Stable Plane Wave Research Articles

The generation of diverse traveling pulses and its solution scheme in an excitable slow-fast dynamics.

In this article, we report on the generation and propagation of traveling pulses in a homogeneous network of diffusively coupled, excitable, slow-fast dynamical neurons. The spatially extended system is modeled using the nearest neighbor coupling theory, in which the diffusion part measures the spatial distribution of coupling topology. We derive analytically the conditions for traveling wave profiles that allow the construction of the shape of traveling nerve impulses. The analytical and numerical results are used to explore the nature of propagating pulses. The symmetric or asymmetric nature of traveling pulses is characterized, and the wave velocity is derived as a function of system parameters. Moreover, we present our results for an extended excitable medium by considering a slow-fast biophysical model with a homogeneous, diffusive coupling that can exhibit various traveling pulses. The appearance of series of pulses is an interesting phenomenon from biophysical and dynamical perspective. Varying the perturbation and coupling parameters, we observe the propagation of activities with various amplitude modulations and transition phases of different wave profiles that affect the speed of pulses in certain parameter regimes. We observe different types of traveling pulses, such as envelope solitons and multi-bump solutions, and show how system parameters and coupling play a major role in the formation of different traveling pulses. Finally, we obtain the conditions for stable and unstable plane waves.

Stable plane waves in nonlocally coupled phase oscillators

We consider a system of nonlocally coupled phase oscillators and perform a linear stability analysis of the plane wave solutions of the system. Consequently, we demonstrate the stability of the solution associated with a particular wavenumber and also the robustness of the stability against the heterogeneity of natural frequencies. The mathematical model is valid in any spatial dimension, and the theoretical results are confirmed via direct numerical simulations.

Propagation Regularity of Air Shock Wave in Turning Roadway

In order to study the propagation law of explosive shock wave in underground turning roadway, the peak overpressure value in the roadway with specific turning angle was compared and analyzed by combining field monitoring experiment with ANSYS numerical simulation. The results show that the blast wave propagates forward in a stable plane wave before turning. Before the explosion air shock wave propagates to the turn, it follows the propagation law in the straight through roadway. After turning, the diffraction and reflection through the wall of the roadway will form a turbulence zone of 10-20m, and then continue to propagate forward in a stable plane wave. The turning roadway has a certain attenuation effect on the propagation of the shock wave. By analyzing the peak overpressure value before and after turning, the attenuation coefficient values of roadways at various turning angles are determined, That is, the attenuation coefficient values corresponding to the turning angles of 30°, 60°, 90°, 120° and 150° are 1.25, 1.31, 1.45, 1.50, 1.65, respectively, and the attenuation coefficient values are fitted with the roadway turning angle formula to obtain the quantitative calculation formula, which can provide reference for the safety of underground personnel, equipment and the design and production of mines.

ERROR CONTROL OF THE VECTORIAL NONDIRECTIVE STABLE PLANE WAVE MULTILEVEL FAST MULTIPOLE ALGORITHM

Novel formulas are presented that allow the rapid estimation of the number of terms L that needs to be taken into account in the translation operator of the vectorial Nondirective Stable Plane Wave Multilevel Fast Multipole Algorithm (NSPWMLFMA). This is especially important for low frequencies, since the L needed for error- controllability can be substantially higher than the L required in the scalar case. Although these formulas were originally derived for use in the NSPWMLFMA, they are equally useful in at least three other fast matrix multiplication methods.

