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ABSTRACT: We consider a parametrically driven damped discrete nonlinear Schr\"odinger
(PDDNLS) equation. Analytical and numerical calculations are performed to
determine the existence and stability of fundamental discrete bright solitons.
We show that there are two types of onsite discrete soliton, namely onsite type
I and II. We also show that there are four types of intersite discrete soliton,
called intersite type I, II, III, and IV, where the last two types are
essentially the same, due to symmetry. Onsite and intersite type I solitons,
which can be unstable in the case of no dissipation, are found to be stabilized
by the damping, whereas the other types are always unstable. Our further
analysis demonstrates that saddle-node and pitchfork (symmetry-breaking)
bifurcations can occur. More interestingly, the onsite type I, intersite type
I, and intersite type III-IV admit Hopf bifurcations from which emerge periodic
solitons (limit cycles). The continuation of the limit cycles as well as the
stability of the periodic solitons are computed through the numerical
continuation software Matcont. We observe subcritical Hopf bifurcations along
the existence curve of the onsite type I and intersite type III-IV. Along the
existence curve of the intersite type I we observe both supercritical and
subcritical Hopf bifurcations.
06/2012;
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ABSTRACT: Traveling solitary waves in the one-dimensional discrete nonlinear Schrödinger equation with saturable onsite nonlinearity are studied. A variational approximation (VA) for the solitary waves is derived in analytical form. The stability is also studied by means of the VA, demonstrating that the solitons are stable, which is consistent with previously published results. Then, the VA is applied to predict parameters of traveling solitons with non-oscillatory tails (embedded solitons, ESs). Two-soliton bound states are considered too. The separation distance between the solitons forming the bound state is derived by means of the VA. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton–Raphson method. In general, good agreement between the analytical and numerical results is obtained. In particular, we demonstrate the relevance of the analytical prediction of characteristics of the embedded solitons.
Journal of Physics A Mathematical and Theoretical 02/2012; 45(7):075207. · 1.56 Impact Factor
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ABSTRACT: Travelling solitary waves in the one-dimensional discrete nonlinear
Schr\"{o}dinger equation (DNLSE) with saturable onsite nonlinearity are
studied. A variational approximation (VA) for the solitary waves is derived in
an analytical form. The stability is also studied by means of the VA,
demonstrating that the solitons are stable, which is consistent with previously
published results. Then, the VA is applied to predict parameters of travelling
solitons with non-oscillatory tails (\textit{embedded solitons}, ESs).
Two-soliton bound states are considered too. The separation distance between
the solitons forming the bound state is derived by means of the VA. A numerical
scheme based on the discretization of the equation in the moving coordinate
frame is derived and implemented using the Newton--Raphson method. In general,
a good agreement between the analytical and numerical results is obtained. In
particular, we demonstrate the relevance of the analytical prediction of
characteristics of the embedded solitons.
01/2012;
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ABSTRACT: We study homoclinic snaking in the cubic-quintic Swift–Hohenberg equation (SHE) close to the onset of a subcritical pattern-forming instability. Application of the usual multiple-scales method produces a leading-order stationary front solution, connecting the trivial solution to the patterned state. A localized pattern may therefore be constructed by matching between two distant fronts placed back-to-back. However, the asymptotic expansion of the front is divergent, and hence should be truncated. By truncating optimally, such that the resultant remainder is exponentially small, an exponentially small parameter range is derived within which stationary fronts exist. This is shown to be a direct result of the 'locking' between the phase of the underlying pattern and its slowly varying envelope. The locking mechanism remains unobservable at any algebraic order, and can only be derived by explicitly considering beyond-all-orders effects in the tail of the asymptotic expansion, following the method of Kozyreff and Chapman as applied to the quadratic-cubic SHE (Chapman and Kozyreff 2009 Physica D 238 319–54, Kozyreff and Chapman 2006 Phys. Rev. Lett. 97 44502). Exponentially small, but exponentially growing, contributions appear in the tail of the expansion, which must be included when constructing localized patterns in order to reproduce the full snaking diagram. Implicit within the bifurcation equations is an analytical formula for the width of the snaking region. Due to the linear nature of the beyond-all-orders calculation, the bifurcation equations contain an analytically indeterminable constant, estimated in the previous work by Chapman and Kozyreff using a best fit approximation. A more accurate estimate of the equivalent constant in the cubic-quintic case is calculated from the iteration of a recurrence relation, and the subsequent analytical bifurcation diagram compared with numerical simulations, with good agreement.
