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Mielke A, Sørensen MP, Wyller J. Memory effects in disease modelling through kernel estimates with oscillatory time history. J Math Biol 2024; 88:57. [PMID: 38578546 PMCID: PMC10997723 DOI: 10.1007/s00285-024-02080-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/16/2023] [Revised: 12/05/2023] [Accepted: 03/10/2024] [Indexed: 04/06/2024]
Abstract
We design a linear chain trick algorithm for dynamical systems for which we have oscillatory time histories in the distributed time delay. We make use of this algorithmic framework to analyse memory effects in disease evolution in a population. The modelling is based on a susceptible-infected-recovered SIR-model and on a susceptible-exposed-infected-recovered SEIR-model through a kernel that dampens the activity based on the recent history of infectious individuals. This corresponds to adaptive behavior in the population or through governmental non-pharmaceutical interventions. We use the linear chain trick to show that such a model may be written in a Markovian way, and we analyze the stability of the system. We find that the adaptive behavior gives rise to either a stable equilibrium point or a stable limit cycle for a close to constant number of susceptibles, i.e. locally in time. We also show that the attack rate for this model is lower than it would be without the dampening, although the adaptive behavior disappears as time goes to infinity and the number of infected goes to zero.
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Affiliation(s)
- Adam Mielke
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, Bldg. 324, 2800, Kongens Lyngby, Denmark
| | - Mads Peter Sørensen
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, Bldg. 324, 2800, Kongens Lyngby, Denmark.
| | - John Wyller
- Department of Mathematics, Norwegian University of Life Sciences, P.O. Box 5003, 1432, Ås, Norway
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Bergland H, Burlakov E, Wyller J. The Dynamics of Pasture-Herbivores-Carnivores with Sigmoidal Density Dependent Harvesting. Bull Math Biol 2023; 85:104. [PMID: 37726494 DOI: 10.1007/s11538-023-01210-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2023] [Accepted: 09/01/2023] [Indexed: 09/21/2023]
Abstract
We investigate biomass-herbivore-carnivore (top predator) interactions in terms of a tritrophic dynamical systems model. The harvesting rates of the herbivores and the top predators are described by means of a sigmoidal function of the herbivores density and the top predator density, respectively. The main focus in this study is on the dynamics as a function of the natural mortality and the maximal harvesting rate of the top predators. We identify parameter regimes for which we have non-existence of equilibrium points as well as necessary conditions for the existence of such states of the modelling framework. The system does not possess any finite equilibrium states in the regime of high herbivore mortality. In the regime of a high consumption rate of the herbivores and low mortality rates of the top predator, an asymptotically stable finite equilibrium state exists. For this positive equilibrium to exist the mortality of the top predator should not exceed a certain threshold level. We also detect regimes producing coexistence of equilibrium states and their respective stability properties. In the regime of negligible harvesting of the top predator level, we observe a finite window of the natural top predator mortality rates for which oscillations in the top predator-, the herbivore- and the biomass level take place. The lower and upper bound of this window correspond to two Hopf bifurcation points. We also identify a bifurcation diagram using the top predator harvesting rate as a control variable. Using this diagram we detect several saddle node- and Hopf bifurcation points as well as regimes for which we have coexistence of interior equilibrium states, bistability and relaxation type of oscillations.
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Affiliation(s)
- Harald Bergland
- School of Business and Economics, Campus Harstad, University of Tromsø - The Arctic University of Norway, Havnegata 5, 9404, Harstad, Norway
| | - Evgenii Burlakov
- X-Bio Institute, University of Tyumen, Volodarskogo 6, Tyumen, Russia, 625003.
- Research Institute of Mathematics, Physics and Computer Sciences, Derzhavin Tambov State University, Internatsionalnaya 33, Tambov, Russia, 392000.
| | - John Wyller
- Department of Mathematics, Norwegian University of Life Sciences, Drøbakveien 31, 1432, Ås, Norway
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Kolodina K, Wyller J, Oleynik A, Sørensen MP. Pattern formation in a 2-population homogenized neuronal network model. J Math Neurosci 2021; 11:9. [PMID: 34173912 PMCID: PMC8236021 DOI: 10.1186/s13408-021-00107-1] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/22/2020] [Accepted: 06/09/2021] [Indexed: 06/13/2023]
Abstract
We study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of [Formula: see text]-stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.