Embedding Calderón Multiplicative Preconditioners in Multilevel Fast Multipole Algorithms

Calderon preconditioners have recently been demonstrated to be very successful in stabilizing the electric field integral equation (EFIE) for perfect electric conductors at lower frequencies. Previous authors have shown that, by using a dense matrix preconditioner based on the Calderon identities, the low frequency instability is removed while still maintaining the inherent accuracy of the EFIE. It was also demonstrated that the spectral properties of the Caldero-n preconditioner are conserved during discretization if the EFIE operator is discretized with Rao-Wilton-Glisson expansion functions and the preconditioner with Buffa-Christiansen expansion functions. In this article we will show how the Calderon multiplicative preconditioner (CMP) can be combined with fast multipole methods to accelerate the numerical solution, leading to an overall complexity of O(N logN) for the entire iterative solution. At low frequencies, where the CMP is most useful, the traditional multilevel fast multipole algorithm (MLFMA) is unstable and we apply the nondirectional stable plane wave MLFMA (NSPWMLFMA) that resolves the low frequency breakdown of the MLFMA. The combined algorithm will be called the CMP-NSPWMLFMA. Applying the CMP-NSPWMLFMA at open surfaces or very low frequencies leads to certain problems, which will be discussed in this article.

SIMULATION AND EXPERIMENTAL VERIFICATION OF W-BAND FINITE FREQUENCY SELECTIVE SURFACES ON INFINITE BACKGROUND WITH 3D FULL WAVE SOLVER NSPWMLFMA

We present the design, processing and testing of a W-band finite by infinite and a finite by finite Grounded Frequency Selective Surfaces (FSSs) on infinite background. The 3D full wave solver Nondirective Stable Plane Wave Multilevel Fast Multipole Algorithm (NSPWMLFMA) is used to simulate the FSSs. As NSPWMLFMA solver improves the complexity matrix-vector product in an iterative solver from O(N(2)) to O(N log N) which enables the solver to simulate finite arrays with faster execution time and manageable memory requirements. The simulation results were verified by comparing them with the experimental results. The comparisons demonstrate the accuracy of the NSPWMLFMA solver. We fabricated the corresponding FSS arrays on quartz substrate with photolithographic etching techniques and characterized the vector S-parameters with a free space Millimeter Wave Vector Network Analyzer (MVNA).

Zero-refractoriness spirals in phase-coupled excitable media

Effects of excitability and coupling strength on plane and spiral waves in two-dimensional excitable lattices modeled by phase-coupled elements are investigated. The corresponding phase diagrams for stable plane waves and spiral waves are obtained by simulations. The parameters capable of supporting stable spiral waves are sorted out together with the spiral rotation frequencies. This discrete model corresponds to an excitable medium with zero refractoriness and in the continuum limit supports zero-core spiral waves. The associated wave propagating behaviors are also discussed analytically and verified.

Parabolic Resonance: A Route to Hamiltonian Spatiotemporal Chaos

We show that initial data near an unperturbed stable plane wave can evolve into a regime of spatiotemporal chaos in the slightly forced conservative periodic one-dimensional nonlinear Schrödinger equation. Statistical measures are employed to demonstrate that this spatiotemporal chaos is intermittent: there are windows in time for which the solution gains spatial coherence. The parameters and initial profiles that lead to such intermittency are predicted by utilizing a novel geometrical description of the integrable unforced equation.

Weakly nonlinear analysis of short-wave elliptical instability

Two-dimensional inviscid flows with elliptical streamlines have linearly unstable three-dimensional modes close to the center of the ellipse. The linear instability equilibrates in the presence of enough viscosity, yielding a continuous set of neutrally stable plane waves known as Craik–Criminale waves (CCWs). For initial disturbance in a discrete superposition of plane waves our weakly nonlinear analysis yields a cubic ordinary differential equation for the slow-time evolution of the neutrally stable plane-wave amplitudes. In the subcritical region we find that all solutions eventually vanish, while in the supercritical region the amplitudes either saturate or have exponential growth; the existence of a finite-time blowup at cubic order being unlikely. The saturation to a steady state is possible only for special initial conditions, and for generic initial conditions the amplitudes saturate to a chaotic state. In contrast, a single CCW with exponential growth is still unstable, but it loses stability to cycles of growth and decay among the plane waves.

