Amir Sagiv, Department of Applied Mathematics, Columbia University

Floquet Hamiltonians - effective gaps and resonant decay

Floquet topological insulators are an emerging category of materials whose properties are transformed by time-periodic forcing. Can their properties be understood from their first-principles continuum models, i.e., from a driven Schrodinger equation?  

First, we study the transformation of graphene from a conductor into an insulator under a time-periodic magnetic potential. We show that the dynamics of certain wave-packets are governed by a Dirac equation, which has a spectral gap property. This gap is then carried back to the original Schrodinger equation in the form of an “effective gap” - a new and physically-relevant relaxation of a spectral gap.

Next, we consider periodic media with a localized defect, and ask whether edge/defect modes remain stable under forcing. In a model of planar waveguides, we see how such modes decay and disappear due to resonant coupling with the radiation modes.

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Igor Rumanov, Department of Applied Mathematics, �鶹ӰԺ

(2+1)-dimensional Whitham systems: 2dNLS and KP vs. ‘hydrodynamic’ systems in one and two spatial dimensions

The main result of this talk is the recently obtained Whitham modulation system for the (2+1)-dimensional nonlinear Schroedinger equation (2dNLS) (joint work with M. J. Ablowitz and J. T. Cole). The first applications demonstrating its validity are the linear stability analysis of plane periodic traveling waves with its help and its KP Whitham limit. The need for finding solutions of such systems raises a number of interesting questions.

I will review the 2dNLS and KP Whitham systems in the context of the theory of previously known hydrodynamic integrable systems solvable by the generalized hodograph method. Understanding the 2dNLS Whitham system may be achieved by studying its multiple interesting reductions including the KP Whitham system. While the KP Whitham system is simpler and supposed to be integrable, finding its general solutions is still a challenge. The theory of hydrodynamic reductions may help as well as a better understanding of the well-known (1+1)-dimensional Whitham systems.

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Samuel Ryskamp, Department of Applied Mathematics, �鶹ӰԺ

Modulation theory for Miles resonance and Mach reflection

Mach reflection occurs when a sufficiently large amplitude line soliton interacts with a barrier at a sufficiently small angle. A Y-shaped resonant triad is then formed consisting of two smaller amplitude solitons and a larger “Mach” stem. In this talk, I will present a new description of these Y-shaped "Miles" solitons, traveling wave solutions to the Kadomtsev-Petviashvili II (KPII) equation, as discontinuous shock solutions to an infinite family of soliton modulation equations.

The fully two-dimensional soliton modulation equations, valid in the zero-dispersion limit of the KPII equation, are first shown to reduce to a one-dimensional system. In this same limit, the rapid transition from the larger Y soliton stem to the two smaller legs compresses to a travelling discontinuity, which is a multivalued, weak solution to the one-dimensional modulation equations. These results are then used to analytically describe the dynamics of the Mach reflection problem where a line soliton is incident upon an inward oblique corner, reformulated as a Riemann problem for the modulation equations. Modulation theory results show excellent agreement with direct KPII numerical simulation, and they also reveal a remarkable symmetry with the dynamics of a soliton incident upon an outward oblique corner (the Mach expansion problem).  More broadly, this local description of Y-solitons within the context of modulation theory enables the construction of approximate solutions to more complex problems.

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Yi Zhu, Department of Mathematical Sciences, Tsinghua University, China

Three-fold Weyl points for the periodic Schrödinger operator 

Weyl points are degenerate points on the spectral bands at which energy bands intersect conically. They are the origins of many novel physical phenomena and have attracted much attention recently. In this talk, we investigate the existence of such points in the spectrum of the 3-dimensional Schrödinger operator H = −Δ+V (x) with V (x) being in a large class of periodic potentials.  To the best of our knowledge, this is the first result on the existence of Weyl points for a broad family of 3d continuous Schrödinger equations.  Indeed, we give very general conditions on the potentials which ensure the existence of 3-fold Weyl points on the associated energy bands. Different from 2-dimensional Dirac points where two adjacent band surfaces touch each other conically, the 3-fold Weyl points are conically intersection points of two energy bands with an extra band sandwiched in between.  We give the required conditions and  provide a comprehensive proof of such 3-fold Weyl points, which extends the Fefferman-Weinstein's strategy on the analysis of conical spectral points (JAMS 2012) to a higher dimension and to higher multiplicities. This talk is based on the joint work with H. Guo and M. Zhang at Tsinghua university.

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Patrick Sprenger, Department of Mathematics, North Carolina State University

Traveling wave solutions of the Kawahara equation

The Kawahara equation is an asymptotic model of weakly nonlinear wave phenomena when third and fifth order dispersion are in balance. The model equation consists of the Korteweg-de Vries equation with an additional fifth-order spatial derivative term. Traveling wave solutions of the Kawahara equation satisfy a fifth order ODE that can be integrated once, revealing the Hamiltonian structure of the resulting fourth order equation. Periodic and solitary traveling waves have generated an extensive literature related to Hamiltonian dynamics, but the fourth order traveling wave ODE allows for more general solutions including traveling waves that asymptote to distinct periodic wavetrains at infinity. Generically, these heterclinic traveling wave solutions represent a heteroclinic connections between two hyperbolic periodic orbits on a level-set of the spatial Hamiltonian and are identified by the intersection of the stable/unstable manifolds of the far-field periodic orbits.

