| 1. |
- Aurell, Erik, et al.
(author)
-
On the dynamics of a self-gravitating medium with random and non-random initial conditions
- 2001
-
In: Physica D. - 0167-2789 .- 1872-8022. ; 148:04-mar, s. 272-288
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Journal article (peer-reviewed)abstract
- The dynamics of a 1D self-gravitating medium with initial density almost uniform is studied. Numerical experiments are performed with ordered and with Gaussian random initial conditions. The phase space portraits art shown to be qualitatively similar to shock waves, in particular with initial conditions of Brownian type. The PDF of the mass distribution is investigated.
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| 2. |
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| 3. |
- Dankowicz, H., et al.
(author)
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On the origin and bifurcations of stick-slip oscillations
- 2000
-
In: Physica D. - 0167-2789 .- 1872-8022. ; 136:04-mar, s. 280-302
-
Journal article (peer-reviewed)abstract
- A recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis. Particular emphasis is put on the bifurcations associated with the appearance of stick-slip oscillations. In the model it is found that the existence of these oscillations is a result of a periodic orbit straddling a discontinuity in the first derivative of the vector field. A local analysis tool is developed to discuss the stability of such an orbit and its bifurcations due to changes in system parameters. The analysis tool is found to be highly efficient at quantitatively predicting the location and type of bifurcations. It is argued that the method and the general results are applicable to a large class of friction models containing similar discontinuities and thus, hopefully, to actual frictional dynamics. (C)2000 Elsevier Science B.V. All rights reserved.
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| 4. |
- di Bernardo, M., et al.
(author)
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Bifurcations of dynamical systems with sliding : derivation of normal-form mappings
- 2002
-
In: Physica D. - 0167-2789 .- 1872-8022. ; 170:04-mar, s. 175-205
-
Journal article (peer-reviewed)abstract
- This paper is concerned with the analysis of so-called sliding bifurcations in n-dimensional piecewise-smooth dynamical systems with discontinuous vector field. These novel bifurcations occur when the system trajectory interacts with regions on the discontinuity set where sliding is possible. The derivation of appropriate normal-form maps is detailed. It is shown that the leading-order term in the map depends on the particular bifurcation scenario considered. This is in turn related to the possible bifurcation scenarios exhibited by a periodic orbit undergoing one of the sliding bifurcations discussed in the paper. A third-order relay system serves as a numerical example.
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| 5. |
- Djehiche, Boualem
(author)
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Global solution of the pressureless gas equation with viscosity
- 2002
-
In: Physica D. - 0167-2789 .- 1872-8022. ; 163:3-4, s. 184-190
-
Journal article (peer-reviewed)abstract
- We construct a global weak solution to a d-dimensional system of zero-pressure gas dynamics modified by introducing a finite artificial viscosity. We use discrete approximations to the continuous gas and make particles move along trajectories of the normalized simple symmetric random walk with deterministic drift. The interaction of these particles is given by a sticky particle dynamics. We show that a subsequence of these approximations converges to a weak solution of the system of zero-pressure gas dynamics in the sense of distributions. This weak solution is interpreted in terms of a random process solution of a nonlinear stochastic differential equation. We get a weak solution of the inviscid system by tending the viscosity to zero.
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| 6. |
- Gaite, José A., et al.
(author)
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Nonlinear spherical gravitational downfall of gas onto a solid ball : Analytic and numerical results
- 2003
-
In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 183:1-2, s. 102-116
-
Journal article (peer-reviewed)abstract
- The process of downfall of initially homogeneous gas onto a solid ball due to the ball's gravity (relevant in astrophysical situations) is studied with a combination of analytic and numerical methods. The initial explicit solution soon becomes discontinuous and gives rise to a shock wave. Afterwards, there is a crossover between two intermediate asymptotic similarity regimes, where the shock wave propagates outwards according to two self-similar laws, initially accelerating and eventually decelerating and vanishing, leading to a static state. The numerical study allows one to investigate in detail this dynamical problem and its time evolution, verifying and complementing the analytic results on the initial solution, intermediate self-similar laws and static long-term solution.
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| 7. |
- Lennholm, Erik, et al.
(author)
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Revisiting Salerno's sine-Gordon model of DNA : Active regions and robustness
- 2003
-
In: Physica D. - 0167-2789 .- 1872-8022. ; 177:1-4, s. 233-241
-
Journal article (peer-reviewed)abstract
- We return to a simple model of DNA-transcription, first investigated by Salerno more than 10 years ago. One conjecture that time was that the promoter-regions were "dynamically active" in the sense that a stationary kink solution to the discrete sine-Gordon equation spontaneously starts to move when positioned in certain regions. Here we explore the whole genome of the bacteriophage T7, which is the same that was used in the first studies. We find that the regions in the promoters where the DNA-binding molecules attach have no special significance, while the start of the RNA-coding regions are dynamically active on a significant level. The results are checked to be robust by imposing an external disturbance in the form of a thermostat, simulating a constant temperature. © 2002 Elsevier Science B.V. All rights reserved.
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| 8. |
- Morgante, A.M., et al.
