Nonlinear Waves Seminar

Nalini Joshi, Department of Mathematics, Sydney University When Applied Mathematics Collided with Algebra Imagine walking from one tile to another on a lattice defined by reflections associated with an affine Coxeter or Weyl group. Examples include triangular or hexagonal lattices on the plane. Recently, it was discovered that translations on...

Justin Cole, Department of Applied Mathematics, University of Colorado Boulder Soliton Dynamics in the Korteweg-de Vries Equation with Nonzero Boundary Conditions Inspired by recent experiments, the Korteweg-de Vries equation with nonzero Dirichlet boundary conditions is considered. Two types of boundary data are examined: step up (which generates a rarefaction wave)...

n Patrick Sprenger, Department of Applied Mathematics, University of Colorado Boulder Generalized Riemann problems in dispersive hydrodynamics Nonlinear, dispersive wave phenomena are observed in a variety of physical contexts in nature and the laboratory. Mathematically, the dynamics are described by a dispersive hydrodynamic system---conservation laws modified by dispersion. Oftentimes, a...

Justin Cole Department of Applied Mathematics, University of Colorado Boulder Soliton Dynamics in the Korteweg-de Vries Equation with Nonzero Boundary Conditions Inspired by recent experiments, the Korteweg-de Vries equation with nonzero Dirichlet boundary conditions is considered. Two types of boundary data are examined: step up (which generates a rarefaction wave)...

S. Chakravarty, Department of Mathematics, University of Colorado Colorado Springs A class of rational solutions of the KPI equation I will revisit a class of KP "lump" solutions which were studied by Ablowitz & Villarroel as rational potentials of the non-stationary Schroedinger equation. Unlike other KPI lump solutions (simple lumps)...

Patrick Weidman Department of Mechanical Engineering, University of Colorado Boulder Steady flow of one uniformly rotating fluid layer above another immiscible uniformly rotating fluid layer. The steady laminar flow of two immiscible, uniformly rotating fluid layers is studied and exact similarity solutions of the axisymmetric Navier-Stokes equations in cylindrical polar...

Whitham theory and dispersive shock waves (DSWs) for the radial nonlinear Schroedinger (rNLS) equation. Dispersive shock waves of the defocusing rNLS equation in two spatial dimensions are studied. This equation arises naturally in Bose-Einstein condensates, water waves and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable and...

Experimental Observation of Spin-Wave Fractals A fractal is a shape made of parts each of which is similar to the whole in some way. One can group fractals into two main categories, (i) exact fractals in which the same feature replicates itself on successively smaller scales and (ii) statistical fractals...

Whitham modulation theory - developments and open problems I discuss the development of Whitham modulation theory as a nonlinear WKB method successfully used to describe the behavior of nonlinear dispersive waves. Recent advances include the appearance of the Whitham theory for (2+1)-dimensional evolution equations of Kadomtsev-Petviashvili (KP) type. The systems...

Generalized dispersion relation predicts harmonic generation in strongly nonlinear systems In recent work, we have derived an exact nonlinear dispersion relation for elastic wave propagation in a thin rod (linearly nondispersive) and a thick rod (linearly dispersive). The derivation is generally applicable to any type of nonlinearity, geometric (related to...