretofore simply referred to as optical vortices) where the phase circulation exists purely in spatial dimensions. Principle among these differences is the energy flow about the vortex. Along the temporal dimension, energy flow scales with the material dispersion, allowing for both saddle and circular flow in regularly and anomalously dispersive media respectively. Moving forward, we hope to identify and characterize the complex pulse dynamics of post collapse beams using changes to the phase topology associated with the creation and destruction of vortices.**Linearization of the Gross-Pitaevskii Equation around a Vacuum State and the Black Soliton: Low Frequency Effects in 1D**

Numann Malik (Brown University)

We consider a cubic Nonlinear Schrodinger (NLS) equation subject to non-vanishing boundary conditions. Linearizing around a vacuum state solution and the black soliton, yields evolution equations in matrix form (we assume the perturbation is small and complex-valued). We study the long-time asymptotics by carrying out a spectral and Fourier analysis to derive a formula for the propagators using the resolvent kernel. This is obtained from explicit squared Jost solutions.

This work is motivated by research conducted several years ago where it was shown that a shelf develops around dark solitons which propagate with speed determined by the background intensity. The focusing NLS exhibits the much better understood bright solitons (pulses that decay rapidly at infinity). In the defocusing regime, decaying pulses broaden and bright solitons do not exist. Instead solitons can be found as localized dips in intensity that decay off of a continuous wave background. These dark solitons are termed black when the intensity of the dip goes to zero, and gray otherwise. The experimental observations of dark solitons in both fibre optics and planar waveguides sparked significant interest in the asymptotic analysis of their propagation.**Phase Separation from Directional Quenching**

Rafael Monteiro da Silva (University of Minnesota, Twin Cities)

We study the effect of directional quenching on patterns formed in simple bistable systems such as the Allen-Cahn and the Cahn-Hilliard equation on the plane. We model directional quenching as an externally triggered change in system parameters, changing the system from monostable to bistable across an interface. We are then interested in patterns forming in the bistable region, in particular as the trigger progresses and increases the bistable region. We find existence and non-existence results of single interfaces and striped patterns. Joint work with Arnd Scheel.**Second Harmonic Imaging in Random Media**

Wei Li (University of Minnesota, Twin Cities)

We consider the imaging of small nonlinear scatterers in random media. This problem is analyzed in weakly scattering media which respond linearly to light. We show that for propagation distances within a few transport mean free paths, robust images can be constructed by the coherent interferometry (CINT) imaging functions. We also show that imaging the quadratic susceptibility with CINT yields better result, because that the CINT imaging function for the linear susceptibility has noisy peaks in a region that depends on the geometry of the aperture and the cone of incident directions.**Wavenumber Selection via Spatial Parameter Step**

Jasper Weinrich-Burd (University of Minnesota, Twin Cities)

The Swift-Hohenberg (SH) equation is a prototypical, pattern-forming model equation for Turing instability phenomena. We study the SH equation with a space-dependent parameter that renders the medium stable in $x0$. We construct a family of steady-state solutions that represent a transition in space from a homogeneous state to a striped state. Solutions in the family vanish as $x\to -\infty$ and are asymptotically periodic as $x\to \infty$. Wavenumbers appearing asymptotically in the family are restricted to a neighborhood of 1 with width depending linearly on the size of the step.**The Kawahara Equation in Nonlinear Optics**

Mark Hoefer (University of Colorado)