(author)
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Standing wave instabilities in a chain of nonlinear coupled oscillators
- 2002
-
In: Physica D. - 0167-2789 .- 1872-8022. ; 162:1-2, s. 53-94
-
Journal article (peer-reviewed)abstract
- We consider existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators. Specifically, we consider Klein-Gordon (KG) chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site potentials, as well as discrete nonlinear Schrödinger (DNLS) chains approximating the small-amplitude dynamics of KG chains with weak inter-site coupling. The SWs are constructed as exact time-periodic multibreather solutions from the anticontinuous limit of uncoupled oscillators. In the validity regime of the DNLS approximation these solutions can be continued into the linear phonon band, where they merge into standard harmonic SWs. For SWs with incommensurate wave vectors, this continuation is associated with an inverse transition by breaking of analyticity. When the DNLS approximation is not valid, the continuation may be interrupted by bifurcations associated with resonances with higher harmonics of the SW. Concerning the stability, we identify one class of SWs which are always linearly stable close to the anticontinuous limit. However, approaching the linear limit all SWs with non-trivial wave vectors become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. Investigating the dynamics resulting from these instabilities, we find two qualitatively different regimes for wave vectors smaller than or larger than p/2, respectively. In one regime persisting breathers are found, while in the other regime the system rapidly thermalizes. © 2002 Elsevier Science B.V. All rights reserved.
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| 9. |
- Schiebold, Cornelia
(author)
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An operator-theoretic approach to the Toda lattice equation
- 1998
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In: Physica D. - 0167-2789 .- 1872-8022. ; 122:1-4, s. 37-61
-
Journal article (peer-reviewed)abstract
- We treat the Toda lattice equation with operator methods and derive an explicit solution formula in terms of determinants. As an application, we investigate solutions which are given by special settings. In the finite-dimensional case matrices in Jordan canonical form give rise to a new class of solutions. Within this class the well-known N-soliton solutions can be recovered by the special choice of diagonal matrices. Moreover, using diagonal operators we get solutions depending on an infinite number of parameters. We comprehensively discuss the case involving diagonal operators and show that it can be reduced to a very particular situation.
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| 10. |
- Wyller, John, et al.
(author)
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The effect of resonant particles on Alfvén solitons
- 1989
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In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 39:2-3, s. 405-422
-
Journal article (peer-reviewed)abstract
- The derivative nonlinear Schrödinger equation is treated. A perturbation theory for the evolution of an Alfvén soliton is given. This leads to a phase portrait of the changes in the soliton parameters. The phase curves are circles. Energy dissipation is estimated. Numerical calculations are given and results compared.
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| 11. |
- Zorzano, María Paz, et al.
(author)
-
Emergence of synchronous oscillations in neural networks excited by noise
- 2003
-
In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 179:1-2, s. 105-114
-
Journal article (peer-reviewed)abstract
- The presence of noise in nonlinear dynamical systems can play a constructive role, increasing the degree of order and coherence or evoking improvements in the performance of the system. An example of this positive influence in a biological system is the impulse transmission in neurons and the synchronization of a neural network. Integrating numerically the Fokker-Planck (FP) equation we show a self-induced synchronized oscillation. Such an oscillatory state appears in a neural network coupled with a feedback term, when this system is excited by noise and the noise strength is within a certain range
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| 12. |
- Zorzano, María Paz, et al.
(author)
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Numerical solution for Fokker-Planck equations in accelerators
- 1998
-
In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 113:2-4, s. 379-381
-
Journal article (peer-reviewed)abstract
- We present a finite difference scheme to solve numerically the type of Fokker-Planck equations describing the stochastic dynamics of a particle in a storage ring.
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| 13. |
- Aurell, Erik, et al.
(author)
-
The inner structure of Zeldovich pancakes
- 2003
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 186:04-mar, s. 171-184
-
Journal article (peer-reviewed)abstract
- The evolution of a planar perturbation in a Einstein-de Sitter Universe is studied using a previously introduced Lagrangian scheme. An approximate discrete dynamical system is derived, which describes the mass agglomeration process. Quantitative predictions for the late-time mean density profile are obtained therefrom, and validated by numerical simulations. A simple but important result is that the characteristic scale of a mass agglomeration is an increasing function of cosmological time t. For one kind of initial conditions we further find a scaling regime for the density profile of a collapsing object. These results are compared with analogous investigations for the adhesion model (Burgers equation with positive viscosity). We further study the mutual motion of two mass agglomerations, and show that they oscillate around each other for long times, like two heavy particles. Individual particles in the two agglomerations do not mix effectively on the time scale of the inter-agglomeration motion.
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| 14. |
- Berntson, Bjorn K., et al.
(author)
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A focusing–defocusing intermediate nonlinear Schrödinger system
- 2023
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 451
-
Journal article (peer-reviewed)abstract
- We introduce and study a system of coupled nonlocal nonlinear Schrödinger equations that interpolates between the mixed, focusing–defocusing Manakov system on one hand and a limiting case of the intermediate nonlinear Schrödinger equation on the other. We show that this new system, which we call the intermediate mixed Manakov (IMM) system, admits multi-soliton solutions governed by a complexification of the hyperbolic Calogero–Moser (CM) system. Furthermore, we introduce a spatially periodic version of the IMM system, for which our main result is a class of exact solutions governed by a complexified elliptic CM system.
