Asymptotic Wavenumber Research Articles

Target patterns in a 2D array of oscillators with nonlocal coupling

We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction–diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when , the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in .The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be ‘correct’ to all orders in . We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution which was obtained using matched asymptotics.

On the asymptotic wavenumber of spiral waves in systems

In this paper we consider spiral wave solutions of a general class of systems with a small twist parameter q and we prove that the asymptotic wavenumber of the spirals is a -flat function of the perturbation parameter q.

The scattering phase shifts of the Hulthén-type potential plus Yukawa potential

The Hulthen-type potential is perturbed by adding the Yukawa potential. By applying a short-range approximation to the Schrodinger equation containing this model via standard method, the scattering state solutions and the corresponding phase shifts were obtained. Our numerical calculations and graphical solutions show the dependencies of scattering phase shifts on Yukawa potential constant A, asymptotic wave number k, screening parameter \( \alpha\) and angular momentum quantum number l.

The effect of boundaries on the asymptotic wavenumber of spiral wave solutions of the complex Ginzburg–Landau equation

In this paper we consider an oscillatory medium whose dynamics are modeled by the complex Ginzburg–Landau equation. In particular, we focus on n-armed spiral wave solutions of the complex Ginzburg–Landau equation in a disk of radius d with homogeneous Neumann boundary conditions. It is well-known that such solutions exist for small enough values of the twist parameter q and large enough values of d. We investigate the effect of boundaries on the rotational frequency of the spirals, which is an unknown of the problem uniquely determined by the parameters d and q. We show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak. We use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q.

Motion of spiral waves in the complex Ginzburg–Landau equation

Solutions of the general cubic complex Ginzburg–Landau equation comprising multiple spiral waves are considered. For parameters close to the vortex limit, and for a system of spiral waves with well-separated centres, laws of motion of the centres are found which vary depending on the order of magnitude of the separation of the centres. In particular, the direction of the interaction changes from along the line of centres to perpendicular to the line of centres as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wavenumber and frequency are determined. These depend on the positions of the centres of the spirals, and so evolve slowly as the spirals move.

Interaction of Spiral Waves in the Complex Ginzburg-Landau Equation

Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered, and laws of motion for the centers are derived. The direction of the motion changes from along the line of centers to perpendicular to the line of centers as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wave number and frequency are also determined, which evolve slowly as the spirals move.

Weakly radiative spiral waves

The transition from the complex Ginzburg–Landau equation (CGLe) to the nonlinear Schrödinger equation (NLSe) in the analytically tractable weakly radiative limit is highly singular. This is strongly felt in the structure of spiral wave solutions of CGLe. The asymptotic wavenumber of the radiated wave is exponentially small in the effective “radiative” parameter q, but this value is attained only at exponentially large distances. This calls for great caution in application of perturbation schemes to detect various dynamic effects in weakly distorted spirals. The perturbation in q is expected to work well at moderately large, but not on exponentially large distances. After outlining the general perturbation scheme, we explore interactions of oppositely charged spirals and show that no bound pairs can be formed before interaction becomes exponentially weak.

On interaction of spiral waves

Bound states of a pair of interacting spiral waves (vortices) with the opposite sense of rotation are sought for using consistent approximations near the vortex core and in the outer region under conditions when the asymptotic wavenumber is small. The far field solution is constructed by correcting a logarithmic superposition solution to insure the preservation of topological charges. The final result, obtained by matching the inner and outer solutions, shows that the interaction remains attractive at all distances, and bound states are not formed.

Mobility of spiral waves.

We consider interacting rotating spiral waves (vortices) under conditions when the asymptotic wave number is small and apply consistent approximations near the vortex core and in the outer region and asymptotic matching to derive a mobility relationship that connects the propagation velocity of a vortex with relevant characteristics of the extrinsic phase field in its vicinity. The results are applied to the problem of the existence of bound states of a pair of interacting vortices of the opposite charge. It is found that interaction decays exponentially with separation and remains attractive at all distances.

Spiral and concentric waves in active media with soft and hard excitation

An approximation method is presented for describing spiral waves with the generalized Ginzburg-Landau equation, which for weak overcriticality is a universal model of active media with soft excitation of self-oscillation. A comparison with numerical results indicates the this method, which is based on direct matching of asymptotic expansions, is valid for large and small values of the radial variable, and yields satisfactory values for the asymptotic radial wave number of a spiral wave for values of the dispersion coefficients that are not too small. Spiral waves are described with a simple model of an active medium with hard excitation. This description is based on the rigorous method of asymptotic expansion matching for small dispersion coefficients and on the direct matching method for the general case. Both methods are used to show that, in contrast to spiral waves, the asymptotic wave number for concentric waves is an arbitrary parameter, and the group velocity is directed from the periphery to the center.