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Affiliation(s)
- Karina Kolodina
- Faculty of Science and Technology, Norwegian University of Life Sciences, P.O. Box 5003, N-1432, Ås, Norway
| | - John Wyller
- Faculty of Science and Technology, Norwegian University of Life Sciences, P.O. Box 5003, N-1432, Ås, Norway.
| | - Anna Oleynik
- Department of Mathematics, University of Bergen, P.O. Box 7803, N-5020, Bergen, Norway
| | - Mads Peter Sørensen
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800, Kongens Lyngby, Denmark
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Bergland H, Burlakov E, Pedersen PA, Wyller J. Aquaculture, pollution and fishery - dynamics of marine industrial interactions. Ecological Complexity 2020. [DOI: 10.1016/j.ecocom.2020.100853] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Nielsen BF, Wyller J. Ill-Posed Point Neuron Models. J Math Neurosci 2016; 6:7. [PMID: 27129667 PMCID: PMC5396507 DOI: 10.1186/s13408-016-0039-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/17/2015] [Accepted: 04/20/2016] [Indexed: 06/05/2023]
Abstract
We show that point-neuron models with a Heaviside firing rate function can be ill posed. More specifically, the initial-condition-to-solution map might become discontinuous in finite time. Consequently, if finite precision arithmetic is used, then it is virtually impossible to guarantee the accurate numerical solution of such models. If a smooth firing rate function is employed, then standard ODE theory implies that point-neuron models are well posed. Nevertheless, in the steep firing rate regime, the problem may become close to ill posed, and the error amplification, in finite time, can be very large. This observation is illuminated by numerical experiments. We conclude that, if a steep firing rate function is employed, then minor round-off errors can have a devastating effect on simulations, unless proper error-control schemes are used.
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Affiliation(s)
- Bjørn Fredrik Nielsen
- Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P.O. Box 5003, Ås, 1432, Norway.
| | - John Wyller
- Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P.O. Box 5003, Ås, 1432, Norway
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Abstract
Spatial and/or temporal clustering of pathogens will invalidate the commonly used assumption of Poisson-distributed pathogen counts (doses) in quantitative microbial risk assessment. In this work, the theoretically predicted effect of spatial clustering in conventional "single-hit" dose-response models is investigated by employing the stuttering Poisson distribution, a very general family of count distributions that naturally models pathogen clustering and contains the Poisson and negative binomial distributions as special cases. The analysis is facilitated by formulating the dose-response models in terms of probability generating functions. It is shown formally that the theoretical single-hit risk obtained with a stuttering Poisson distribution is lower than that obtained with a Poisson distribution, assuming identical mean doses. A similar result holds for mixed Poisson distributions. Numerical examples indicate that the theoretical single-hit risk is fairly insensitive to moderate clustering, though the effect tends to be more pronounced for low mean doses. Furthermore, using Jensen's inequality, an upper bound on risk is derived that tends to better approximate the exact theoretical single-hit risk for highly overdispersed dose distributions. The bound holds with any dose distribution (characterized by its mean and zero inflation index) and any conditional dose-response model that is concave in the dose variable. Its application is exemplified with published data from Norovirus feeding trials, for which some of the administered doses were prepared from an inoculum of aggregated viruses. The potential implications of clustering for dose-response assessment as well as practical risk characterization are discussed.