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| 15. |
- Borgqvist, Johannes G., et al.
(author)
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Energy translation symmetries and dynamics of separable autonomous two-dimensional ODEs
- 2023
-
In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 454
-
Journal article (peer-reviewed)abstract
- We study symmetries in the phase plane for separable, autonomous two-state systems of ordinary differential equations (ODEs). We prove two main theoretical results concerning the existence and non-triviality of two orthogonal symmetries for such systems. In particular, we show that these symmetries correspond to translations in the internal energy of the system, and describe their action on solution trajectories in the phase plane. In addition, we apply recent results establishing how phase plane symmetries can be extended to incorporate temporal dynamics to these energy translation symmetries. Subsequently, we apply our theoretical results to the analysis of three models from the field of mathematical biology: a canonical biological oscillator model, the Lotka–Volterra (LV) model describing predator–prey dynamics, and the SIR model describing the spread of a disease in a population. We describe the energy translation symmetries in detail, including their action on biological observables of the models, derive analytic expressions for the extensions to the time domain, and discuss their action on solution trajectories.
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| 16. |
- Cetoli, Alberto, 1979-, et al.
(author)
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Dynamics of kicked matter-wave solitons in an optical lattice
- 2009
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In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 238:15, s. 1388-1393
-
Journal article (peer-reviewed)abstract
- We investigate effects of the application of a kick to one-dimensional matter-wave solitons in a self-attractive Bose–Einstein condensate trapped in an optical lattice. The resulting soliton’s dynamics is studied within the framework of the time-dependent nonpolynomial Schrödinger equation. The crossover from the pinning to quasi-free motion crucially depends on the size of the kick, strength of the self-attraction, and parameters of the optical lattice.
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| 17. |
- Charlier, Christophe, et al.
(author)
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Long-time asymptotics for an integrable evolution equation with a 3 x 3 Lax pair
- 2021
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 426
-
Journal article (peer-reviewed)abstract
- We derive a Riemann-Hilbert representation for the solution of an integrable nonlinear evolution equation with a 3 x 3 Lax pair. We use the derived representation to obtain formulas for the long-time asymptotics.
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| 18. |
- Cirillo, Emilio N.M., et al.
(author)
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When diffusion faces drift : consequences of exclusion processes for bi–directional pedestrian flows
- 2020
-
In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 413
-
Journal article (peer-reviewed)abstract
- Stochastic particle-based models are useful tools for describing the collective movement of large crowds of pedestrians in crowded confined environments. Using descriptions based on the simple exclusion process, two populations of particles, mimicking pedestrians walking in a built environment, enter a room from two opposite sides. One population is passive - being unaware of the local environment; particles belonging to this group perform a symmetric random walk. The other population has information on the local geometry in the sense that as soon as particles enter a visibility zone, a drift activates them. Their self-propulsion leads them towards the exit. This second type of species is referred here as active. The assumed crowdedness corresponds to a near-jammed scenario. The main question we ask in this paper is: Can we induce modifications of the dynamics of the active particles to improve the outgoing current of the passive particles? To address this question, we compute occupation number profiles and currents for both populations in selected parameter ranges. Besides observing the more classical faster-is-slower effect, new features appear as prominent like the non-monotonicity of currents, self-induced phase separation within the active population, as well as acceleration of passive particles for large-drift regimes of active particles.
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| 19. |
- Delvenne, Jean-Charles, et al.
(author)
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Finite-time thermodynamics of port-Hamiltonian systems
- 2014
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 267, s. 123-132
-
Journal article (peer-reviewed)abstract
- In this paper, we identify a class of time-varying port-Hamiltonian systems that is suitable for studying problems at the intersection of statistical mechanics and control of physical systems. Those port-Hamiltonian systems are able to modify their internal structure as well as their interconnection with the environment over time. The framework allows us to prove the First and Second Laws of thermodynamics, but also lets us apply results from optimal and stochastic control theory to physical systems. In particular, we show how to use linear control theory to optimally extract work from a single heat source over a finite time interval in the manner of Maxwell's demon. Furthermore, the optimal controller is a time-varying port-Hamiltonian system, which can be physically implemented as a variable linear capacitor and transformer. We also use the theory to design a heat engine operating between two heat sources in finite-time Carnot-like cycles of maximum power, and we compare those two heat engines.
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| 20. |
- di Bernardo, M., et al.
(author)
-
Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems
- 2008
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 237:1, s. 119-136
-
Journal article (peer-reviewed)abstract
- A rich variety of dynamical scenarios has been shown to occur when a fixed point of a non-smooth map undergoes a border-collision. This paper concerns a closely related class of discontinuity-induced bifurcations, those involving equilibria of n-dimensional piecewise-smooth flows. Specifically, transitions are studied which occur when a boundary equilibrium, that is one lying within a discontinuity manifold, is perturbed. It is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios. Conditions to classify among the possible simplest scenarios are given for piecewise-smooth continuous, Filippov and impacting systems. Also, we investigate the possible birth of other attractors (e.g. limit cycles) at a boundary-equilibrium bifurcation.