Nonlinearity 10/2011; 24(12):3323. · 1.39 Impact Factor
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ABSTRACT: We consider a particular type of parametrically driven discrete Klein-Gordon system describing microdevices and nanodevices, with integrated electrical and mechanical functionality. Using a multiscale expansion method we reduce the system to a discrete nonlinear Schrödinger equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental bright and dark discrete solitons admitted by the Klein-Gordon system through the discrete Schrödinger equation. We show that a parametric driving can not only destabilize onsite bright solitons, but also stabilize intersite bright discrete solitons and onsite and intersite dark solitons. Most importantly, we show that there is a range of values of the driving coefficient for which dark solitons are stable, for any value of the coupling constant, i.e., oscillatory instabilities are totally suppressed. Stability windows of all the fundamental solitons are presented and approximations to the onset of instability are derived using perturbation theory, with accompanying numerical results. Numerical integrations of the Klein-Gordon equation are performed, confirming the relevance of our analysis.
Physical Review E 02/2010; 81(2 Pt 2):026207. · 2.26 Impact Factor
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SIAM J. Applied Dynamical Systems. 01/2008; 7:63-78.
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ABSTRACT: The Nikolaevskiy equation has been proposed as a model for seismic waves, electroconvection, and weak turbulence; we show that it can also be used to model transverse instabilities of fronts. This equation possesses a large-scale "Goldstone" mode that significantly influences the stability of spatially periodic steady solutions; indeed, all such solutions are unstable at onset, and the equation exhibits spatiotemporal chaos. In many applications, a weak damping of this neutral mode will be present, and we study the influence of this damping on solutions to the Nikolaevskiy equation. We examine the transition to the usual Eckhaus instability as the damping of the large-scale mode is increased, through numerical calculation and weakly nonlinear analysis. The latter is accomplished using asymptotically consistent systems of coupled amplitude equations. We find that there is a critical value of the damping below which (for a given value of the supercriticality parameter) all periodic steady states are unstable. The last solutions to lose stability lie in a cusp close to the left-hand side of the marginal stability curve.
Physical Review E 12/2007; 76(5 Pt 2):056202. · 2.26 Impact Factor
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ABSTRACT: The influence of a conserved quantity on an oscillatory pattern-forming instability is examined in one space dimension. Amplitude equations are derived which are not only generic for systems with a pseudoscalar conserved quantity (e.g. rotating convection, magnetoconvection) but also applicable to systems with a scalar conserved quantity. The stability properties of both travelling and standing waves are analysed, with particular progress being possible in the limit of long-wavelength perturbations. For both forms of waves, the corresponding modulational stability boundaries are significantly altered by the presence of the conserved quantity; also, new instabilities are generated. For general perturbations, the full stability regions are found numerically. Simulations of the nonlinear governing equations are performed using a pseudo-spectral code; a variety of stable attractors are thus found of varying degrees of complexity. Previously unseen, highly localized, solutions are observed.
Nonlinearity 02/2005; 18(3):1031. · 1.39 Impact Factor
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ABSTRACT: A model system of partial differential equations in two dimensions is derived from the three-dimensional equations for thermal convection in a horizontal fluid layer in a vertical magnetic field. The model consists of an equation of Swift-Hohenberg type for the amplitude of convection, coupled to an equation for a large-scale mode representing the local strength of the magnetic field. The model facilitates both analytical and numerical studies of magnetoconvection in large domains. In particular, we investigate the phenomenon of flux separation, where the domain divides into regions of strong convection with a weak magnetic field and regions of weak convection with a strong field. Analytical predictions of flux separation based on weakly nonlinear analysis are extended into the fully nonlinear regime through numerical simulations. The results of the model are compared with simulations of the full three-dimensional magnetoconvection problem.