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Affiliation(s)
- Vegard Nilsen
- Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, N-1432 Aas, Norway
| | - John Wyller
- Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, N-1432 Aas, Norway
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Abstract
Dose-response models are essential to quantitative microbial risk assessment (QMRA), providing a link between levels of human exposure to pathogens and the probability of negative health outcomes. In drinking water studies, the class of semi-mechanistic models known as single-hit models, such as the exponential and the exact beta-Poisson, has seen widespread use. In this work, an attempt is made to carefully develop the general mathematical single-hit framework while explicitly accounting for variation in (1) host susceptibility and (2) pathogen infectivity. This allows a precise interpretation of the so-called single-hit probability and precise identification of a set of statistical independence assumptions that are sufficient to arrive at single-hit models. Further analysis of the model framework is facilitated by formulating the single-hit models compactly using probability generating and moment generating functions. Among the more practically relevant conclusions drawn are: (1) for any dose distribution, variation in host susceptibility always reduces the single-hit risk compared to a constant host susceptibility (assuming equal mean susceptibilities), (2) the model-consistent representation of complete host immunity is formally demonstrated to be a simple scaling of the response, (3) the model-consistent expression for the total risk from repeated exposures deviates (gives lower risk) from the conventional expression used in applications, and (4) a model-consistent expression for the mean per-exposure dose that produces the correct total risk from repeated exposures is developed.
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Affiliation(s)
- Vegard Nilsen
- Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, N-1432 Aas, Norway
| | - John Wyller
- Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, N-1432 Aas, Norway
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Yousaf M, Kriener B, Wyller J, Einevoll GT. Generation and annihilation of localized persistent-activity states in a two-population neural-field model. Neural Netw 2013; 46:75-90. [PMID: 23708672 DOI: 10.1016/j.neunet.2013.04.012] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2013] [Revised: 04/25/2013] [Accepted: 04/26/2013] [Indexed: 10/26/2022]
Abstract
We investigate the generation and annihilation of persistent localized activity states, so-called bumps, in response to transient spatiotemporal external input in a two-population neural-field model of the Wilson-Cowan type. Such persistent cortical states have been implicated as a biological substrate for short-term working memory, that is, the ability to store stimulus-related information for a few seconds and discard it once it is no longer relevant. In previous studies of the same model it has been established that the stability of bump states hinges on the relative inhibitory constant τ, i.e., the ratio of the time constants governing the dynamics of the inhibitory and excitatory populations: persistent bump states are typically only stable for values of τ smaller than a critical value τcr. We find here that τ is also a key parameter determining whether a transient input can generate a persistent bump state (in the regime where τ<τcr) or not. For small values of τ generation of the persistent states is found to depend only on the overall strength of the transient input, i.e., as long as the magnitude and duration of the excitatory transient input are larger and/or long enough, the persistent state will be activated. For higher values of τ we find that only specific combinations of amplitude and duration leads to persistent activation. For the corresponding annihilation process, no such delicate selectivity on the transient input is observed.
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Affiliation(s)
- M Yousaf
- Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P.O. Box 5003, N-1432 Ås, Norway.
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Norheim ES, Wyller J, Nordlie E, Einevoll GT. A minimal mechanistic model for temporal signal processing in the lateral geniculate nucleus. Cogn Neurodyn 2012; 6:259-81. [PMID: 23730357 PMCID: PMC3368059 DOI: 10.1007/s11571-012-9198-9] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/17/2011] [Revised: 03/03/2012] [Accepted: 03/07/2012] [Indexed: 02/03/2023] Open
Abstract
The receptive fields of cells in the lateral geniculate nucleus (LGN) are shaped by their diverse set of impinging inputs: feedforward synaptic inputs stemming from retina, and feedback inputs stemming from the visual cortex and the thalamic reticular nucleus. To probe the possible roles of these feedforward and feedback inputs in shaping the temporal receptive-field structure of LGN relay cells, we here present and investigate a minimal mechanistic firing-rate model tailored to elucidate their disparate features. The model for LGN relay ON cells includes feedforward excitation and inhibition (via interneurons) from retinal ON cells and excitatory and inhibitory (via thalamic reticular nucleus cells and interneurons) feedback from cortical ON and OFF cells. From a general firing-rate model formulated in terms of Volterra integral equations, we derive a single delay differential equation with absolute delay governing the dynamics of the system. A freely available and easy-to-use GUI-based MATLAB version of this minimal mechanistic LGN circuit model is provided. We particularly investigate the LGN relay-cell impulse response and find through thorough explorations of the model's parameter space that both purely feedforward models and feedback models with feedforward excitation only, can account quantitatively for previously reported experimental results. We find, however, that the purely feedforward model predicts two impulse response measures, the time to first peak and the biphasic index (measuring the relative weight of the rebound phase) to be anticorrelated. In contrast, the models with feedback predict different correlations between these two measures. This suggests an experimental test assessing the relative importance of feedforward and feedback connections in shaping the impulse response of LGN relay cells.