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| 21. |
- Eliasson, Veronica, et al.
(author)
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On cylindrically converging shock waves shaped by obstacles
- 2008
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 237:14-17, s. 2203-2209
-
Journal article (peer-reviewed)abstract
- Motivated by recent experiments, numerical simulations of cylindrically converging shock waves were performed. The converging shocks impinged upon a set of 0-16 regularly space obstacles. For more than two obstacles the resulting diffracted shock fronts formed polygonal shaped patterns near the point of focus. The maximum pressure and temperature as a function of the number of obstacles were studied. The self-similar behavior of cylindrical, triangular and square-shaped shocks was also investigated.
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| 22. |
- Figueras, Jordi-Lluis, et al.
(author)
-
A modified parameterization method for invariant Lagrangian tori for partially integrable Hamiltonian systems
- 2024
-
In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 462
-
Journal article (peer-reviewed)abstract
- In this paper we present an a-posteriori KAM theorem for the existence of an (n - d)-parameter family of d-dimensional isotropic invariant tori with Diophantine frequency vector omega is an element of R-d, of type (gamma,tau), for n degrees of freedom Hamiltonian systems with (n-d) independent first integrals in involution. If the first integrals induce a Hamiltonian action of the n-d)-dimensional torus, then we can produce n-dimensional Lagrangian tori with frequency vector of the form (omega, omega(p)), with omega(p) is an element of Rn-d. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of width rho, and the corresponding error in the functional equation is epsilon. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if gamma(-2) rho(-2) tau(-1) epsilon is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if gamma(-4) rho(-4 tau) epsilon is small enough. The approach is suitable to perform computer assisted proofs.
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| 23. |
- Gustafsson, Björn, et al.
(author)
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Selected topics on quadrature domains
- 2007
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 235:02-jan, s. 90-100
-
Journal article (peer-reviewed)abstract
- This is a selection of facts, old and recent, about quadrature domains. The text, written in the form of a survey, is addressed to non-experts and covers a variety of phenomena related to quadrature domains, such as: the difference between quadrature domains for subharmonic, harmonic and respectively complex analytic functions, geometric properties of the boundary, instability in the reverse Hele-Shaw flow, dependence and nonuniqueness on the quadrature data, interpretation in terms of function theory on Riemann surfaces, a matrix model and a reconstruction algorithm. Also there are some low degree/order examples where computations can be carried out in detail.
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| 24. |
- Hörnquist, Michael, et al.
(author)
-
Discrete breathers in aperiodic diatomic FPU lattices with long range order
- 2000
-
In: Physica D. - 0167-2789 .- 1872-8022. ; 136:1-2, s. 93-124
-
Journal article (peer-reviewed)abstract
- This study presents various aspects of discrete breathers in diatomic FPU lattices with masses varying according to the aperiodic Fibonacci and Thue-Morse sequences. We investigate the existence of time-periodic breathers, starting from the anti-continuous limit and consider excitations of isolated light atoms, hence obtaining the domain of unique existence for these breathers. We also perform a linear stability analysis by studying the Floquet operator. The found exact solutions are used, slightly perturbed, as initial conditions for long-time simulations of the breathers. These breathers turn out to be robust. Finally we consider how initial excitations of two consecutive light atoms evolve. Depending on the properties of the phase space for the two-atom system at the anti-continuous limit, we obtain different behaviour of these excitations. Especially we find that the aperiodic lattices can support localized excitations with a continuous frequency distribution for the timescales we consider, while a periodic lattice is unable to. These excitations are referred to as chaotic breathers. (C)2000 Elsevier Science B.V. All rights reserved.
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| 25. |
- Johansson, Magnus, 1965-
(author)
-
Discrete nonlinear Schrödinger approximation of a mixed Klein-Gordon/Fermi-Pasta-Ulam chain : Modulational instability and a statistical condition for creation of thermodynamic breathers
- 2006
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 216:1 SPEC. ISS., s. 62-70
-
Journal article (peer-reviewed)abstract
- We analyze certain aspects of the classical dynamics of a one-dimensional discrete nonlinear Schrödinger model with inter-site as well as on-site nonlinearities. The equation is derived from a mixed Klein-Gordon/Fermi-Pasta-Ulam chain of anharmonic oscillators coupled with anharmonic inter-site potentials, and approximates the slow dynamics of the fundamental harmonic of discrete small-amplitude modulational waves. We give explicit analytical conditions for modulational instability of travelling plane waves, and find in particular that sufficiently strong inter-site nonlinearities may change the nature of the instabilities from long-wavelength to short-wavelength perturbations. Further, we describe thermodynamic properties of the model using the grand-canonical ensemble to account for two conserved quantities: norm and Hamiltonian. The available phase space is divided into two separated parts with qualitatively different properties in thermal equilibrium: one part corresponding to a normal thermalized state with exponentially small probabilities for large-amplitude excitations, and another part typically associated with the formation of high-amplitude localized excitations, interacting with an infinite-temperature phonon bath. A modulationally unstable travelling wave may exhibit a transition from one region to the other as its amplitude is varied, and thus modulational instability is not a sufficient criterion for the creation of persistent localized modes in thermal equilibrium. For pure on-site nonlinearities the created localized excitations are typically pinned to particular lattice sites, while for significant inter-site nonlinearities they become mobile, in agreement with well-known properties of localized excitations in Fermi-Pasta-Ulam-type chains. © 2006 Elsevier Ltd. All rights reserved.