Physical Review E 07/2004; 69(6 Pt 2):066314. · 2.26 Impact Factor
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ABSTRACT: The effectiveness of a large number of protocols for mixing in a two-dimensional chaotic Stokes flow, according to a variety of measures, is investigated. The degree to which the various mixing measures are correlated is computed, and while no single protocol simultaneously optimises all measures, it is found that a small subset of the protocols perform well against most measures. However, it is difficult to elicit general rules for selecting effective protocols: for example, superficially similar protocols are found to exhibit considerably different mixing capabilities. The results presented here suggest that the selection of effective protocols by `sieving' (i.e., by successively eliminating candidate protocols that fail increasingly discerning mixing measures) may be ineffective in practice.
Journal of Engineering Mathematics 01/2004; 48(2):129-155. · 0.86 Impact Factor
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ABSTRACT: Topological chaos may be used to generate highly effective laminar mixing in a simple batch stirring device. Boyland, Aref & Stremler (2000) have computed a material stretch rate that holds in a chaotic flow, provided it has appropriate topological properties, irrespective of the details of the flow. Their theoretical approach, while widely applicable, cannot predict the size of the region in which this stretch rate is achieved. Here, we present numerical simulations to support the observation of Boyland et
al. that the region of high stretch is comparable with that through which the stirring elements move during operation of the device. We describe a fast technique for computing the velocity field for either inviscid, irrotational or highly viscous flow, which enables accurate numerical simulation of dye advection. We calculate material stretch rates, and find close agreement with those of Boyland et
al., irrespective of whether the fluid is modelled as inviscid or viscous, even though there are significant differences between the flow fields generated in the two cases.
Journal of Fluid Mechanics 10/2003; 493:345 - 361. · 2.46 Impact Factor
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ABSTRACT: We analyze thermal convection in a fluid layer confined between isothermal horizontal boundaries at which the tangential component of the fluid stress vanishes. The layer rotates about an oblique, nearly vertical axis. Using a model set of equations for w, the horizontal planform of the vertical velocity component, and psi, a stream function related to a large-scale vertical vorticity field, we describe the instabilities of convection rolls. We show how the usual Küppers-Lortz instability, which leads to a continual precession of the roll pattern, can be suppressed by the oblique rotation vector. Of particular interest is the small-angle instability of rolls, to perturbations in the form of rolls that are almost aligned with the primary rolls; at finite Prandtl number, this instability is not prevented by the horizontal component of the rotation vector, unless this component is sufficiently strong, in which case stability is confined to small-amplitude rolls near the marginal stability boundary. A one-dimensional instability leading to amplitude-modulated rolls is unaffected by the oblique rotation. Numerical simulations of the model equations are presented, which illustrate the instabilities analyzed.
Physical Review E 02/2003; 67(1 Pt 2):016301. · 2.26 Impact Factor
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ABSTRACT: The large-time asymptotic behavior of a two-stage reaction (A+B-->R, B+R-->S) with initially segregated reactants is described. The concentration of the reactants is found to be significantly less than the initial concentrations in a depletion zone of width proportional to t(1/2), where t is time; the reaction takes place in a thinner zone of width proportional to t(1/6). Similarity solutions for the chemical concentration profiles in the reaction zone are calculated, and are compared with numerical simulations of the full partial differential reaction-diffusion equations. The large-time asymptotic scalings reported here are the same as in the absence of the secondary reaction, but we find that the location of the reaction zone is significantly shifted due to the secondary reaction. The reaction zone may behave in an exotic fashion at large time, moving first one way, then reversing its direction.