A Nondirective Plane Wave MLFMA Stable at Low Frequencies

A novel method, called the nondirective stable plane wave multilevel fast multipole algorithm (NSPWMLFMA), is presented to evaluate the low-frequency (LF) interactions that cannot be handled by the multilevel fast multipole algorithm (MLFMA). It is well known that the MLFMA cannot be used for LF interactions, since it suffers from numerical instability. Contrary to current techniques, the proposed technique is not based on the spectral representation of the Green function. Instead the addition theorem of the MLFMA is manipulated into a form that allows numerically stable translations along the z axis. The translation operator for these translations is derived in closed form. A QR-based method is devised to allow stable translations in all the other directions. Interpolations and anterpolations are also provided, allowing a full multilevel algorithm. Since the NSPWMLFMA is based on the same mathematical foundations as the MLFMA, it requires limited adaptations to existing MLFMA codes. The fact that a QR is needed limits this algorithm to LF interactions. However, a coupling with the MLFMA is straightforward, allowing the easy construction of a broadband algorithm. The DC limit of the algorithm is also presented and it is shown that the algorithm remains valid for static problems. Finally, it is shown that the error introduced in the different steps of the algorithm is controllable, and a single-level vectorial version of the algorithm is applied to a generic scattering application to demonstrate its validity.

A low frequency stable plane wave addition theorem

The multilevel fast multipole algorithm (MLFMA) is a well known and very successful method for accelerating the matrix-vector products required for the iterative solution of Helmholtz problems. The MLFMA has an important drawback, namely its inability to handle scattering problems with a lot of subwavelength detail due to the low frequency (LF) breakdown of the MLFMA. There is a need to extend the MLFMA to LF, since alternative methods are less efficient (multipole methods) or hard to implement (spectral methods). In this paper a new addition theorem will be developed that does not suffer from an LF breakdown. Instead it suffers from a high-frequency (HF) breakdown. The new method relies on a novel set of distributions, the so-called pseudospherical harmonics, closely related to the spherical harmonics. These allow the discretization points and translation operators to be calculated in closed form. Hence the method presented in this paper allows the easy implementation of a method that is stable at LF. Furthermore, a combination of the traditional MLFMA and the new method allows for the construction of a broadband MLFMA.

Phase twisted modes and current reversals in a lattice model of waveguide arrays with nonlinear coupling

We consider a lattice model for waveguide arrays embedded in nonlinear Kerr media. Inclusion of nonlinear coupling results in many phenomena involving complex, phase-twisted, stationary modes. The norm (Poynting power) current of stable plane-wave solutions can be controlled in magnitude and direction, and may be reversed without symmetry-breaking perturbations. Also stable localized phase-twisted modes with zero current exist, which for particular parameter values may be compact and expressed analytically. The model also describes coupled Bose-Einstein condensates.

Efficient fast multipole method for low-frequency scattering

The solution of the Helmholtz and Maxwell equations using integral formulations requires to solve large complex linear systems. A direct solution of those problems using a Gauss elimination is practical only for very small systems with few unknowns. The use of an iterative method such as GMRES can reduce the computational expense. Most of the expense is then computing large complex matrix vector products. The cost can be further reduced by using the fast multipole method which accelerates the matrix vector product. For a linear system of size N, the use of an iterative method combined with the fast multipole method reduces the total expense of the computation to Nlog N. There exist two versions of the fast multipole method: one which is based on a multipole expansion of the interaction kernel expι kr/ r and which was first proposed by V. Rokhlin and another based on a plane wave expansion of the kernel, first proposed by W.C. Chew. In this paper, we propose a third approach, the stable plane wave expansion (SPW-FMM), which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane wave expansion. The computational complexity is Nlog N as with the other methods.

Dynamics of Pattern Formation and Modulational Instabilities in the Quintic Complex Ginzburg–Landau Equation*

We study analytically modulational instability in an extended media, taking the one-dimensional quintic complex Ginzburg–Landau equation model as an example. We show that the quintic term can affect the conditions for modulational instability (MI), and the propagating regime. Numerical simulations demonstrate the validity of analytical predictions. The model presents a rich variety of patterns propagating into the system. The main feature of the travelling shock waves is the propagation of a secondary modulated wave at the same time with a stable plane wave.