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| 26. |
- Lenells, Jonatan, 1981-
(author)
-
Initial-boundary value problems for integrable evolution equations with 3×3 Lax pairs
- 2012
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 241:8, s. 857-875
-
Journal article (peer-reviewed)abstract
- We present an approach for analyzing initial-boundary value problems for integrable equations whose Lax pairs involve 3×3 matrices. Whereas initial value problems for integrable equations can be analyzed by means of the classical Inverse Scattering Transform (IST), the presence of a boundary presents new challenges. Over the last fifteen years, an extension of the IST formalism developed by Fokas and his collaborators has been successful in analyzing boundary value problems for several of the most important integrable equations with 2×2 Lax pairs, such as the Kortewegde Vries, the nonlinear Schrdinger, and the sine-Gordon equations. In this paper, we extend these ideas to the case of equations with Lax pairs involving 3×3 matrices.
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| 27. |
- Lenells, Jonatan, 1981-
(author)
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The derivative nonlinear Schrödinger equation on the half-line
- 2008
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 237:23, s. 3008-3019
-
Journal article (peer-reviewed)abstract
- We analyze the derivative nonlinear Schrödinger equation i qt + qx x = i (| q |2 q)x on the half-line using the Fokas method. Assuming that the solution q (x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x, t dependence and is given in terms of the spectral functions a (ζ), b (ζ) (obtained from the initial data q0 (x) = q (x, 0)) as well as A (ζ), B (ζ) (obtained from the boundary values g0 (t) = q (0, t) and g1 (t) = qx (0, t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q0 (x), g0 (t), g1 (t)} such that there exist spectral functions satisfying the global relation, we show that the function q (x, t) defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.
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| 28. |
- Lenells, Jonatan, 1981-
(author)
-
The solution of the global relation for the derivative nonlinear Schrodinger equation on the half-line
- 2011
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 240:6, s. 512-525
-
Journal article (peer-reviewed)abstract
- We consider initial-boundary value problems for the derivative nonlinear Schrdinger (DNLS) equation on the half-line x>0. In a previous work, we showed that the solution q(x,t) can be expressed in terms of the solution of a RiemannHilbert problem with jump condition specified by the initial and boundary values of q(x,t). However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.
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| 29. |
- Lundmark, Hans, 1970-, et al.
(author)
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A view of the peakon world through the lens of approximation theory
- 2022
-
In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 440
-
Journal article (peer-reviewed)abstract
- Peakons (peaked solitons) are particular solutions admitted by certain nonlinear PDEs, most famously the Camassa-Holm shallow water wave equation. These solutions take the form of a train of peak-shaped waves, interacting in a particle-like fashion. In this article we give an overview of the mathematics of peakons, with particular emphasis on the connections to classical problems in analysis, such as Padé approximation, mixed Hermite-Padé approximation, multi-point Padé approximation, continued fractions of Stieltjes type and (bi)orthogonal polynomials. The exposition follows the chronological development of our understanding, exploring the peakon solutions of the Camassa-Holm, Degasperis-Procesi, Novikov, Geng-Xue and modified Camassa-Holm (FORQ) equations. All of these paradigm examples are integrable systems arising from the compatibility condition of a Lax pair, and a recurring theme in the context of peakons is the need to properly interpret these Lax pairs in the sense of Schwartz's theory of distributions. We trace out the path leading from distributional Lax pairs to explicit formulas for peakon solutions via a variety of approximation-theoretic problems, and we illustrate the peakon dynamics with graphics.
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| 30. |
- Lyons, Rainey, et al.
(author)
-
A Continuum Model for Morphology Formation from Interacting Ternary Mixtures : Simulation Study of the Formation and Growth of Patterns
- 2023
-
In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 453
-
Journal article (peer-reviewed)abstract
- Our interest lies in exploring the ability of a coupled nonlocal system of two quasilinear parabolic partial differential equations to produce phase separation patterns. The obtained patterns are referred here as morphologies. Our target system is derived in the literature as the rigorous hydrodynamic limit of a suitably scaled interacting particle system of Blume–Capel–type driven by Kawasaki dynamics. The system describes in a rather implicit way the interaction within a ternary mixture that is the macroscopic counterpart of a mix of two populations of interacting solutes in the presence of a background solvent. Our discussion is based on the qualitative behavior of numerical simulations of finite volume approximations of smooth solutions to our system and their quantitative postprocessing in terms of two indicators (correlation and structure factor calculations). Our results show many similar qualitative features (e.g. general shape and approximate coarsening rates) which have been observed in previous works on the stochastic Blume–Capel dynamics with three interacting species. The properties of the obtained morphologies (shape, connectivity, and so on) can play a key role in, e.g., the design of the active layer for efficient organic solar cells.
