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Fred Cooper

Publications and source records attributed to Fred Cooper.

At least 37 records · Page 2Linked to original sources
arXiv2019

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 $f(x) =r\cos(K x)$, 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 $λ=2 π/K$, and the soliton width $l_s$, which is a function of the nonlinearity $κ$ as well as the initial frequency $ω_0$ of the solitary wave, there is a transition (at fixed $ω_0$) 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. {\bf 52} 285201) where we showed that for $λ\gg l_s$ the soliton oscillates in an effective potential, while for $λ\ll l_s$ 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 $ω_0$. Since the solitary wave gets narrower for fixed $ω_0$ as a function of $κ$, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase $κ$.

nlin.PS↗
arXiv2019

Composite Molecules and Decoupling in Reaction Diffusion Models

The Gray-Scott model can be thought of as an effective theory at large spatiotemporal scales coming from a more fundamental theory valid at shorter spatiotemporal scales. The more fundamental theory includes a composite molecule which is trilinear in the molecules of the Gray-Scott model as was shown in the recent derivation of the Gray-Scott model from the master equation. Here we show that at a classical level, ignoring the fluctuations describable in a Langevin description, the late time dynamics of the more fundamental theory leads to the same pattern formation as found in the Gray-Scott model with suitable choices of the parameters describing the diffusion of the composite molecule.

cond-mat.stat-mech↗
arXiv2018

Exact solutions of a generalized variant of the derivative nonlinear Schrodinger equation in a Scarff II external potential and their stability properties

We obtain exact solitary wave solutions of a variant of the generalized derivative nonlinear Schrodinger\equation in 1+1 dimensions with arbitrary values of the nonlinearity parameter $κ$ in a Scarf-II potential. This variant of the usual derivative nonlinear Schrodinger equation has the properties that for real external potentials, the dynamics is derivable from a Lagrangian. The solitary wave and trapped solutions have the same form as those of the usual derivative nonlinear Schrodinger equation. We show that the solitary wave solutions are orbitally stable for $κ\leq 1$ We find new exact nodeless solutions to the bound states in the external complex potential which are related to the static solutions of the equation. We also use a collective coordinate approximation to analyze the stability of the trapped solutions when the external potential is real.

nlin.PS↗
arXiv2017

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 $\{ \, \rmi \, \partial_t + \partial_x^2 + g |ψ(x,t)|^2 - V^{+}(x) \, \} \, ψ(x,t) = 0$, 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.

nlin.PS↗
arXiv2017

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 $\frac{g^2}{2} (\barΨ Ψ)^2$ 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.

nlin.PS↗
arXiv2017

Stability of new exact solutions of the nonlinear Schrodinger equation in a Poschl-Teller external potential

We discuss the stability properties of the solutions of the general nonlinear \Schrodinger\ equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time ($\PT$) symmetric superpotential $W(x)$ that we considered earlier \cite{PhysRevE.92.042901}. In particular we consider the nonlinear partial differential equation $ \{ i \, \partial_t + \partial_x^2 - V(x) + g | ψ(x,t) |^{2κ} \} \, ψ(x,t) = 0 \>, $ for arbitrary nonlinearity parameter $κ$, where $g= \pm1$ and $V$ is the well known P{ö}schl-Teller potential which we allow to be repulsive as well as attractive. Using energy landscape methods, linear stability analysis as well as a time dependent variational approximation, we derive consistent analytic results for the domains of instability of these new exact solutions as a function of the strength of the external potential and $κ$. For the repulsive potential (and $g=+1$) we show that there is a translational instability which can be understood in terms of the energy landscape as a function of a stretching parameter and a translation parameter being a saddle near the exact solution. In this case, numerical simulations show that if we start with the exact solution, the initial wave function breaks into two pieces traveling in opposite directions. If we explore the slightly perturbed solution situations, a 1\% change in initial conditions can change significantly the details of how the wave function breaks into two separate pieces. For the attractive potential (and $g=+1$), changing the initial conditions by 1 \% modifies the domain of stability only slightly. For the case of the attractive potential and negative $g$ perturbed solutions merely oscillate with the oscillation frequencies predicted by the variational approximation.

