Long-time Solution Research Articles

Long-time solutions of scalar nonlinear hyperbolic reaction equations incorporating relaxation I. The reaction function is a bistable cubic polynomial

We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general formuττ+uτ=uxx+ε(F(u)+F(u)τ), in which x and τ represent dimensionless distance and time respectively and ε>0 is a parameter related to the relaxation time. Furthermore the reaction function, F(u), is given by the bistable cubic polynomial,F(u)=u(1−u)(u−μ), in which 0<μ<1/2 is a parameter. The initial data is given by a simple step function with u(x,0)=1 for x≤0 and u(x,0)=0 for x>0. It is established, via the method of matched asymptotic expansions, that the large-time structure of the solution to the initial-value problem involves the evolution of a propagating wave front which is either of reaction–diffusion or of reaction–relaxation type. The one exception to this occurs when μ=12 in which case the large time attractor for the solution of the initial-value problem is a stationary state solution of kink type centred at the origin.

A remark on the Bismut–Ricci form on 2-step nilmanifolds

In this note, we observe that, on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure, the (1,1)-part of the Bismut–Ricci form is seminegative definite. As an application, we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long-time solution to the pluriclosed flow in 2-step nilmanifolds.

Dynamics from a mathematical model of a two-state gas laser

Motivated by recent work in the area, we consider the behavior of solutions to a nonlinear PDE model of a two-state gas laser. We first review the derivation of the two-state gas laser model, before deriving a non-dimensional model given in terms of coupled nonlinear partial differential equations. We then classify the steady states of this system, in order to determine the possible long-time asymptotic solutions to this model, as well as corresponding stability results, showing that the only uniform steady state (the zero motion state) is unstable, while a linear profile in space is stable. We then provide numerical simulations for the full unsteady model. We show for a wide variety of initial conditions that the solutions tend toward the stable linear steady state profiles. We also consider traveling wave solutions, and determine the unique wave speed (in terms of the other model parameters) which allows wave-like solutions to exist. Despite some similarities between the model and the inviscid Burger’s equation, the solutions we obtain are much more regular than the solutions to the inviscid Burger’s equation, with no evidence of shock formation or loss of regularity.

Resonant oscillations in open axisymmetric tubes

We study the behaviour of the isentropic flow of a gas in both a straight tube of constant cross section and a cone, open at one end and forced at or near resonance at the other. A continuous transition between these configurations is provided through the introduction of a geometric parameter k associated with the opening angle of the cone where the tube corresponds to \(k=0\). The primary objective is to find long-time resonant and near-resonant approximate solutions for the open tube, i.e. \(k\rightarrow 0\). Detailed analysis for both the tube and cone in the limit of small forcing \((O(\varepsilon ^{3}))\) is carried out, where \(\varepsilon ^{3}\) is the Mach number of the forcing function and the resulting flow has Mach number \(O(\varepsilon )\). The resulting approximate solutions are compared with full numerical simulations. Interesting distinctions between the cone and the tube emerge. Depending on the damping and detuning, the responses for the tube are continuous and of \(O(\varepsilon )\). In the case of the cone, the resonant response involves an amplification of the fundamental resonant mode, usually called the dominant first-mode approximation. However, higher modes must be included for the tube to account for the nonlinear generation of higher-order resonances. Bridging these distinct solution behaviours is a transition layer of \(O(\varepsilon ^{2})\) in k. It is found that an appropriately truncated set of modes provides the requisite modal approximation, again comparing well to numerical simulations.

General Linearized Theory of Quantum Fluctuations around Arbitrary Limit Cycles.

The theory of Gaussian quantum fluctuations around classical steady states in nonlinear quantum-optical systems (also known as standard linearization) is a cornerstone for the analysis of such systems. Its simplicity, together with its accuracy far from critical points or situations where the nonlinearity reaches the strong coupling regime, has turned it into a widespread technique, being the first method of choice in most works on the subject. However, such a technique finds strong practical and conceptual complications when one tries to apply it to situations in which the classical long-time solution is time dependent, a most prominent example being spontaneous limit-cycle formation. Here, we introduce a linearization scheme adapted to such situations, using the driven Van der Pol oscillator as a test bed for the method, which allows us to compare it with full numerical simulations. On a conceptual level, the scheme relies on the connection between the emergence of limit cycles and the spontaneous breaking of the symmetry under temporal translations. On the practical side, the method keeps the simplicity and linear scaling with the size of the problem (number of modes) characteristic of standard linearization, making it applicable to large (many-body) systems.