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| 31. |
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| 32. |
- Morimoto, Masaki, et al.
(author)
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Assessments of epistemic uncertainty using Gaussian stochastic weight averaging for fluid-flow regression
- 2022
-
In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 440
-
Journal article (peer-reviewed)abstract
- We use Gaussian stochastic weight averaging (SWAG) to assess the epistemic uncertainty associated with neural-network-based function approximation relevant to fluid flows. SWAG approximates a posterior Gaussian distribution of each weight, given training data, and a constant learning rate. Having access to this distribution, it is able to create multiple models with various combinations of sampled weights, which can be used to obtain ensemble predictions. The average of such an ensemble can be regarded as the 'mean estimation', whereas its standard deviation can be used to construct 'confidence intervals', which enable us to perform uncertainty quantification (UQ) with regard to the training process of neural networks. We utilize representative neural-network-based function approximation tasks for the following cases: (i) a two-dimensional circular-cylinder wake; (ii) the DayMET dataset (maximum daily temperature in North America); (iii) a three-dimensional square-cylinder wake; and (iv) urban flow, to assess the generalizability of the present idea for a wide range of complex datasets. SWAG-based UQ can be applied regardless of the network architecture, and therefore, we demonstrate the applicability of the method for two types of neural networks: (i) global field reconstruction from sparse sensors by combining convolutional neural network (CNN) and multi-layer perceptron (MLP); and (ii) far-field state estimation from sectional data with two-dimensional CNN. We find that SWAG can obtain physically-interpretable confidence-interval estimates from the perspective of epistemic uncertainty. This capability supports its use for a wide range of problems in science and engineering.
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| 33. |
- Ohlsson, Fredrik, et al.
(author)
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On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations
- 2024
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In: Physica D. - : Elsevier. - 0167-2789 .- 1872-8022. ; 461
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Journal article (peer-reviewed)abstract
- We consider the relationship between symmetries of two-dimensional autonomous dynamical systems in two common formulations; as a set of differential equations for the derivative of each state with respect to time, and a single differential equation in the phase plane representing the dynamics restricted to the state space of the system. Both representations can be analysed with respect to their symmetries, and we establish the correspondence between the set of infinitesimal generators of the respective formulations. We show that every generator of a symmetry of the autonomous system induces a well-defined vector field generating a symmetry in the phase plane and, conversely, that every symmetry generator in the phase plane can be lifted to a generator of a symmetry of the original system, which is unique up to constant translations in time. We exemplify the lift of symmetries in two cases; a mass conserved linear model and a non-linear oscillator.
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| 34. |
- Shi, Guodong, et al.
(author)
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A graph-theoretic approach on optimizing informed-node selection in multi-agent tracking control
- 2014
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In: Physica D-Nonlinear Phenomena. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 267, s. 104-111
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Journal article (peer-reviewed)abstract
- A graph optimization problem for a multi-agent leader follower problem is considered. In a multi-agent system with n followers and one leader, each agent's goal is to track the leader using the information obtained from its neighbors. The neighborhood relationship is defined by a directed communication graph where k agents, designated as informed agents, can become neighbors of the leader. This paper establishes that, for any given strongly connected communication graph with k informed agents, all agents will converge to the leader. In addition, an upper bound and a lower bound of the convergence rate are obtained. These bounds are shown to explicitly depend on the maximal distance from the leader to the followers. The dependence between this distance and the exact convergence rate is verified by empirical studies. Then we show that minimizing the maximal distance problem is a metric k-center problem in classical combinatorial optimization studies, which can be approximately solved. Numerical examples are given to illustrate the properties of the approximate solutions. (C) 2013 Elsevier B.V. All rights reserved.
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| 35. |
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| 36. |
- Öster, Michael, et al.
(author)
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Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling
- 2009
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In: Physica D. - : Elsevier BV. - 0167-2789 .- 1872-8022. ; 238:1, s. 88-99
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Journal article (peer-reviewed)abstract
- A two-dimensional nonlinear Schrodinger lattice with nonlinear coupling, modelling a square array of weakly coupled linear optical waveguides embedded in a nonlinear Kerr material, is studied. We find that despite a vanishing energy difference (Peierls-Nabarro barrier) of fundamental stationary modes the mobility of localized excitations is very poor. This is attributed to a large separation in parameter space of the bifurcation points of the involved stationary modes. At these points the stability of the fundamental modes is changed and an asymmetric intermediate solution appears that connects the points. The control of the power flow across the array when excited with plane waves is also addressed and shown to exhibit great flexibility that may lead to applications for power-coupling devices. In certain parameter regimes, the direction of a stable propagating plane-wave current is shown to be continuously tunable by amplitude variation (with fixed phase gradient). More exotic effects of the nonlinear coupling terms like compact discrete breathers and vortices, and stationary complex modes with nontrivial phase relations are also briefly discussed. Regimes of dynamical linear stability are found for all these types of solutions.