nlin.PS↗
arXiv2016

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 self-interaction in the presence of external forces as well as damping of the form $γ^0 f(x,t) - i μγ^0 Ψ$, where both $f, \{f_j = r_i e^{i K_j x} \}$ 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 $K_j=K, ~ j=1,2$ 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 $K_1 \neq K_2$ which dramatically modifies the behavior of the solitary wave in the presence of these external forces.

nlin.PS↗
arXiv2016

Variational Approach to studying solitary waves in the nonlinear Schrodinger equation with Complex Potentials

We discuss the behavior of solitary wave solutions of the nonlinear Schr{ö}dinger equation (NLSE) as they interact with complex potentials, using a four parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope of $dp(t)/dv(t)$ becomes negative when plotted parametrically as a function of time, where $p(t)$ is the momentum of the solitary wave and $v(t)$ the velocity.

nlin.PS↗
arXiv2016

Stability of exact solutions of the nonlinear Schroedinger equation in an external potential having supersymmetry and parity-time symmetry

We discuss the stability properties of the solutions of the general nonlinear Schroedinger equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time (PT) symmetric superpotential $W(x)$ that we considered earlier [Kevrekedis et al Phys. Rev. E 92, 042901 (2015)]. In particular we consider the nonlinear partial differential equation $\{ i \partial_t + \partial_x^2 - V^{-}(x) +| ψ(x,t) |^{2κ} \} \, ψ(x,t) = 0$, for arbitrary nonlinearity parameter $κ$. We study the bound state solutions when $V^{-}(x) = (1/4- b^2)$ sech$^2(x)$, which can be derived from two different superpotentials $W(x)$, one of which is complex and $PT$ symmetric. Using Derrick's theorem, as well as a time dependent variational approximation, we derive exact analytic results for the domain of stability of the trapped solution as a function of the depth $b^2$ of the external potential. We compare the regime of stability found from these analytic approaches with a numerical linear stability analysis using a variant of the Vakhitov-Kolokolov (V-K) stability criterion. The numerical results of applying the V-K condition give the same answer for the domain of stability as the analytic result obtained from applying Derrick's theorem. Our main result is that for $κ>2$ a new regime of stability for the exact solutions appears as long as $b > b_{crit}$, where $b_{crit}$ is a function of the nonlinearity parameter $κ$. In the absence of the potential the related solitary wave solutions of the NLSE are unstable for $κ>2$.

nlin.PS↗
arXiv2016

Approximate Analytic Solutions to Coupled Nonlinear Dirac Equations

We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalar-scalar self interactions $\frac{ g_1^2}{2} ( {\bpsi} ψ)^2 + \frac{ g_2^2}{2} ( {\bphi} ϕ)^2 + g_3^2 ({\bpsi} ψ) ( {\bphi} ϕ)$ as well as vector-vector interactions of the form $\frac{g_1^2 }{2} (\bpsi γ_μ ψ)(\bpsi γ^μ ψ)+ \frac{g_2^2 }{2} (\bphi γ_μ ϕ)(\bphi γ^μ ϕ) + g_3^2 (\bpsi γ_μ ψ)(\bphi γ^μ ϕ). $ Writing the two components of the assumed solitary wave solution of these equations in the form $ψ= e^{-i ω_1 t} \{R_1 \cos θ, R_1 \sin θ\}$, $ϕ= e^{-i ω_2 t} \{R_2 \cos η, R_2\sin η\}$, and assuming that $ θ(x),η(x)$ have the {\it same} functional form they had when $g_3$=0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for $R_i(x)$ which are valid for small values of $g_3^2/ g_2^2 $ and $g_3^2/ g_1^2$. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at $x \rightarrow \pm \infty$.

nlin.PS↗
arXiv2015

Solitary waves of a PT-symmetric Nonlinear Dirac equation

In the present work, we consider a prototypical example of a PT-symmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the PT-phase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the continuation in the PT-symmetric model of the solutions of the corresponding Hamiltonian model and find that the solutions can be continued robustly as stable ones all the way up to the PT-transition threshold. In the latter, they degenerate into linear waves. We also examine the dynamics of the model. Given the stability of the waveforms in the PT-exact phase we consider them as initial conditions for parameters outside of that phase. We find that both oscillatory dynamics and exponential growth may arise, depending on the size of the corresponding "quench". The former can be characterized by an interesting form of bi-frequency solutions that have been predicted on the basis of the SU(1,1) symmetry. Finally, we explore some special, analytically tractable, but not PT-symmetric solutions in the massless limit of the model.