Low-order dynamical system model of a fully developed turbulent channel flow

A reduced order model of a turbulent channel flow is composed from a direct numerical simulation database hosted at the Johns Hopkins University. Snapshot proper orthogonal decomposition (POD) is used to identify the Hilbert space from which the reduced order model is obtained, as the POD basis is defined to capture the optimal energy content by mode. The reduced order model is defined by coupling the evolution of the dynamic POD mode coefficients through their respective time derivative with a least-squares polynomial fit of terms up to third order. Parameters coupling the dynamics of the POD basis are defined in analog to those produced in the classical Galerkin projection. The resulting low-order dynamical system is tested for a range of basis modes demonstrating that the non-linear mode interactions do not lead to a monotonic decrease in error propagation. A basis of five POD modes accounts for 50% of the integrated turbulence kinetic energy but captures only the largest features of the turbulence in the channel flow and is not able to reflect the anticipated flow dynamics. Using five modes, the low-order model is unable to accurately reproduce Reynolds stresses, and the root-mean-square error of the predicted stresses is as great as 30%. Increasing the basis to 28 modes accounts for 90% of the kinetic energy and adds intermediate scales to the dynamical system. The difference between the time derivatives of the random coefficients associated with individual modes and their least-squares fit is amplified in the numerical integration leading to unstable long-time solutions. Periodic recalibration of the dynamical system is undertaken by limiting the integration time to the range of the sampled data and offering the dynamical system new initial conditions. Renewed initial conditions are found by pushing the mode coefficients in the end of the integration time toward a known point along the original trajectories identified through a least-squares projection. Under the recalibration scheme, the integration time of the dynamical system can be extended to arbitrarily large values provided that modified initial conditions are offered to the system. The low-order dynamical system composed with 28 modes employing periodic recalibration reconstructs the spatially averaged Reynolds stresses with similar accuracy as the POD-based turbulence description. Data-driven reduced order models like the one undertaken here are widely implemented for control applications, derive all necessary parameters directly from the input, and compute predictions of system dynamics efficiently. The speed, flexibility, and portability of the reduced order model come at the cost of strict data requirements; the model identification requires simultaneous realizations of mode coefficients and their time derivatives, which may be difficult to achieve in some investigations.

Longtime existence of the Kähler-Ricci flow on ℂⁿ

We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on C n \mathbb {C}^n without assuming the initial metric has bounded curvature, thus extending results in an earlier work of the authors. We prove the existence of a longtime bounded curvature solution emerging from any complete U ( n ) U(n) -invariant Kähler metric with non-negative holomorphic bisectional curvature, and that the solution converges as t → ∞ t\to \infty to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general Kähler metrics on C n \mathbb {C}^n which are not necessarily U ( n ) U(n) -invariant.

Combating food pathogens using sodium benzoate functionalized silver nanoparticles: synthesis, characterization and antimicrobial evaluation

Herein, we report the antimicrobial efficacy of sodium benzoate-functionalized silver nanoparticles synthesized via a simple method and its potential prospects as food preservatives. Sodium benzoate (SB) is a commonly known food preservative and is capable of reducing the silver salt to silver nanoparticles in high yield under facile conditions. The functionalization of SB imparts enough stability to the nanoparticles to remain dispersed in aqueous solution for long time, without the need of any other additional stabilizing agent. The nanoparticles were characterized using high-performance liquid chromatography (HPLC), atomic force microscopy (AFM), zeta potential, Fourier transform infrared spectroscopy (FT-IR), proton nuclear magnetic resonance (1H-NMR) and UV–visible spectroscopy. In order to explore its role as food preservative, the antimicrobial activity of sodium benzoate-capped silver nanoparticles (SB-SNPs) was tested against various food-borne pathogenic bacteria and the results clearly indicated that the synthesized nanoparticles were highly active against the food pathogen strains even at very low concentrations as compared to sodium benzoate alone.

Foam front displacement in improved oil recovery in systems with anisotropic permeability

A foam front propagating through an oil reservoir is considered in the context of foam improved oil recovery. Specifically the evolution of the shape of a foam front in a strongly anisotropic reservoir (vertical permeability much smaller than horizontal permeability) is determined via the pressure-driven growth model. The shape of the foam front is demonstrated to be extremely close to that predicted in the limiting case of a reservoir with no vertical permeability whatsoever, in particular any deviations from this shape are found to be second order in the ratio of vertical to horizontal permeabilities. Material points used to represent the foam front shape are shown to exhibit a uniform downward vertical motion, with a vertical velocity component which is proportional to the ratio of vertical to horizontal permeabilities. As the material points in question migrate downwards, they are replaced by new material points arriving from higher up, representing a long-time asymptotic solution for the front shape. This long-time asymptotic shape is sensitive to the ratio of vertical to horizontal permeabilities, with the foam front sweeping the reservoir less effectively as this ratio decreases.