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| 37. |
- Carlen, E., et al.
(author)
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Kinetic hierarchy and propagation of chaos in biological swarm models
- 2013
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In: Physica D: Nonlinear Phenomena. - : Elsevier BV. - 0167-2789. ; 260, s. 90-111
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Journal article (peer-reviewed)abstract
- We consider two models of biological swarm behavior. In these models, pairs of particles interact to adjust their velocities one to each other. In the first process, called 'BDG', they join their average velocity up to some noise. In the second process, called 'CL', one of the two particles tries to join the other one's velocity. This paper establishes the master equations and BBGKY hierarchies of these two processes. It investigates the infinite particle limit of the hierarchies at large time scales. It shows that the resulting kinetic hierarchy for the CL process does not satisfy propagation of chaos. Numerical simulations indicate that the BDG process has similar behavior to the CL process. (c) 2012 Elsevier B.V. All rights reserved.
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| 38. |
- Einarsson, Jonas, et al.
(author)
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Orientational dynamics of weakly inertial axisymmetric particles in steady viscous flows
- 2014
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In: Physica D : Non-linear phenomena. - : Elsevier BV. - 0167-2789. ; 278, s. 79-85
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Journal article (peer-reviewed)abstract
- The orientational dynamics of weakly inertial axisymmetric particles in a steady flow is investigated. We derive an asymptotic equation of motion for the unit axial vector along the particle symmetry axis, valid for small Stokes number St, and for any axisymmetric particle in any steady linear viscous flow. This reduced dynamics is analysed in two ways, both pertain to the case of a simple shear flow. In this case inertia induces a coupling between precession and nutation. This coupling affects the dynamics of the particle, breaks the degeneracy of the Jeffery orbits, and creates two limiting periodic orbits. We calculate the leading-order Floquet exponents of the limiting periodic orbits and show analytically that prolate objects tend to a tumbling orbit, while oblate objects tend to a log-rolling orbit, in agreement with previous analytical and numerical results. Second, we analyse the role of the limiting orbits when rotational noise is present. We formulate the Fokker-Planck equation describing the orientational distribution of an axisymmetric particle, valid for small St and general Peclet number Pe. Numerical solutions of the Fokker-Planck equation, obtained by means of expansion in spherical harmonics, show that stationary orientational distributions are close to the inertia-free case when PeSt << 1, whereas they are determined by inertial effects, though small, when Pe >> 1/St >> 1. (C) 2014 The Authors. Published by Elsevier B.V.
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| 39. |
- Groves, M. D., et al.
(author)
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Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity
- 2008
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In: Physica D: Nonlinear Phenomena. - : Elsevier BV. - 0167-2789. ; 237:10-12, s. 1530-1538
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Journal article (peer-reviewed)abstract
- This paper presents an existence theory for small-amplitude Stokes and solitary-wave solutions to the classical water-wave problem in the absence of surface tension and with an arbitrary distribution of vorticity. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial coordinate is the time-like variable. A centre-manifold technique is used to reduce the system to a locally equivalent Hamiltonian system with one degree of freedom for values of a dimensionless parameter a near its critical value alpha*. The phase portrait of the reduced system contains a homoclinic orbit for alpha < alpha* and a family of periodic orbits for alpha > alpha*; the corresponding solutions of the water-wave problem are respectively a solitary wave of elevation and a family of Stokes waves. (c) 2008 Elsevier B.V. All rights reserved.
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| 40. |
- Homburg, A J, et al.
(author)
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Accumulations of T-points in a model for solitary pulses in an excitable reaction-diffusion medium
- 2005
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In: Physica D: Nonlinear Phenomena. - : Elsevier BV. - 0167-2789. ; 201:3-4, s. 212-229
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Journal article (peer-reviewed)abstract
- We consider a family of differential equations that describes traveling waves in a reaction-diffusion equation modeling oxidation of carbon oxide on a platinum surface, near the onset of spatio-temporal chaos. The organizing bifurcation for the bifurcation structure with small carbon oxide pressures, turns out to be a codimension 3 bifurcation involving a homoclinic orbit to an equilibrium undergoing a transcritical bifurcation. We show how infinitely many T-point bifurcations of multi loop heteroclinic cycles occur in the unfolding.
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| 41. |
- Jeffrey, Mike, et al.
(author)
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Preface to VSI: Advances in nonsmooth dynamics
- 2023
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In: Physica D: Nonlinear Phenomena. - 0167-2789. ; 453
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Journal article (other academic/artistic)abstract
- This Special Issue on nonsmooth dynamics brings together recent developments in nonsmooth dynamics, from applications in control engineering and mechanics, economics, climate modelling, physiological modelling, medicine, ecology and epidemiology, and others, to theory of novel forms of unpredictability and nonlinearity, chaos and bifurcations, and the study of higher dimensions.
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| 42. |
- Kanno, Takahiro, et al.
