Speed-of-light pulses in the massless nonlinear Dirac equation with a potential

We consider the massless nonlinear Dirac (NLD) equation in $1+1$ dimension with scalar-scalar self-interaction $\frac{g^2}{2} (\bar{\Psi} \Psi)^2$ in the presence of three external electromagnetic potentials $V(x)$, a potential barrier, a constant potential, and a potential well. By solving numerically the NLD equation, we find that, for all three cases, after a short transit time, the initial pulse breaks into two pulses which are solutions of the massless linear Dirac equation traveling in opposite directions with the speed of light. During this splitting the charge and the energy are conserved, whereas the momentum is conserved when the solutions possess specific symmetries. For the case of the constant potential, we derive exact analytical solutions of the massless NLD equation that are also solutions of the massless linearized Dirac equation.Comment: 11 pages, 7 figure

Credit Availability for Potential Irrigators in North Dakota

Agricultural Finance, Resource /Energy Economics and Policy,

Nonlinear Dirac equation solitary waves under a spinor force with different components

We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar selfinteraction in the presence of external forces as well as damping of the form γͦf(x-t) - ιμγͦψ, where both f, {fj = rieiKjx} and ψ are two-component spinors. We develop an approximate variational approach using collective coordinates (CC) for studying the time dependent response of the solitary waves to these external forces. In our previous paper we assumed Kj = K, j = 1,2 which allowed a transformation to a simplifying coordinate system, and we also assumed the "small" component of the external force was zero. Here we include the effects of the small component and also the case K1 ≠ K2 which dramatically modi es the behavior of the solitary wave in the presence of these external forces.United States Department of EnergySanta Fe InstituteNational Natural Science Foundation of China (Nos. 11471025 and 11421101)Alexander von Humboldt Foundation (Germany) through Research Fellowship for Experienced Researchers SPA 1146358 STPMinisterio de Economía y Competitividad (Spain) through FIS2014-54497-PJunta de Andalucía (Spain) under Projects No. FQM207Excellent Grant P11-FQM-7276Mathematical Institute of the University of Seville (IMUS)Theoretical Division and Center for Nonlinear Studies at Los Alamos National LaboratoryPlan Propio of the University of Sevill

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

We consider the nonlinear Dirac equation in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{\kappa+1} ({\bar \Psi} \Psi)^{\kappa+1}$ and with mass $m$. Using the exact analytic form for rest frame solitary waves of the form $\Psi(x,t) = \psi(x) e^{-i \omega t}$ for arbitrary $\kappa$, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schr\"odinger equation. In particular we study the validity of a version of Derrick's theorem, the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed 4th-order operator splitting integration method. For different ranges of $\kappa$ we map out the stability regimes in $\omega$. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time $t_c$, it takes for the instability to set in, is an exponentially increasing function of $\omega$ and $t_c$ decreases monotonically with increasing $\kappa$.Comment: 35 pages, 13 figure

Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter κ is analyzed, when the external force is periodic in space and given by f(x) = r cos(Kx), both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force λ = 2π/K, and the soliton width ls, which is a function of the nonlinearity κ as well as the initial frequency ω0 of the solitary wave, there is a transition (at fixed ω0) from trapped to unbound behavior of the soliton, which depends on the parameters r and K of the external force and the nonlinearity parameter κ. We previously studied this phenomena when κ = 1 (2019 J. Phys. A: Math. Theor. 52 285201) where we showed that for λ ≫ ls the soliton oscillates in an effective potential, while for λ ≪ ls it moves uniformly as a free particle. In this paper we focus on the κ dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of r and K when κ = 1/2,1,2 at fixed value of the frequency ω0. Since the solitary wave gets narrower for fixed ω0 as a function of κ, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase κ.Ministerio de Economía y Competitividad of Spain FIS2017-89349-PMinisterio de Ciencia, Innovación y Universidades of Spain PGC2018-093998-B-I0

Effect of Hydrocarbon Production and Depressurization on Subsidence and Possible Fault Reactivation: Port Acres-Port Arthur Field Area, Southeast, Texas

Subsidence has been extensive in the coastal area of southeast Texas. Despite enormous hydrocarbon production in the area, however, most subsidence has been attributed more to regional shallow groundwater withdrawal than to hydrocarbon production. The impact of hydrocarbon production on subsidence can be accurately quantified only where the effects of groundwater withdrawal are minimal. The Port Acres and Port Arthur field area satisfies this requirement. More than 380 Bcf of gas has been produced from the Port Acres and Port Arthur field area. Pressure in the Hackberry reservoir declined from an original 9,000 psi to less than 3,000 psi by the 1970s, then to less than 2,000 psi by the 1980s. The pressure drop from 9,000 to 1,000 psi could produce a maximum subsidence of 6 percent at reservoir depth. Assuming an average gas column of 50 to 120 ft, estimated compaction of the Hackberry reservoir is 2 to 5 ft (0.61 to 1.63 m), which is consistent with reported surface subsidence.Bureau of Economic Geolog

Locating the source of projectile fluid droplets

The ill-posed projectile problem of finding the source height from spattered droplets of viscous fluid is a longstanding obstacle to accident reconstruction and crime scene analysis. It is widely known how to infer the impact angle of droplets on a surface from the elongation of their impact profiles. However, the lack of velocity information makes finding the height of the origin from the impact position and angle of individual drops not possible. From aggregate statistics of the spatter and basic equations of projectile motion, we introduce a reciprocal correlation plot that is effective when the polar launch angle is concentrated in a narrow range. The vertical coordinate depends on the orientation of the spattered surface, and equals the tangent of the impact angle for a level surface. When the horizontal plot coordinate is twice the reciprocal of the impact distance, we can infer the source height as the slope of the data points in the reciprocal correlation plot. If the distribution of launch angles is not narrow, failure of the method is evident in the lack of linear correlation. We perform a number of experimental trials, as well as numerical calculations and show that the height estimate is insensitive to aerodynamic drag. Besides its possible relevance for crime investigation, reciprocal-plot analysis of spatter may find application to volcanism and other topics and is most immediately applicable for undergraduate science and engineering students in the context of crime-scene analysis.Comment: To appear in the American Journal of Physics (ms 23338). Improved readability and organization in this versio

Response of exact solutions of the nonlinear Schrodinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry

We discuss the effect of small perturbation on nodeless solutions of the nonlinear \Schrodinger\ equation in 1+1 dimensions in an external complex potential derivable from a parity-time symmetric superpotential that was considered earlier [Phys.~Rev.~E 92, 042901 (2015)]. In particular we consider the nonlinear partial differential equation \{ \, \rmi \, \partial_t + \partial_x^2 + g |\psi(x,t)|^2 - V^{+}(x) \, \} \, \psi(x,t) = 0, where V^{+}(x) = \qty( -b^2 - m^2 + 1/4 ) \, \sech^2(x) - 2 i \, m \, b \, \sech(x) \, \tanh(x) represents the complex potential. Here we study the perturbations as a function of $b$ and $m$ using a variational approximation based on a dissipation functional formalism. We compare the result of this variational approach with direct numerical simulation of the equations. We find that the variational approximation works quite well at small and moderate values of the parameter $b m$ which controls the strength of the imaginary part of the potential. We also show that the dissipation functional formalism is equivalent to the generalized traveling wave method for this type of dissipation.Comment: 18 pages, 6 figure