arXiv subjects

John F. Dawson

Publications and source records attributed to John F. Dawson.

At least 19 recordsLinked to original sources

arXiv2026

Two-Parameter Family of Nonlinear Dirac Equations With Scalar-Scalar plus Vector-Vector Interactions

We obtain exact solutions of the nonlinear Dirac equation in $1+1$ dimensions of the form $ψ(x,t) = e^{-iωt} ψ(x)$ for scalar-scalar (SS) plus vector-vector (VV) interaction with interaction Lagrangian given by $$L_{I} = \frac{g^2}{κ+1}[(\barψ ψ)^{κ+1} +\frac{1}{p} (\barψ γ_μψ\barψ γ^μ ψ)^{κ+1}]$$ where $p > 0$ but arbitrary otherwise. We look for solutions with $0 < ω< m$ where $ω, m$ are frequency and mass, respectively. We find solutions for all values of $ω$ in this range. We compute the charge $Q$ and the energy $E$ for each of the solitary wave solutions and explore the region in the ($p, κ$) parameter space in which solitary wave bound states exist (i.e., for which $E/Q < m$). We show that for all the cases while both $E$ and $Q$ depend on the coupling constant $g$, their ratio $E/Q$ is independent of $g$. We further find that in case $pκ\le 1$, the charge density for all the solitary waves have only single hump while for $p κ> 1$ there is a transition from double to single hump and we determine it as a function of $ω/m$. We notice that for all $p$ there is a transition at $κ=2$ in the behavior of $E/Q$ as a function of $ω$ which we speculate is related to the onset of instability of the solutions at $κ=2$. We obtain the nonrelativistic reduction of the two-parameter family to a non-relativistic modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the single hump olitary waves in the domain of validity of the modified NLSE.

nlin.PS↗

arXiv2025

Modeling pandemics

We review several models for pandemics that plagued the USA and the world in the past decade. Methods of data fitting are reviewed and several types of microscopic and rate equation models are discussed and numerically solved. This paper was written in March of 2021 and newer data is now available; however some of the models and techniques we used to numerically study these models are still of interest. Several appendices discuss in detail these models.

stat.AP↗

arXiv2025

Solitary waves in the complementary generalized ABS model

We obtain exact solutions of the nonlinear Dirac equation in 1+1 dimension of the form $ Ψ(x,t) =Φ(x) \rme^{-\rmi ωt}$ where the nonlinear interactions are a combination of vector-vector and scalar-scalar interactions with the interaction Lagrangian given by $L_I = \frac{g^2}{(κ+1)}[\barψ γ_μψ\barψ γ^μ ψ]^{(κ+1)/2} - \frac{g^2}{q(κ+1)}(\barψ ψ)^{κ+1}$, where $κ>0$ and $q>1$. This is the complement of the generalization of the ABS model \cite{abs} that we recently studied \cite{ak} and denoted as the gABS model. We show that like the gABS model, in the complementary gABS models the solitary wave solutions also exist in the entire $(κ, q)$ plane and further in both models energy of the solitary wave divided by its charge is {\it independent} of the coupling constant $g$. However, unlike the gABS model here all the solitary waves are single humped, any value of $0 < ω< m$ is allowed and further unlike the gABS model, for this complementary gABS model the solitary wave bound states exist only in case $κ\le κ_c$, where $κ_c$ depends on the value of $q$. Here $ω$ and $m$ denote frequency and mass, respectively. We discuss the regions of stability of these solutions as a function of $ω,q,κ$ using the Vakhitov-Kolokolov criterion. Finally we discuss the non-relativistic reduction of the two-parameter family of this complementary generalized ABS model to a modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the solitary waves in the domain of validity of the modified NLSE.

nlin.PS↗

arXiv2025

Solitary waves in a Two Parameter Family of Generalized Nonlinear Dirac Equations in $1+1$ Dimensions