nlin.PS↗
arXiv2015

Auxiliary Field Loop Expansion of the Effective Action for Stochastic Partial Differential Equations

We present an alternative to the perturbative diagrammatic approach for studying stochastic dynamics. Our approach is based on an auxiliary field loop expansion for the path integral representation for the generating functional of the noise induced correlation functions. We derive two different effective actions, one based on the Onsager-Machlup (OM) approach, and the other on the Martin-Siggia-Rose (MSR) response function approach. In particular we determine the leading order approximation for the effective action and effective potential for arbitrary spatial dimensions for several simple systems. These include the Kardar-Parisi-Zhang (KPZ) equation, the chemical reaction annihilation and diffusion process $A+A \rightarrow 0$, and the Ginzburg-Landau (GL) model for spin relaxation. We show how to obtain the effective potential of the OM approach from the effective potential in the MSR approach. For the KPZ equation we find that our approximation, which is non-perturbative and obeys broken symmetry Ward identities, does not lead to the appearance of a fluctuation induced symmetry breakdown. This contradicts the results of earlier studies. We also obtain some of the renormalization group flows directly from the effective potential and compare our results with exact and perturbative results.

cond-mat.stat-mech↗
arXiv2015

Nonlinear Dirac equation solitary waves in the presence of external driving forces

We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar self-interaction in the presence of external forces as well as damping of the form $ f(x,t) - i μγ^0 Ψ$, where both $f$ 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. This approach predicts intrinsic oscillations of the solitary waves, i.e. the amplitude, width and phase all oscillate with the same frequency. The translational motion is also affected, because the soliton position oscillates around a mean trajectory. We then compare the results of the variational approximation with numerical simulations of the NLD equation, and find a good agreement, if we take into account a certain linear excitation with specific wavenumber that is excited together with the intrinsic oscillations such that the momentum in a transformed NLD equation is conserved. We also solve explicitly the CC equations of the variational approximation in the non-relativistic regime for a homogeneous external force and obtain excellent agreement with the numerical solution of the CC equations.

nlin.PS↗
arXiv2014

Auxiliary Field Loop Expansion for the Effective Action for Stochastic Partial Differential Equations I

Using a path integral formulation for correlation functions of stochastic partial differential equations based on the Onsager-Machlup approach, we show how, by introducing a composite auxiliary field one can generate an auxiliary field loop expansion for the correlation functions which is similar to the one used in the $1/N$ expansion for an $O(N)$ scalar quantum field theory. We apply this formalism to the Kardar Parisi Zhang (KPZ) equation, and introduce the composite field $σ= \fracλ{2} \nabla ϕ\cdot \nabla ϕ$ by inserting a representation of the unit operator into the path integral which enforces this constraint. In leading order we obtain a self-consistent mean field approximation for the effective action similar to that used for the Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein Condensate (BEC) theories of dilute Fermi and Bose gases. This approximation, though related to a self-consistent Gaussian approximation, preserves all symmetries and broken symmetries. We derive the leading order in the auxiliary field (LOAF) effective potential and compare our results to the one loop in the fluctuation strength ${\cal A}$ approximation. We find, contrary to what is found in the one loop and self-consistent Gaussian approximation schemes that in the LOAF approximation there is no fluctuation induced symmetry breaking as a function of the coupling constant in any dimension $d$.

cond-mat.stat-mech↗
arXiv2014

Auxiliary Field Loop expansion for the Effective Action for Stochastic Partial Differential equations II

We extend our discussion of effective actions for stochastic partial differential equations to systems that give rise to a Martin-Siggia-Rose (MSR) type of action. This type of action naturally arises when one uses the many-body formalism of Doi and Peliti to describe reaction-diffusion models which undergo transitions into the absorbing state and which are described by a Master equation. These models include predator prey models, and directed percolation models as well as chemical kinetic models. For classical dynamical systems with external noise it is always possible to construct an MSR action. Using a path integral representation for the generator of the correlation functions, we show how, by introducing a composite auxiliary field, one can generate an auxiliary field loop expansion for the effective action for both types of systems. As a specific example of the Doi-Peliti formalism we determine the effective action for the chemical reaction annihilation and diffusion process $A+A \rightarrow 0$. For the external noise problem we evaluate the effective action for the Cole-Hopf form of the Kardar-Parisi Zhang (KPZ) equation as well as for the Ginzburg Landau model of spin relaxation. We determine for arbitrary spatial dimension $d$, the renormalized effective potential in leading order in the auxiliary field loop expansion (LOAF) and also determine the renormalization group equation for the running of the reaction rate (coupling constant) for arbitrary $d$. We compare our results with known perturbative and non-perturbative results for the renormalization group equations.