PERIODIC NUCLEATION SOLUTIONS OF THE REAL GINZBURG–LANDAU EQUATION IN A FINITE BOX

We study the stationary solutions of the real Ginzburg–Landau equation with periodic boundary conditions in a finite box. We show explicitly how to construct nucleation solutions allowing transitions between stable plane waves.

Existence and stability of singular heteroclinic orbits for the Ginzburg - Landau equation

The existence and stability of travelling waves of the Ginzburg - Landau equation is considered. The waves in question exist on a slow manifold in phase space and connect two stable plane waves with different wave numbers. The existence of these waves is proven via the use of the methods of geometric singular perturbation theory. Topological methods are used to prove the linear stability of the waves. The waves are shown to be nonlinearly stable in polynomially weighted spaces. Even though the Ginzburg - Landau equation possesses both a rotational invariance and a spatial translation invariance, small perturbations of the wave decay to the wave itself, and not to a translate of the wave. 35K57, 35Q35, 58F19

A variety of stable persistent waves in intrinsically bistable reaction-diffusion systems. From one-dimensional periodic waves to one-armed and two-armed rotating spiral waves

Abstract Various types of non-trivial stable persistent waves in intrinsically bistable reaction-diffusion equations are studied with an emphasis on initiation problems of these waves. The reaction-diffusion equation of Rinzel and Keller's type allows us to obtain exact plane wave solutions and their dispersion relation curves with three branches in most cases. Coexistence of six plane wave solutions is also discussed. On the basis of the knowledge of the stable plane wave solutions, the initiation problem for multi-armed rotating spiral waves is discussed. Numerical simulations show that a one-armed spiral wave maintains its wave pattern, and that a two-armed spiral wave survives for a long time before it eventually dies out.

First observation of self-focusing of nonlinear second sound in superfluid helium near T lambda.

Nonlinear second sound was studied in a large undisturbed sample of superfluid helium at temperatures near ${T}_{\ensuremath{\lambda}}$. Since the source of second sound was finite sized, the second-sound pulses were expected to show the rarefaction and amplitude attenuation characteristic of diffracting three-dimensional pulses. It was found instead that moderately high amplitude pulses propagating far from reflecting walls self-focused into stable plane wave pulses as the temperature approached the critical temperature ${T}_{\ensuremath{\lambda}}$. Self-focusing is a nonlinear phenomenon previously unreported for a sound wave.

Spatiotemporal intermittency regimes of the one-dimensional complex Ginzburg-Landau equation

Completing an earlier work, the 'phase diagram' of the one-dimensional complex Ginzburg-Landau equation is presented. In the Benjamin-Feir stable region, spatiotemporal intermittency regimes are identified which consist of patches of linearly stable plane waves separated by localized objects with a well defined dynamics. The simplest of these structures are shown to be members of a family of exact solutions discovered by Nozaki and Bekki (1985). The problem of the determination of the parameter domain of existence of spatiotemporal intermittency is discussed. In particular, given the inadequacy of the quantities usually measured to determine spatiotemporal disorder, only a rather crude determination of the limit of spatiotemporal intermittency is proposed, awaiting further knowledge on the nature, stability and interactions of the localized objects involved. In the transition region, asymptotic states with an irregular, frozen spatial structure are shown to occur. Finally, the disordered regimes observed in the Benjamin-Feir unstable region are reviewed and argued to be of the spatiotemporal intermittency type in some cases.

Mode formation and beating in the transverse pattern dynamics in a laser

Numerical studies of a model introduced by Lugiato, Oldano and Narducci for the interaction among transverse modes in a laser reveal time-dependent solutions in the region of “cooperative frequency locking” (CFL), which are unexpected from linear stability analysis. Spectral analysis reveals that time-dependent solutions are predominantly beating between two CFL solutions. Both CFL solutions and time-dependent solutions coexist stably with a stable plane wave solution in some regions.