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DOE OSTI2021

Solitons in lattice field theories via tight-binding supersymmetry

Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems (such as the Korteweg-de Vries equation), non-perturbative solutions of various large-N field theories (such as the Gross-Neveu model), and closely related solitonic solutions to the Bogoliubov-de Gennes equations in the theory of superconductivity. These solutions rely on the inverse scattering method, which reduces these seemingly unrelated problems to identifying reflectionless potentials of an auxiliary one-dimensional quantum scattering problem. There are several ways of constructing these potentials, one of which is quantum mechanical supersymmetry (SUSY). In this paper, motivated by recent experimental platforms, we generalize this framework to develop a theory of lattice solitons. We first briefly review the classical inverse scattering method in the continuum limit, focusing on the Korteweg-de Vries (KdV) equation and SU(N) Gross-Neveu model in the large N limit. We then generalize this methodology to lattice versions of interacting field theories. Our analysis hinges on the use of trace identities, which are relations connecting the potential of an equation of motion to the scattering data. For a discrete Schrödinger operator, such trace identities had been known as far back as Toda; however, we derive a new set of identities for the discrete Dirac operator. We then use these identities in a lattice Gross-Neveu and chiral Gross-Neveu (Nambu-Jona-Lasinio) model to show that lattice solitons correspond to reflectionless potentials associated with the discrete scattering problem. These models are of significance as they are equivalent to a mean-field theory of a lattice superconductor. To explicitly construct these solitons, we generalize supersymmetric quantum mechanics to tight-binding models. We show that a matrix transformation exists that maps a tight-binding model to an isospectral one which shares the same structure and scattering properties. The corresponding soliton solutions have both modulated hopping and onsite potential, the former of which has no analogue in the continuum limit. We explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2026

From the Great Wave of Translation to the Force between Quarks

Here, the chance observation of a novel traveling wave in a canal led over time to the formulation of a nonlinear wave equation—the Korteweg–de Vries equation—that describes strikingly robust disturbances now called solitons. The figure of an isolated soliton corresponds to a reflectionless potential that supports a single bound state in the one-dimensional Schrödinger equation. An appropriate combination of individual solitons yields a symmetric reflectionless potential that supports multiple bound states. Thus, the KdV equation opens the path to solving the inverse scattering problem for a collection of bound states. Applied to the quarkonium spectra, this formalism allows the construction of reflectionless approximations to the confining potentials that account for the force between quarks, and to tests of the flavor-independence of the interquark interaction.

Inverse Scattering↗

NASA NTRS1991

Experiment and theoretical study of the propagation of high power microwave pulse in air breakdown environment

In the study of the propagation of high power microwave pulse, one of the main concerns is how to minimize the energy loss of the pulse before reaching the destination. In the very high power region, one has to prevent the cutoff reflection caused by the excessive ionization in the background air. A frequency auto-conversion process which can lead to reflectionless propagation of powerful EM pulses in self-generated plasmas is studied. The theory shows that under the proper conditions the carrier frequency, omega, of the pulse will indeed shift upward with the growth of plasma frequency, omega(sub pe). Thus, the plasma during breakdown will always remain transparent to the pulse (i.e., omega greater than omega(sub pe)). A chamber experiment to demonstrate the frequency auto-conversion during the pulse propagation through the self-generated plasma is then conducted in a chamber. The detected frequency shift is compared with the theoretical result calculated y using the measured electron density distribution along the propagation path of the pulse. Good agreement between the theory and the experiment results is obtained.

Kuo, S. P.↗

NASA NTRS1992

Frequency up-conversion of a high-power microwave pulse propagating in a self-generated plasma

In the study of the propagation of a high-power microwave pulse, one of the main concerns is how to minimize the energy loss of the pulse before reaching the destination. A frequency autoconversion process that can lead to reflectionless propagation of powerful electromagnetic pulses in self-generated plasmas is studied. The theory shows that, under the proper condition, the carrier frequency omega of the pulse shifts upward during the growth of local plasma frequency omega(pe). Thus, the self-generated plasma remains underdense to the pulse. A chamber experiment to demonstrate the frequency autoconversion during the pulse propagation through the self-generated plasma is conducted. The detected frequency shift is compared with the theoretical result calculated by using the measured electron density distribution along the propagation path of the pulse. Good agreement is obtained.

Kuo, S. P.↗

NASA NTRS1990

Observation of frequency up-conversion in the propagation of a high-power microwave pulse in a self-generated plasma

A chamber experiment is conducted to study the propagation of a high-power microwave pulse. The results show that the pulse is experiencing frequency up-shift while ionizing the background air if the initial carrier frequency of the pulse is higher than the electron plasma frequency at the incident boundary. Such a frequency autoconversion process may lead to reflectionless propagation of a high-power microwave pulse through the atmosphere.

Kuo, S. P.↗

NASA NTRS2000

High-order Two-way Artificial Boundary Conditions for Nonlinear Wave Propagation with Backscattering

When solving linear scattering problems, one typically first solves for the impinging wave in the absence of obstacles. Then, by linear superposition, the original problem is reduced to one that involves only the scattered waves driven by the values of the impinging field at the surface of the obstacles. In addition, when the original domain is unbounded, special artificial boundary conditions (ABCs) that would guarantee the reflectionless propagation of waves have to be set at the outer boundary of the finite computational domain. The situation becomes conceptually different when the propagation equation is nonlinear. In this case the impinging and scattered waves can no longer be separated, and the problem has to be solved in its entirety. In particular, the boundary on which the incoming field values are prescribed, should transmit the given incoming waves in one direction and simultaneously be transparent to all the outgoing waves that travel in the opposite direction. We call this type of boundary conditions two-way ABCs. In the paper, we construct the two-way ABCs for the nonlinear Helmholtz equation that models the laser beam propagation in a medium with nonlinear index of refraction. In this case, the forward propagation is accompanied by backscattering, i.e., generation of waves in the direction opposite to that of the incoming signal. Our two-way ABCs generate no reflection of the backscattered waves and at the same time impose the correct values of the incoming wave. The ABCs are obtained for a fourth-order accurate discretization to the Helmholtz operator; the fourth-order grid convergence is corroborated experimentally by solving linear model problems. We also present solutions in the nonlinear case using the two-way ABC which, unlike the traditional Dirichlet boundary condition, allows for direct calculation of the magnitude of backscattering.

Fibich, Gadi↗

DOE OSTI2021

Wormhole as a waveguide for non-relativistic quantum particles

Herein we consider a static wormhole as a waveguide and determine the conditions for full transmission through the wormhole waveguide for a quantum particle satisfying Schrodinger equation. We find that the waveguide is transparent when the angular momentum L of the quantum particle is 0 and for $L\ne 0$ when the de Broglie wavelength of the quantum particle is a multiple depending on the inverse of the angular momentum times the throat radius of the wormhole.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