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Nonlinear Sciences

New submissions

[ total of 9 entries: 1-9 ]
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New submissions for Mon, 25 Sep 17

[1]  arXiv:1709.07568 [pdf, other]
Title: A dynamical systems model of unorganised segregation
Comments: 17 pages (inc references), 9 figures
Subjects: Adaptation and Self-Organizing Systems (nlin.AO)

We consider Schelling's bounded neighbourhood model (BNM) of unorganised segregation of two populations from the perspective of modern dynamical systems theory. We derive a Schelling dynamical system and carry out a complete quantitative analysis of the system for the case of a linear tolerance schedule in both populations. In doing so, we recover and generalise Schelling's qualitative results. For the case of unlimited population movement, we derive exact formulae for regions in parameter space where stable integrated population mixes can occur. We show how neighbourhood tipping can be adequately explained in terms of basins of attraction. For the case of limiting population movement, we derive exact criteria for the occurrence of new population mixes and identify the stable cases. We show how to apply our methodology to nonlinear tolerance schedules, illustrating our approach with numerical simulations. We associate each term in our Schelling dynamical system with a social meaning. In particular we show that the dynamics of one population in the presence of another can be summarised as follows
{rate of population change} = {intrinsic popularity of neighbourhood} - {finite size of neighbourhood} - {presence of other population}
By approaching the dynamics from this perspective, we have a complementary approach to that of the tolerance schedule.

Cross-lists for Mon, 25 Sep 17

[2]  arXiv:1709.07309 (cross-list from math-ph) [pdf, ps, other]
Title: Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials
Comments: 17 pages
Subjects: Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)

For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $\pi$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},\pi)$-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of $\mathfrak{g}$-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of $(\mathfrak{g},\pi)$-type with the coefficients being the Pl\"ucker coordinates. As an application, we provide a way of computing polynomial tau functions for the DS hierarchy. For $\mathfrak{g}$ of low rank, we give several examples of polynomial tau functions, and use them to detect bilinear equations for the DS hierarchy.

[3]  arXiv:1709.07746 (cross-list from math.AP) [pdf, ps, other]
Title: Control of blow-up singularities for nonlinear wave equations
Journal-ref: Evolution Equations and Control Theory, vol. 2, no. 4 (Dec. 2013) 667-677
Subjects: Analysis of PDEs (math.AP); Pattern Formation and Solitons (nlin.PS)

While the global boundary control of nonlinear wave equations that exhibit blow-up is generally impossible, we show on a typical example, motivated by laser breakdown, that it is possible to control solutions with small data so that they blow up on a prescribed compact set bounded away from the boundary of the domain. This is achieved using the representation of singular solutions with prescribed blow-up surface given by Fuchsian reduction. We outline on this example simple methods that may be of wider applicability.

[4]  arXiv:1709.07760 (cross-list from gr-qc) [pdf, ps, other]
Title: Soliton stars in the breather limit
Journal-ref: Classical and Quantum Gravity, 25 (2008) 245004
Subjects: General Relativity and Quantum Cosmology (gr-qc); Analysis of PDEs (math.AP); Pattern Formation and Solitons (nlin.PS)

This paper presents an asymptotic reduction of the Einstein-Klein-Gordon system with real scalar field ("soliton star problem"). A periodic solution of the reduced system, similar to the sine-Gordon breather, is obtained by a variational method. This tallies with numerical computations. As a consequence, a time-periodic redshift for sources close to the center of the star is obtained.