cond-mat.stat-mech↗
arXiv2014

Effects of intrinsic noise on a cubic autocatalytic reaction diffusion system

Starting from our recent chemical master equation derivation of the model of an autocatalytic reaction-diffusion chemical system with reactions $U+2V {\stackrel {λ_0}{\rightarrow}}~ 3 V;$ and $V {\stackrel μ{\rightarrow}}~P$, $U {\stackrel ν{\rightarrow}}~ Q$, we determine the effects of intrinsic noise on the momentum-space behavior of its kinetic parameters and chemical concentrations. We demonstrate that the intrinsic noise induces $n \rightarrow n$ molecular interaction processes with $n \geq 4$, where $n$ is the number of molecules participating of type $U$ or $V$. The momentum dependences of the reaction rates are driven by the fact that the autocatalytic reaction (inelastic scattering) is renormalized through the existence of an arbitrary number of intermediate elastic scatterings, which can also be interpreted as the creation and subsequent decay of a three body composite state $σ= ϕ_u ϕ_v^2$, where $ϕ_i$ corresponds to the fields representing the densities of $U$ and $V$. Finally, we discuss the difference between representing $σ$ as a composite or an elementary particle (molecule) with its own kinetic parameters. In one dimension we find that while they show markedly different behavior in the short spatio-temporal scale, high momentum (UV) limit, they are formally equivalent in the large spatio-temporal scale, low momentum (IR) regime. On the other hand in two dimensions and greater, due to the effects of fluctuations, there is no way to experimentally distinguish between a fundamental and composite $σ$. Thus in this regime $σ$ behave as an entity unto itself suggesting that it can be effectively treated as an independent chemical species.

cond-mat.stat-mech↗
arXiv2014

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}{κ+1} ({\bar Ψ} Ψ)^{κ+1}$ and with mass $m$. Using the exact analytic form for rest frame solitary waves of the form $Ψ(x,t) = ψ(x) e^{-i ωt}$ for arbitrary $ κ$, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger 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 $κ$ we map out the stability regimes in $ω$. 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 $ω$ and $t_c$ decreases monotonically with increasing $κ$.

nlin.PS↗
arXiv2013

Composite Bound States and Broken U(1) symmetry in the Chemical Master Equation derivation of the Gray-Scott Model

We give a first principles derivation of the stochastic partial differential equations that describe the chemical reactions of the Gray-Scott model (GS): $U+2V {\stackrel λ{\rightarrow}} 3 V;$ and $V {\stackrel μ{\rightarrow}} P$, $U {\stackrel ν{\rightarrow}} Q$, with a constant feed rate for $U$. We find that the conservation of probability ensured by the chemical master equation leads to a modification of the usual differential equations for the GS model which now involves two composite fields and also intrinsic noise terms. One of the composites is $ψ_1 = ϕ_v^2$, where $ < ϕ_v >_η = v$ is the concentration of the species $V$ and the averaging is over the internal noise $η_{u,v,ψ_1}$. The second composite field is the product of three fields $ χ= λϕ_u ϕ_v^2$ and requires a noise source to ensure probability conservation. A third composite $ψ_2 = ϕ_{u} ϕ_{v}$ can be also be identified from the noise-induced reactions. The Hamiltonian that governs the time evolution of the many-body wave function, associated with the master equation, has a broken U(1) symmetry related to particle number conservation. By expanding around the (broken symmetry) zero energy solution of the Hamiltonian (by performing a Doi shift) one obtains from our path integral formulation the usual reaction diffusion equation, at the classical level. The Langevin equations that are derived from the chemical master equation have multiplicative noise sources for the density fields $ϕ_u, ϕ_v, χ$ that induce higher order processes such as $n \rightarrow n$ scattering for $n > 3$. The amplitude of the noise acting on $ ϕ_v$ is itself stochastic in nature.

cond-mat.stat-mech↗