Asymptotic Green’s functions for time-fractional diffusion equation and their application for anomalous diffusion problem

Asymptotic Green’s functions for short and long times for time-fractional diffusion equation, derived by simple and heuristic method, are provided in case if fractional derivative is presented in Caputo sense. The applicability of the asymptotic Green’s functions for solving the anomalous diffusion problem on a semi-infinite rod is demonstrated. The initial value problem for longtime solution of the time-fractional diffusion equation by Green’s function approach is resolved.

Modulations of viscous fluid conduit periodic waves

Modulated periodic interfacial waves along a conduit of viscous liquid are explored using nonlinear wave modulation theory and numerical methods. Large-amplitude periodic-wave modulation (Whitham) theory does not require integrability of the underlying model equation, yet often either integrable equations are studied or the full extent of Whitham theory is not developed. Periodic wave solutions of the nonlinear, dispersive, non-integrable conduit equation are characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the nonlinear Schrödinger (NLS) equation are derived, depending on the carrier wavenumber. Dark and bright envelope solitons are found to persist in long-time numerical solutions of the conduit equation, providing numerical evidence for the existence of strongly nonlinear, large-amplitude envelope solitons. Due to non-convex dispersion, modulational instability for periodic waves above a critical wavenumber is predicted and observed. In the large-amplitude regime, structural properties of the Whitham modulation equations are computed, including strict hyperbolicity, genuine nonlinearity and linear degeneracy. Bifurcating from the NLS critical wavenumber at zero amplitude is an amplitude-dependent elliptic region for the Whitham equations within which a maximally unstable periodic wave is identified. The viscous fluid conduit system is a mathematically tractable, experimentally viable model system for wide-ranging nonlinear, dispersive wave dynamics.

Long-time asymptotic solution structure of Camassa-Holm equation subject to an initial condition with non-zero reflection coefficient of the scattering data

In this article, we numerically revisit the long-time solution behavior of the Camassa-Holm equation ut − uxxt + 2ux + 3uux = 2uxuxx + uuxxx. The finite difference solution of this integrable equation is sought subject to the newly derived initial condition with Delta-function potential. Our underlying strategy of deriving a numerical phase accurate finite difference scheme in time domain is to reduce the numerical dispersion error through minimization of the derived discrepancy between the numerical and exact modified wavenumbers. Additionally, to achieve the goal of conserving Hamiltonians in the completely integrable equation of current interest, a symplecticity-preserving time-stepping scheme is developed. Based on the solutions computed from the temporally symplecticity-preserving and the spatially wavenumber-preserving schemes, the long-time asymptotic CH solution characters can be accurately depicted in distinct regions of the space-time domain featuring with their own quantitatively very different solution behaviors. We also aim to numerically confirm that in the two transition zones their long-time asymptotics can indeed be described in terms of the theoretically derived Painlevé transcendents. Another attempt of this study is to numerically exhibit a close connection between the presently predicted finite-difference solution and the solution of the Painlevé ordinary differential equation of type II in two different transition zones.

Development of a symplectic and phase error reducing perturbation finite-difference advection scheme

ABSTRACTThe aim of this work is to develop a new scheme for solving the pure advection equation. This scheme formulated within the perturbation finite-difference context not only conserves symplecticity but also preserves the numerical dispersion relation equation. The employed symplectic integrator of second-order accuracy in time enables calculation of a long-time accurate solution in the sense that the Hamiltonian is conserved at all times. The generalized high-order spatially accurate perturbation difference scheme optimizes numerical phase accuracy through the minimization of the difference between the numerical and exact dispersion relation equations. Our proposed new class of phase error reducing perturbation difference schemes can in addition locally capture discontinuities underlying the concept of applying a shope/flux limiter. The high-order spatial accuracy can be recovered in a smooth region. Besides the Fourier analysis of the discretization errors, anisotropy and dispersion analyses are both conducted on the dispersion-relation and symplecticity-preserving pure advection scheme to shed light on the distinguished nature of the proposed scheme. Numerical tests are carried out and the results compare well with the exact solutions, demonstrating the efficiency, accuracy, and the discontinuity-resolving ability using the proposed class of high-resolution perturbation finite-difference schemes.

Mean curvature flow of contractions between Euclidean spaces

We consider the mean curvature flow of the graphs of maps between Euclidean spaces of arbitrary dimension. If the initial map is Lipschitz and satisfies a length-decreasing condition, we show that the mean curvature flow has a smooth long-time solution for \(t>0\). Further, we prove uniform decay estimates for the mean curvature vector and for all higher-order derivatives of the defining map.