[5]  arXiv:1709.07784 (cross-list from physics.flu-dyn) [pdf]
Title: Momentary increase in reactive surface area in dissolving carbonate
Comments: 32 pages, 8 figures
Subjects: Fluid Dynamics (physics.flu-dyn); Soft Condensed Matter (cond-mat.soft); Pattern Formation and Solitons (nlin.PS); Geophysics (physics.geo-ph)

The quantification of surface area between mineral and reactive fluid is essential in environmental applications of reactive transport modelling. This quantity evolves with microstructures and is difficult to predict because the mechanisms for the generation and destruction of reactive surface remain elusive. The challenge of accounting for the inherent heterogeneities of natural porous media in numerical simulation further complicates the problem. Here we first show a direct observation of reactive surface generation in chalk under circumneutral to alkaline pH using in situ X-ray microtomography. The momentary increase of reactive surface area cannot be explained by a change in fluid accessibility or by surface roughening stemming from mineralogical heterogeneity. We then combine greyscale nanotomography data with numerical simulations to show that similar temporal behaviour can be observed over a wide range of pH as porous media dissolve in imposed flow field. We attribute the observation to the coupling between fluid flow and mineral dissolution and argue that the extent of surface generation is strongly correlated with the advective penetration depth of reactants. To conclude, we demonstrate the applicability of using a macroscopic Damk\"ohler number as an indicator for the phenomenon and discuss its environmental significance beyond geologic carbon storage.

[6]  arXiv:1709.07787 (cross-list from math.AP) [pdf, ps, other]
Title: Breather Solutions of the Nonlinear Wave Equation
Journal-ref: Communications in Pure and Applied Mathematics, 44 (1991) 789--818
Subjects: Analysis of PDEs (math.AP); General Relativity and Quantum Cosmology (gr-qc); Pattern Formation and Solitons (nlin.PS)

We construct series solutions to all orders for breathers of Klein-Gordon equations, in powers of an amplitude parameter epsilon, under a sign condition on the coefficients of the expansion of the nonlinearity. All terms may be computed thanks to the properties of a 1D Schr\"odinger equation with two-soliton potential. We prove that this series is free of poles in the unit disc of the complex epsilon plane if and only if the equation is essentially equivalent to the sine-Gordon equation. This appears to be the only result that explains both what makes the sine-Gordon equation special from a PDE point of view, and at the same time, why approximate, long-lived breathers exist. In a related paper (Classical and Quantum Gravity, 25 (2008) 245004 ) we have extended this approach to address the soliton star problem in General Relativity.

[7]  arXiv:1709.07850 (cross-list from physics.flu-dyn) [pdf, other]
Title: Nonlinear stage of Benjamin-Feir instability in forced/damped deep water waves
Comments: 8 pages, 4 figures submitted to Physics of Fluids
Subjects: Fluid Dynamics (physics.flu-dyn); Pattern Formation and Solitons (nlin.PS)

We study a three-wave truncation of a recently proposed damped/forced high-order nonlinear Schr\"odinger equation (DF-HONLS) for deep-water gravity waves under the effect of wind and viscosity. The evolution of the norm (wave-action) and spectral mean of the full model are well captured by the reduced dynamics. Three regimes are found for the wind-viscosity balance: we classify them according to the attractor in the phase-plane of the degenerate four-wave mixing truncated system and to the shift of the spectral mean. Downshift can coexist with both forcing and damping, i.e. attraction to period-1 or period-2 solutions. Upshift is associated to stronger winds, i.e. to a net forcing where the attractor is always a period-1 solution. The applicability of our classification to experiments in long wave-tanks is verified.

Replacements for Mon, 25 Sep 17

[8]  arXiv:1603.01880 (replaced) [pdf, ps, other]
Title: Optimal sequence memory in driven random networks
Subjects: Neurons and Cognition (q-bio.NC); Chaotic Dynamics (nlin.CD)
[9]  arXiv:1611.01164 (replaced) [pdf, other]
Title: Sensitive Dependence of Optimal Network Dynamics on Network Structure
Comments: Accepted for publication in PRX; 19 pages, 9 figures; substantial new materials added since last version
Subjects: Adaptation and Self-Organizing Systems (nlin.AO); Disordered Systems and Neural Networks (cond-mat.dis-nn); Physics and Society (physics.soc-ph)
[ total of 9 entries: 1-9 ]
[ showing up to 2000 entries per page: fewer | more ]

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