We obtain exact solutions of the nonlinear Dirac equation in 1+1 dimension of the form $Ψ(x,t) = Φ(x) e^{-i ωt}$ where the nonlinear interactions are a combination of vector-vector (V-V) and scalar-scalar (S-S) interactions with the interaction Lagrangian given by $L_I= \frac{g^2}{(κ+1)}(\barψ ψ)^{κ+1} -\frac{g^2}{p(κ+1)}[\barψ γ_μ ψ\barψ γ^μ ψ]^{(κ+1)/2}$. This generalizes the model of ABS (N.V. Alexeeva, I.V. Barashenkov and A. Saxena, Annals Phys. {\bf 403}, 198, (2019)) by having the arbitrary nonlinearity parameter $κ>0$ and by replacing the coefficient of the V-V interaction by the arbitrary positive parameter $p>1$ which alters the relative weights of the vector-vector and the scalar-scalar interactions. We show that the solitary wave solutions exist in the entire allowed $(κ,p)$ plane for $ω/m > 1/p^{1/(κ+1)} $, for frequency $ω$ and mass $m$. These solutions have the property that their energy divided by their charge is $\it {independent} $ of the coupling constant $g$. As $ω$ increases, there is a transition from the double humped to the single humped solitons. We discuss the regions of stability of these solutions as a function of $ω,p,κ$ using the Vakhitov-Kolokolov criterion. Finally we discuss the non-relativistic reduction of the 2-parameter family of generalized ABS models to a modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the solitary waves in the domain of validity of the modified NLSE.

nlin.PS↗

arXiv2024

Solitary waves in the coupled nonlinear massive Thirring as well as coupled Soler models with arbitrary nonlinearity

Motivated by the recent introduction of an integrable coupled massive Thirring model by Basu-Mallick et al, we introduce a new coupled Soler model. Further we generalize both the coupled massive Thirring and the coupled Soler model to arbitrary nonlinear parameter $κ$ and obtain exact solitary wave solutions in both cases. Remarkably, it turns out that in both the models, because of the conservation laws of charge and energy, the exact solutions we find seem to not depend on how we parameterize them, and the charge density of these solutions is related to the charge density of the single field solutions found earlier by a subset of the present authors. In both the models, a nonrelativistic reduction of the equations leads to the same conclusion that the solutions are proportional to those found in the one component field case.

nlin.PS↗

arXiv2023

Exact trapped $N$-soliton solutions of the nonlinear Schrödinger equation using the inverse problem method

In this work, we show the application of the ``inverse problem'' method to construct exact $N$ trapped soliton-like solutions of the nonlinear Schrödinger or Gross-Pitaevskii equation (NLSE and GPE, respectively) in one, two, and three spatial dimensions. This method is capable of finding the external (confining) potentials which render specific assumed waveforms exact solutions of the NLSE for both attractive ($g<0$) and repulsive ($g>0$) self-interactions. For both signs of $g$, we discuss the stability with respect to self-similar deformations and translations. For $g<0$, a critical mass $M_c$, or equivalently the number of particles, for instabilities to arise can often be found analytically. On the other hand, for the case with $g>0$ corresponding to repulsive self interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates (BEC), the bound solutions are found to be always stable. For $g<0$, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped $N$-soliton solutions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.

nlin.PS↗

arXiv2023

Uniform Bose-Einstein Condensates as Kovaton solutions of the Gross-Pitaevskii Equation through a Reverse-Engineered Potential

In this work, we consider a ``reverse-engineering'' approach to construct confining potentials that support exact, constant density kovaton solutions to the classical Gross-Pitaevskii equation (GPE) also known as the nonlinear Schrödinger equation (NLSE). In the one-dimensional case, the exact solution is the sum of stationary kink and anti-kink solutions, i.e. a kovaton, and in the overlapping region, the density is constant. In higher dimensions, the exact solutions are generalizations of this wave function. In the absence of self-interactions, the confining potential is similar to a smoothed out finite square well with minima also at the edges. When self-interactions are added, a term proportional to $\pm g ψ^{\ast}ψ$ gets added to the confining potential and $\pm g M$, where $M$ is the norm, gets added to the total energy. In the realm of stability analysis, we find (linearly) stable solutions in the case with repulsive self-interactions which also are stable to self-similar deformations. For attractive interactions, however, the minima at the edges of the potential get deeper and a barrier in the center forms as we increase the norm. This leads to instabilities at a critical value of $M$ (related to the number of particles in the BEC). Comparing the stability criteria from Derrick's theorem and Bogoliubov-de Gennes analysis stability results, we find that both predict stability for repulsive self-interactions and instability at a critical mass $M$ for attractive interactions. However, the numerical analysis gives a much lower critical mass. The numerical analysis shows further that the initial instabilities violate the symmetry $x\rightarrow-x$ assumed by Derrick's theorem.

nlin.PS↗

arXiv2022

Stability of exact solutions of the $(2+1)$-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter $κ$

In this work, we consider the nonlinear Schrödinger equation (NLSE) in $2+1$ dimensions with arbitrary nonlinearity exponent $κ$ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability over their "mass" (i.e., the $L^2$ norm) and the parameter $κ$ is explored. We observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all $κ$ as long as their mass is less than a critical value $M^{\ast}(κ)$. Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick's theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve $M^{\ast}(κ)$ vs. $κ$ lies below the two curves found by Derrick's theorem and the 4CC approximation. In the absence of the external potential, $κ=1$ demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for $κ<1$, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the "frustrated" blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension $d$.