Physical Review E 05/2001; 63(5 Pt 1):051102. · 2.26 Impact Factor
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ABSTRACT: Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near onset. Near a stationary bifurcation, the usual Ginzburg--Landau equation for the amplitude of the pattern is then coupled to an equation for the large-scale mode. These amplitude equations show that for certain parameters all roll-type solutions are unstable. This new instability differs from the Eckhaus instability in that it is amplitude-driven and is supercritical. Beyond the stability boundary, there exist stable stationary solutions in the form of strongly modulated patterns. The envelope of these modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, is closely approximated by a sech profile. Numerical simulations indicate that as the modulation becomes more pronounced, the envelope broadens. A number of applications are considered, including convection with fixed-flux boundaries and convection in a magnetic field, resulting in new instabilities for these systems. Comment: 28 pages, to appear in Nonlinearity
06/2000;
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ABSTRACT: Convection rolls in a rotating layer can become unstable to
the Küppers–Lortz
instability. When the horizontal boundaries are stress free and the Prandtl number is
finite, this instability diverges in the limit where the perturbation rolls make a small
angle with the original rolls. This divergence is resolved by taking full account of
the resonant mode interactions that occur in this limit: it is necessary to include two
roll modes and a large-scale mean flow in the perturbation. It is found that rolls of
critical wavelength whose amplitude is of order ε are always unstable to rolls oriented
at an angle of order ε2/5. However, these rolls are unstable to perturbations at an
infinitesimal angle if the Taylor number is greater than 4π4.
Unlike the Küppers–Lortz instability, this new instability at infinitesimal angles does not depend on the
direction of rotation; it is driven by the flow along the axes of the rolls. It is
this instability that dominates in the limit of rapid rotation. Numerical simulations
confirm the analytical results and indicate that the instability is subcritical, leading to
an attracting heteroclinic cycle. We show that the small-angle instability grows more
rapidly than the skew-varicose instability.
Journal of Fluid Mechanics 01/2000; 403:153 - 172. · 2.46 Impact Factor
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ABSTRACT: We describe a new instability of rolls in three related convection problems. The instability results in a large-scale modulation of the amplitude of the convection (in contrast to a phase instability such as that of Eckhaus) and may be supercritical, with strongly amplitude-modulated rolls predicted at onset. Essential to the instability is the presence of a conserved quantity (whose existence is contingent upon an appropriate choice of boundary conditions), which gives rise to a corresponding slowly evolving large-scale mode. In rotating convection between stress-free boundaries, the conserved quantity is the velocity component along the axes of the rolls; in magnetoconvection it is the magnetic flux through the layer. In rotating magnetoconvection both quantities may be conserved. In each case the appropriate amplitude equations to describe convection near onset consist of a modulation equation of Ginzburg–Landau type for the amplitude of the rolls, coupled to a large-scale modulation equation for each conserved quantity. The large-scale mode(s), although damped according to linear theory, can destabilise the rolls at onset. It is found that rolls are unstable for sufficiently small Prandtl number in rotating convection, and for sufficiently small magnetic diffusivity in magnetoconvection. Throughout we consider only the onset of convection through a stationary bifurcation, but we find, remarkably, that in rotating magnetoconvection stable travelling waves may be found at onset due to this new instability.
Physica D: Nonlinear Phenomena.
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ABSTRACT: We describe the influence of a conserved quantity on the stability properties of roll, square and hexagonal patterns near stationary onset. The appropriate systems of amplitude equations are analysed: they comprise equations for the amplitudes of the pattern modes, together with an additional coupled equation governing the evolution of a large-scale mode associated with the conserved quantity. We find that the presence of this large-scale mode may result in the destabilisation of all regular roll, square or hexagonal patterns, leading to amplitude modulation and strong localisation of the pattern. Our analytical results are complemented throughout by numerical simulations of a model partial differential equation, to illustrate the modulational instabilities and their nonlinear development.
Physica D: Nonlinear Phenomena.
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ABSTRACT: We develop a class of numerical methods for stiff systems, based on the method of exponential time differencing. We describe schemes with second- and higher-order accuracy, introduce new Runge–Kutta versions of these schemes, and extend the method to show how it may be applied to systems whose linear part is nondiagonal. We test the method against other common schemes, including integrating factor and linearly implicit methods, and show how it is more accurate in a number of applications. We apply the method to both dissipative and dispersive partial differential equations, after illustrating its behavior using forced ordinary differential equations with stiff linear parts.
Journal of Computational Physics.