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Affiliation(s)
- Eivind S. Norheim
- CIGENE, Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, PO Box 5003, 1432 Aas, Norway
| | - John Wyller
- CIGENE, Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, PO Box 5003, 1432 Aas, Norway
| | - Eilen Nordlie
- CIGENE, Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, PO Box 5003, 1432 Aas, Norway
| | - Gaute T. Einevoll
- CIGENE, Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, PO Box 5003, 1432 Aas, Norway
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Yousaf M, Einevoll GT, Tetzlaff T, Wyller J. Persistent localized activity in a two-population neural-field model with spatio-temporal external input. BMC Neurosci 2011. [PMCID: PMC3240497 DOI: 10.1186/1471-2202-12-s1-p377] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022] Open
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Norheim E, Wyller J, Nordlie E, Einevoll G. Feedback and feedforward contributions to temporal signal processing in the lateral geniculate nucleus. Front Neuroinform 2009. [DOI: 10.3389/conf.neuro.11.2009.08.089] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/13/2022] Open
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Rotabakk BT, Wyller J, Lekang OI, Sivertsvik M. A mathematical method for determining equilibrium gas composition in modified atmosphere packaging and soluble gas stabilization systems for non-respiring foods. J FOOD ENG 2008. [DOI: 10.1016/j.jfoodeng.2007.08.010] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Nordbø O, Wyller J, Einevoll GT. Neural network firing-rate models on integral form: effects of temporal coupling kernels on equilibrium-state stability. Biol Cybern 2007; 97:195-209. [PMID: 17602240 DOI: 10.1007/s00422-007-0167-z] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2007] [Accepted: 05/25/2007] [Indexed: 05/16/2023]
Abstract
Firing-rate models describing neural-network activity can be formulated in terms of differential equations for the synaptic drive from neurons. Such models are typically derived from more general models based on Volterra integral equations assuming exponentially decaying temporal coupling kernels describing the coupling of pre- and postsynaptic activities. Here we study models with other choices of temporal coupling kernels. In particular, we investigate the stability properties of constant solutions of two-population Volterra models by studying the equilibrium solutions of the corresponding autonomous dynamical systems, derived using the linear chain trick, by means of the Routh-Hurwitz criterion. In the four investigated synaptic-drive models with identical equilibrium points we find that the choice of temporal coupling kernels significantly affects the equilibrium-point stability properties. A model with an alpha-function replacing the standard exponentially decaying function in the inhibitory coupling kernel is in most of our examples found to be most prone to instability, while the opposite situation with an alpha-function describing the excitatory kernel is found to be least prone to instability. The standard model with exponentially decaying coupling kernels is typically found to be an intermediate case. We further find that stability is promoted by increasing the weight of self-inhibition or shortening the time constant of the inhibition.
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Affiliation(s)
- Oyvind Nordbø
- Department of Mathematical Sciences and Technology and Center for Integrative Genetics, Norwegian University of Life Sciences, P. O. Box 5003, 1432, As, Norway.
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Abstract
We investigate the propagation of partially coherent beams in spatially nonlocal nonlinear media with a logarithmic type of nonlinearity. We derive analytical formulas for the evolution of the beam parameters and conditions for the formation of nonlocal incoherent solitons.