nlin.PS↗

arXiv2021

Stability of exact solutions of a nonlocal and nonlinear Schrödinger equation with arbitrary nonlinearity

This work focuses on the study of solitary wave solutions to a nonlocal, nonlinear Schrödinger system in $1$+$1$ dimensions with arbitrary nonlinearity parameter $κ$. Although the system we study here was first reported by Yang (Phys. Rev. E, 98 (2018), 042202) for the fully integrable case $κ=1$, we extend its considerations and offer criteria for soliton stability and instability as a function of $κ$. In particular, we show that for $κ<2$ the solutions are stable whereas for $κ>2$ they are subject to collapse or blowup. At the critical point of $κ=2$, there is a critical mass necessary for blowup or collapse. Furthermore, we show there is a simple one-component nonlocal Lagrangian governing the dynamics of the system which is amenable to a collective coordinate approximation. To that end, we introduce a trial wave function with two collective coordinates to study the small oscillations around the exact solution. We obtain analytical expressions for the small oscillation frequency for the width parameter in the collective coordinate approximation. We also discuss a four collective coordinate approximation which in turn breaks the symmetry of the exact solution by allowing for translational motion. The ensuing oscillations found in the latter case capture the response of the soliton to a small translation. Finally, our results are compared with numerical simulations of the system.

nlin.PS↗

arXiv2020

Behavior of solitary waves of coupled nonlinear Schrödinger equations subjected to complex external periodic potentials with anti-$\mathcal{PT}$ symmetry

We discuss the response of both moving and trapped solitary wave solutions of a nonlinear two-component nonlinear Schrödinger system in 1+1 dimensions to an anti-$\mathcal{PT}$ external periodic complex potential. The dynamical behavior of perturbed solitary waves is explored by conducting numerical simulations of the nonlinear system and using a collective coordinate variational approximation. We present case examples corresponding to choices of the parameters and initial conditions involved therein. The results of the collective coordinate approximation are compared against numerical simulations where we observe qualitatively good agreement between the two. Unlike the case for a single-component solitary wave in a complex periodic $\mathcal{PT}$-symmetric potential, the collective coordinate equations do not have a small oscillation regime, and initially the height of the two components changes in opposite directions often causing instability. We find that the dynamic stability criteria we have used in the one-component case is proven to be a good indicator for the onset of dynamic instabilities in the present setup.

nlin.PS↗

arXiv2020

Stability and response of trapped solitary wave solutions of coupled nonlinear Schrödinger equations in an external, $\mathcal{PT}$- and supersymmetric potential

We present trapped solitary wave solutions of a coupled nonlinear Schrödinger system in $1$+$1$ dimensions in the presence of an external, supersymmetric and complex $\mathcal{PT}$-symmetric potential. The Schrödinger system this work focuses on possesses exact solutions whose existence, stability, and spatio-temporal dynamics are investigated by means of analytical and numerical methods. Two different variational approximations are considered where the stability and dynamics of the solitary waves are explored in terms of eight and twelve time-dependent collective coordinates. We find regions of stability for specific potential choices as well as analytic expressions for the small oscillation frequencies in the collective coordinate approximation. Our findings are further supported by performing systematic numerical simulations of the nonlinear Schrödinger system.

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

Efficient Determination of Reverberation Chamber Time Constant

Determination of the rate of energy loss in a reverberation chamber is fundamental to many different measurements such as absorption cross-section, antenna efficiency, radiated power, and shielding effectiveness. Determination of the energy decay time-constant in the time domain by linear fitting the power delay profile, rather than using the frequency domain quality-factor, has the advantage of being independent of the radiation efficiency of antennas used in the measurement. However, determination of chamber time constant by linear regression suffers from several practical problems, including a requirement for long measurement times. Here we present a new nonlinear curve fitting technique that can extract the time-constant with typically 60% fewer samples of the chamber transfer function for the same measurement uncertainty, which enables faster measurement of chamber time constant by sampling fewer chamber transfer function, and allows for more robust automated data post-processing. Nonlinear curve fitting could have economic benefits for test-houses, and also enables accurate broadband measurements on humans in about ten minutes for microwave exposure and medical applications. The accuracy of the nonlinear method is demonstrated by measuring the absorption cross-section of several test objects of known properties. The measurement uncertainty of the method is verified using Monte-Carlo methods.

physics.class-ph↗

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

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

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↗

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↗