(author)
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Chaotic electrical activity of living beta-cells in the mouse pancreatic islet
- 2007
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In: Physica D: Nonlinear Phenomena. - : Elsevier BV. - 0167-2789. ; 226:2, s. 107-116
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Journal article (peer-reviewed)abstract
- To test for chaotic dynamics of the insulin producing beta-cell and explore its biological role, we observed the action potentials with the perforated patch clamp technique, for isolated cells as well as for intact cells of the mouse pancreatic islet. The time series obtained were analyzed using nonlinear diagnostic algorithms associated with the surrogate method. The isolated cells exhibited short-term predictability and visible determinism, in the steady state response to 10 mM glucose, while the intact cells did not. In the latter case, determinism became visible after the application of a gap junction inhibitor. This tendency was enhanced by the stimulation with tolbutamide. Our observations suggest that, thanks to the integration of individual chaotic dynamics via gap junction coupling, the beta-cells will lose memory of fluctuations occurring at any instant in their electrical activity more rapidly with time. This is likely to contribute to the functional stability of the islet against uncertain perturbations. (c) 2006 Elsevier B.V All rights reserved.
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| 43. |
- Kroon, Lars, et al.
(author)
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The appearance of gap solitons in a nonlinear Schrodinger lattice
- 2010
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In: PHYSICA D-NONLINEAR PHENOMENA. - : Elsevier BV. - 0167-2789. ; 239:5, s. 269-278
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Journal article (peer-reviewed)abstract
- We study the appearance of discrete gap solitons in a nonlinear Schrodinger model with a periodic on-site potential that possesses a gap evacuated of plane-wave solutions in the linear limit. For finite lattices supporting an anti-phase (q = pi/2) gap edge phonon as an anharmonic standing wave in the nonlinear regime, gap solitons are numerically found to emerge via pitchfork bifurcations from the gap edge. Analytically, modulational instabilities between pairs of bifurcation points on this "nonlinear gap boundary" are found in terms of critical gap widths, turning to zero in the infinite-size limit, which are associated with the birth of the localized soliton as well as discrete multisolitons in the gap. Such tunable instabilities can be of relevance in exciting soliton states in modulated arrays of nonlinear optical waveguides or Bose-Einstein condensates in periodic potentials. For lattices whose gal) edge phonon only asymptotically approaches the anti-phase solution, the nonlinear gap boundary splits in a bifurcation scenario leading to the birth of the discrete gap soliton as a continuable orbit to the gap edge in the linear limit. The instability-induced dynamics of the localized soliton in the gap regime is found to thermalize according to the Gibbsian equilibrium distribution, while the spontaneous formation of persisting intrinsically localized modes (discrete breathers) from the extended out-gap soliton reveals a phase transition of the solution.
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| 44. |
- Peikert, R., et al.
(author)
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Comment on "Second derivative ridges are straight lines and the implications for computing Lagrangian Coherent Structures, Physica D 2012.05.006"
- 2013
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In: Physica D: Nonlinear Phenomena. - : Elsevier. - 0167-2789. ; 242:1, s. 65-66
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Journal article (peer-reviewed)abstract
- The finite-time Lyapunov exponent (FTLE) has become a standard tool for analyzing unsteady flow phenomena, partly since its ridges can be interpreted as Lagrangian coherent structures (LCS). While there are several definitions for ridges, a particular one called second derivative ridges has been introduced in the context of LCS, but subsequently received criticism from several researchers for being over-constrained. Among the critics are Norgard and Bremer [Physica D 2012.05.006], who suggest furthermore that the widely used definition of height ridges was a part of the definition of second derivative ridges, and that topological separatrices were ill-suited for describing ridges. We show that (a) the definitions of height ridges and second derivative ridges are not directly related, and (b) there is an interdisciplinary consensus throughout the literature that topological separatrices describe ridges. Furthermore, we provide pointers to practically feasible and numerically stable ridge extraction schemes for FTLE fields.
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| 45. |
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| 46. |
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| 47. |
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| 48. |
- Öster, Michael, et al.
(author)
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Nonlocal and nonlinear dispersion in a nonlinear Schrödinger-type equation: exotic solitons and short-wavelength instabilities
- 2004
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In: Physica D: Nonlinear Phenomena. - : Elsevier BV. - 0167-2789. ; 198:1-2, s. 29-50
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Journal article (peer-reviewed)abstract
- We study the continuum limit of a nonlinear Schrödinger lattice model with both on-site and inter-site nonlinearities, describing weakly coupled optical waveguides or Bose–Einstein condensates. The resulting continuum nonlinear Schrödinger-type equation includes both nonlocal and nonlinear dispersion. Looking for stationary solutions, the equation is reduced to an ordinary differential equation with a rescaled spectral parameter and a single parameter interpolating between the nonlocality and the nonlinear dispersion. It is seen that these two effects give a similar behaviour for the solutions. We find smooth solitons and, beyond a critical value of the spectral parameter, also nonanalytic solitons in the form of peakons and capons. The existence of the exotic solitons is connected to the special properties of the phase space of the equation. Stability is investigated numerically by calculating eigenvalues and eigenfunctions of the linearized problem, and we particularly find that with both nonlocal and nonlinear dispersion simultaneously present, all solutions are unstable with respect to a break-up into short-wavelength oscillations.
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