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Affiliation(s)
- Wiesław Królikowski
- Laser Physics Centre, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, Australian Capital Territory 0200, Australia
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Królikowski W, Bang O, Nikolov NI, Neshev D, Wyller J, Rasmussen JJ, Edmundson D. Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media. ACTA ACUST UNITED AC 2004. [DOI: 10.1088/1464-4266/6/5/017] [Citation(s) in RCA: 323] [Impact Index Per Article: 16.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Wyller J, Krolikowski W, Bang O, Rasmussen JJ. Generic features of modulational instability in nonlocal Kerr media. Phys Rev E Stat Nonlin Soft Matter Phys 2002; 66:066615. [PMID: 12513437 DOI: 10.1103/physreve.66.066615] [Citation(s) in RCA: 176] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2002] [Indexed: 11/07/2022]
Abstract
The modulational instability (MI) of plane waves in nonlocal Kerr media is studied for a general response function. Several generic properties are proven mathematically, with emphasis on how new gain bands are formed through a bifurcation process when the degree of nonlocality, sigma, passes certain bifurcation values and how the bandwidth and maximum of each individual gain band depends on sigma. The generic properties of the MI gain spectrum, including the bifurcation phenomena, are then demonstrated for the exponential and rectangular response functions. For a focusing nonlinearity the nonlocality tends to suppress MI, but can never remove it completely, irrespectively of the shape of the response function. For a defocusing nonlinearity the stability properties depend sensitively on the profile of the response function. For response functions with a positive-definite spectrum, such as Gaussians and exponentials, plane waves are always stable, whereas response functions with spectra that are not positive definite (such as the rectangular) will lead to MI if sigma exceeds a certain threshold. For the square response function, in both the focusing and defocusing case, we show analytically and numerically how new gain bands that form at higher wave numbers when sigma increases will eventually dominate the existing gain bands at lower wave numbers and abruptly change the length scale of the periodic pattern that may be observed in experiments.
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Affiliation(s)
- John Wyller
- Department of Mathematical Sciences, Agricultural University of Norway, P. O. Box 5065, N-1432 As, Norway
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Bang O, Krolikowski W, Wyller J, Rasmussen JJ. Collapse arrest and soliton stabilization in nonlocal nonlinear media. Phys Rev E Stat Nonlin Soft Matter Phys 2002; 66:046619. [PMID: 12443361 DOI: 10.1103/physreve.66.046619] [Citation(s) in RCA: 45] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2002] [Revised: 07/12/2002] [Indexed: 05/24/2023]
Abstract
We investigate the properties of localized waves in cubic nonlinear materials with a symmetric nonlocal nonlinear response of arbitrary shape and degree of nonlocality, described by a general nonlocal nonlinear Schrödinger type equation. We prove rigorously by bounding the Hamiltonian that nonlocality of the nonlinearity prevents collapse in, e.g., Bose-Einstein condensates and optical Kerr media in all physical dimensions. The nonlocal nonlinear response must be symmetric and have a positive definite Fourier spectrum, but can otherwise be of completely arbitrary shape and degree of nonlocality. We use variational techniques to find the soliton solutions and illustrate the stabilizing effect of nonlocality.
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Affiliation(s)
- Ole Bang
- Department of Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
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Krolikowski W, Bang O, Rasmussen JJ, Wyller J. Modulational instability in nonlocal nonlinear Kerr media. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 64:016612. [PMID: 11461433 DOI: 10.1103/physreve.64.016612] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2001] [Indexed: 05/23/2023]
Abstract
We study modulational instability (MI) of plane waves in nonlocal nonlinear Kerr media. For a focusing nonlinearity we show that, although the nonlocality tends to suppress MI, it can never remove it completely, irrespective of the particular profile of the nonlocal response function. For a defocusing nonlinearity the stability properties depend sensitively on the response function profile: for a smooth profile (e.g., a Gaussian) plane waves are always stable, but MI may occur for a rectangular response. We also find that the reduced model for a weak nonlocality predicts MI in defocusing media for arbitrary response profiles, as long as the intensity exceeds a certain critical value. However, it appears that this regime of MI is beyond the validity of the reduced model, if it is to represent the weakly nonlocal limit of a general nonlocal nonlinearity, as in optics and the theory of Bose-Einstein condensates.
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Affiliation(s)
- W Krolikowski
- Australian Photonics Cooperative Research Centre, Laser Physics Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia
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Hesthaven JS, Rasmussen JJ, Bergé L, Wyller J. Numerical studies of localized wavefields governed by the Raman-extended derivative nonlinear Schrödinger equation. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/30/23/019] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Bergé L, Mezentsev VK, Rasmussen JJ, Wyller J. Formation of stable solitons in quadratic nonlinear media. Phys Rev A 1995; 52:R28-R31. [PMID: 9912329 DOI: 10.1103/physreva.52.r28] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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