The Hamiltonian structure of the traveling wave ODE accompanied by numerical computations of Floquet multipliers of periodic orbits reveals many such families of traveling waves. Moreover, each family bifurcates from a single elliptic, periodic traveling wave solution whose Floquet multipliers coalesce at +1. Analysis of the traveling wave Hamiltonian and numerical computations of traveling waves will be complemented by the interpretation of these solutions in terms of Whitham modulation theory. Within this framework, such heteroclinic traveling wave solutions of related models have been interpreted as discontinuous shock solutions of the Whitham modulation equations. 

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Speaker:  Dr. Thibault Congy

Affiliation:  University of Northumbria, Newcastle, UK

Title:  Dispersive Riemann problem for the Benjamin-Bona-Mahony equation

Abstract:

The Benjamin-Bona-Mahony (BBM) equation $u_t + uu_x = u_{xxt}$ as a model for unidirectional, weakly nonlinear dispersive shallow water wave propagation is asymptotically equivalent to the celebrated Korteweg-de Vries (KdV) equation while providing more satisfactory short-wave behavior in the sense that the linear dispersion relation is bounded for the BBM equation, but unbounded for the KdV equation. However, the BBM dispersion relation is nonconvex, a property that gives rise to a number of intriguing features markedly different from those found in the KdV equation, providing the motivation for the study of the BBM equation as a distinct dispersive regularization of the Hopf equation.

The dynamics of the smoothed step initial value problem or dispersive Riemann problem for BBM equation are studied using asymptotic methods and numerical simulations. I will present the emergent wave phenomena for this problem which can be split into two categories: classical and nonclassical. Classical phenomena include dispersive shock waves and rarefaction waves, also observed in convex KdV-type dispersive hydrodynamics. Nonclassical features are due to nonconvex dispersion and include the generation of two-phase linear wavetrains, expansion shocks, solitary wave shedding, dispersive Lax shocks, DSW implosion and the generation of incoherent solitary wavetrains.

This presentation is based on a joint work with G. A. El, M. Shearer and M. Hoefer, available at: 

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Transverse Instability of Rogue Waves

Rogue waves, or “freak waves”, are large amplitude waves that suddenly appear and then disappear. Originally the subject of folklore, these waves have now been successfully observed in numerous physical systems such as deep water waves and fiber optics. A typical rogue wave model is the one space, one time (1+1) nonlinear Schrodinger (NLS) equation and the Peregrine soliton solution which has a peak height three times that of the background. However, in deep water a more complete description is that of the 2+1 hyperbolic NLS equation with two significant transverse dimensions. It is shown that the Peregrine soliton is transversely unstable to both long and short wavelength perturbations of finite size. Moreover, the instability spectrum is found to coincide with that of the background plane wave.

Recorded video of this seminar is available (click ).

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Analytical study of Floquet topological insulators 

The search for novel phenomena in photonic waveguides centers on engineering systems that feature unique dispersive properties often involving spectral degeneracies. From optical graphene to unidirectional invisibility to the anomalous quantum Hall effect, spectral degeneracies are a driving factor even when the system has been perturbed away from the degenerate case. Recently this has extended the study of topological (global) properties of the spectrum. Here longitudinal driving of the waveguides produce topological insulators and protected edge modes. This talk will give an introduction to topological photonics and the analytical tools capable of deriving reduced dynamical systems to model the Floquet spectrum. These tool range from tight-binding approximations to multiple-scales analysis and provide an approach that will be applicable in a wide range of waveguide arrays with nontrivial topologies. Key topolical constants like the Chern number are obtained as well as governing equations for the envelope dynamics in the presence of Kerr nonlinearity. 

Recorded video of this seminar is available (click ).

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Vera Hur, Department of Mathematics, University of Illinois

Stokes waves in a constant vorticity flow: theory and numerics

Stokes in the 1800s made many contributions about periodic waves at the surface of water, under the influence of gravity, propagating in permanent form a long distance at a practically constant velocity. In an irrotational flow, for instance, he observed that crests become sharper and troughs flatter as the amplitude increases, and that the so-called wave of greatest height, or extreme wave, possesses a 120 degree's angle at the crest. The irrotational flow assumption is justified in many situations, and facilitates rigorous analysis and numerical computation. However, rotational effects are significant in many others. I will review recent progress in a constant vorticity flow. Numerical findings include folds and gaps in the wave speed vs. amplitude plane, and a profile enclosing multiple bubbles of fluids. I will discuss analytical and numerical applications if time permits.

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Thibault Congy; Department of Mathematics, Physics, and Electrical Engineering; University of Northumbria; Newcastle, UK

Bidirectional soliton gas

The soliton structure plays a fundamental role in many physical systems due to its fundamental feature: its shape remains unchanged after the collision with another soliton in the case of integrable dynamics. Such particle-like behaviour has been at the origin of a new mathematical object: the soliton gas, consisting of an incoherent collection of solitons for which phases (positions) and spectral parameters (e.g. amplitudes) are randomly distributed. The study of soliton gas involves the description of the gas dynamics as well as the corresponding modulation of nonlinear wave field statistics, which makes the soliton gas a particularly interesting embodiment of the particle-wave duality of solitons.

Motivated by the recent realisation of bidirectional soliton gases in a shallow water experiment, we investigate soliton gases of two bidirectional integrable dynamics: the nonlinear Schrödinger equation and the Kaup-Boussinesq equation. Using a physical approach, we derive the so-called kinetic equation that governs the gas dynamics for both integrable systems. We notably show that the structure of the kinetic equation depends on the "isotropic" or "anisotropic" nature of solitons interaction.  Additionally we derive expressions for statistical moments of the physical fields (e.g. mean water level). As an illustration of the theory, we solve numerically the gas shock tube problem describing the collision of two "cold" soliton gases.  An excellent agreement with the relevant exact solutions of the kinetic equations is observed.

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