Wave propagation across interfaces induced by different interaction exponents in ordered and disordered Hertz-like granular chains

We study solitary wave propagation in 1D granular crystals with Hertz-like interaction potentials. We consider interfaces between media with different exponents in the interaction potential. For an interface with increasing interaction potential exponent along the propagation direction we obtain mainly transmission with delayed secondary transmitted and reflected pulses. For interfaces with decreasing interaction potential exponent we observe both significant reflection and transmission of the solitary wave, where the transmitted part of the wave forms a multipulse structure. We also investigate impurities consisting of beads with different interaction exponents compared to the media they are embedded in, and we find that the impurities cause both reflection and transmission, including the formation of multipulse structures, independent of whether the exponent in the impurities is smaller than in the surrounding media. We explain wave propagation effects at interfaces and impurities in terms of quasi-particle collisions. Next we consider wave propagation along Hertz-like granular chains of beads in the presence of disorder and periodicity in the interaction exponents present in the Hertz-like potential, modelling, for instance, inhomogeneity in the contact geometry between beads in the granular chain. We find that solitary waves in media with randomised interaction exponents (which models disorder in the contact geometry) experience exponential decay, where the dependence of the decay rate is similar to the case of randomised bead masses. In the periodic case of chains with interaction exponents alternating between two fixed values, we find qualitatively different propagation properties depending on the choice of the two exponents. In particular, we find regimes with either exponential decay or stable solitary wave propagation with pairwise collective behaviour.Comment: 33 pages, 28 figure

Reduction of dimension for nonlinear dynamical systems

We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for computing the solution when using a variety of analytical approaches. In the case where this reduction is possible, we employ differential elimination to obtain the reduced system. While analytical, the approach is algorithmic, and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}. In other cases, the reduction cannot be performed strictly in terms of differential operators, and one obtains integro-differential operators, which may still be useful. In either case, one can use the reduced equation to both approximate solutions for the state variables and perform chaos diagnostics more efficiently than could be done for the original higher-dimensional system, as well as to construct Lyapunov functions which help in the large-time study of the state variables. A number of chaotic and hyperchaotic dynamical systems are used as examples in order to motivate the approach.Comment: 16 pages, no figure

Decay of helical Kelvin waves on a quantum vortex filament

We study the dynamics of helical Kelvin waves moving along a quantum vortex filament driven by a normal fluid flow. We employ the vector form of the quantum local induction approximation (LIA) due to Schwarz. For an isolated filament, this is an adequate approximation to the full Hall-Vinen-Bekarevich-Khalatnikov dynamics. The motion of such Kelvin waves is both translational (along the quantum vortex filament) and rotational (in the plane orthogonal to the reference axis). We first present an exact closed form solution for the motion of these Kelvin waves in the case of a constant amplitude helix. Such solutions exist for a critical wave number and correspond exactly to the Donnelly-Glaberson instability, so perturbations of such solutions either decay to line filaments or blow-up. This leads us to consider helical Kelvin waves which decay to line filaments. Unlike in the case of constant amplitude helical solutions, the dynamics are much more complicated for the decaying helical waves, owing to the fact that the rate of decay of the helical perturbations along the vortex filament is not constant in time. We give an analytical and numerical description of the motion of decaying helical Kelvin waves, from which we are able to ascertain the influence of the physical parameters on the decay, translational motion along the filament, and rotational motion, of these waves (all of which depend nonlinearly on time). One interesting finding is that the helical Kelvin waves do not decay uniformly. Rather, suchwaves decay slowly for small time scales, and more rapidly for large time scales. The rotational and translational velocity of the Kelvin waves depend strongly on this rate of decay, and we find that the speed of propagation of a helical Kelvin wave along a quantum filament is large for small time while the wave asymptotically slows as it decays. The rotational velocity of such Kelvin waves along the filament will increase over time, asymptotically reaching a finite value. These decaying Kelvin waves correspond to wave number below the critical value for the Donnelly-Glaberson instability, and hence our results on the Schwarz quantum LIA correspond exactly to what one would expect from prior work on the Donnelly-Glaberson instability

Computing semi-commuting differential operators in one and multiple variables

We discuss the concept of what we refer to as semi-commuting linear differential operators. Such operators hold commuting operators as a special case. In particular, we discuss a heuristic by which one may construct such operators. Restricting to the case in which one such operator is of degree two, we construct families of linear differential operators semi-commuting with some named operators governing special functions (with a focus on the hypergeometric case, as it holds many other cases as reductions); operators commuting with such special degree two operators will necessarily be contained in these families. In the partial differential operator case, we consider examples in the form of the wave equation with a variable wave speed, and then hypergeometric operators on two variables (such operators define Appell functions)

General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation

In his study of superfluid turbulence in the low-temperature limit, Svistunov [ Superfluid turbulence in the low-temperature limit, Phys. Rev. B 52, 3647 (1995)] derived a Hamiltonian equation for the self-induced motion of a vortex filament. Under the local induction approximation (LIA), the Svistunov formulation is equivalent to a nonlinear dispersive partial differential equation. In this paper, we consider a family of rotating vortex filament solutions for the LIA reduction of the Svistunov formulation, which we refer to as the 2D LIA (since it permits a potential formulation in terms of two of the three Cartesian coordinates). This class of solutions holds the well-known Hasimoto-type planar vortex filament [H. Hasimoto, Motion of a vortex filament and its relation to elastica, J. Phys. Soc. Jpn. 31, 293 (1971)] as one reduction and helical solutions as another. More generally, we obtain solutions which are periodic in the space variable. A systematic analytical study of the behavior of such solutions is carried out. In the case where vortex filaments have small deviations from the axis of rotation, closed analytical forms of the filament solutions are given. A variety of numerical simulations are provided to demonstrate the wide range of rotating filament behaviors possible. Doing so, we are able to determine a number of vortex filament structures not previously studied. We find that the solution structure progresses from planar to helical, and then to more intricate and complex filament structures, possibly indicating the onset of superfluid turbulence

Exact solution for the self-induced motion of a vortex filament in the arc-length representation of the local induction approximation

We review two formulations of the fully nonlinear local induction equation approximating the self-induced motion of the vortex filament (in the local induction approximation), corresponding to the Cartesian and arc-length coordinate systems. The arc-length representation put forth by Umeki [Theor. Comput. Fluid Dyn. 24, 383 (2010)] results in a type of 1 + 1 derivative nonlinear Schrodinger (NLS) equation describing the motion of such a vortex filament. We obtain exact stationary solutions to this derivative NLS equation; such exact solutions are a rarity. These solutions are periodic in space and we determine the nonlinear dependence of the period on the amplitude