Numerical study of long-time Camassa–Holm solution behavior for soliton transport

In this paper a three-step solution scheme is employed to numerically explore the long-time solution behavior of the Camassa–Holm equation. In the present u−P−α formation, we conduct modified equation analysis to eliminate several leading discretization error terms and perform Fourier analysis for minimizing the wave-like type of error. A three-point seventh-order spatially accurate combined compact upwind scheme is developed for the approximation of first-order derivative term. For the purpose of retaining Hamiltonian and multi-symplectic geometric structures in the non-dissipative Camassa–Holm equation, the adopted time integrator conserves symplecticity. Another main emphasis of this study is to numerically shed light on the scenario of the soliton transport.

Behavioral crowds: Modeling and Monte Carlo simulations toward validation

Abstract A mesoscopic model of behavioral crowds is developed within the framework of the kinetic theory for active particles. An analytic long-time equilibrium solution is obtained which gives a fundamental density-velocity diagram consistent with the empirical evidence. Numerical simulations based on a Monte Carlo particle method show that the proposed model has the capability to qualitatively depict emerging behaviors and to provide a realistic description of the crowd dynamics in complex evacuation scenarios.

The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere

The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

Analysis of pressure falloff tests of non-Newtonian power-law fluids in naturally-fractured bounded reservoirs

Abstract Application of non-Newtonian Power-law fluids (e.g. polymeric solutions) for production enhancement in petroleum reservoirs has increased over the last three decades. These fluids are often injected as viscous solutions to improve mobility ratio and enhance oil recovery during chemical flooding. As part of the flooding operation, surfactant (or micellar) solutions are first injected at the leading edge of the flood to reduce interfacial tension between water and oil. Subsequently, a slug of polymer solution is injected ahead of normal water to increase viscosity of the water, improve volumetric sweep efficiency and accelerate oil production. Analysis of pressure tests conducted pre and post injection, to evaluate mobility of these fluids, is more demanding than conventional techniques, which were developed strictly for Newtonian fluids. In naturally-fractured reservoirs, flow of non-Newtonian fluids is more complex due to fracture-matrix interaction which is usually resonated in the pressure footprints. Some models have been developed to aid interpretation of pressure tests, but boundary effects on down-hole measurements due to structural discontinuity and presence of an active aquifer, have not been thoroughly investigated. This article presents an analytic technique for interpreting pressure falloff tests of non-Newtonian Power-law fluids in wells that are located near boundaries in dual-porosity reservoirs. First, dimensionless pressure solutions are obtained and Stehfest inversion algorithm is used to develop new type curves. Subsequently, long-time analytic solutions are presented and interpretation procedure is proposed using direct synthesis. Two examples, including real field data from a heavy oil reservoir in Colombian eastern plains basin, are used to validate and demonstrate application of this technique. Results agree with conventional type-curve matching procedure. The approach proposed in this study avoids the use of type curves, which is prone to human errors. It provides a better alternative for direct estimation of formation and flow properties from falloff data.

Smooth solutions of the one-dimensional compressible Euler equation with gravity

We study one-dimensional motions of polytropic gas governed by the compressible Euler equations. The problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized approximation at this equilibrium gives time periodic solutions. But it is difficult to justify the existence of long-time true solutions for which this time periodic solution is the first approximation. The situation is in contrast to the problem of free motions without gravity. The reason is that the usual iteration method for quasilinear hyperbolic problem cannot be used because of the loss of regularities which causes from the touch with the vacuum. Due to this reason, we try to find a family of solutions expanded by a small parameter and apply the Nash–Moser Theorem to justify this expansion. Note that the application of Nash–Moser Theorem is necessary for the sake of conquest of the trouble with loss of regularities, and the justification of the applicability requires a very delicate analysis of the problem.

Time-dependent Hermitian Yang–Mills flow on surfaces

We investigate a generalisation of Hermitian Yang–Mills flow in which the base metric itself is allowed to depend on the time-parameter. For technical reasons we restrict our attention to the case of a holomorphic vector bundles E over compact Riemann surfaces which is assumed to be slope stable with respect to a fixed Kahler class \(\tau \) on the base. We show that if \(\omega (t)\) is a family of Kahler metrics in \(\tau \) converging to a limit metric \(\omega _\infty \) at an exponential rate, then starting at a smooth initial Hermitian metric \(h_0\) on E, the Hermitian Yang–Mills flow with respect to the time-dependent metric \(\omega (t)\) admits a unique long-time solution h(t) converging exponentially to a \(\omega _\infty \)-Hermite–Einstein metric. The proof utilises an extension of Donadson’s techniques used to treat the case where the base metrics does not depend on the time-parameter.