We study the structure of stationary patterns in bistable lattice dynamical systems posed on rings with a symmetric coupling structure in the regime of small coupling strength. We show that sparse coupling (for instance, nearest-neighbour or next-nearest-neighbour coupling) and all-to-all coupling lead to significantly different solution branches. In particular, sparse coupling leads to snaking branches with many saddle-node bifurcations, while all-to-all coupling leads to branches with six saddle nodes, regardless of the size of the number of nodes in the graph.
{"title":"Snaking bifurcations of localized patterns on ring lattices","authors":"Moyi Tian;Jason J Bramburger;Björn Sandstede","doi":"10.1093/imamat/hxab023","DOIUrl":"10.1093/imamat/hxab023","url":null,"abstract":"We study the structure of stationary patterns in bistable lattice dynamical systems posed on rings with a symmetric coupling structure in the regime of small coupling strength. We show that sparse coupling (for instance, nearest-neighbour or next-nearest-neighbour coupling) and all-to-all coupling lead to significantly different solution branches. In particular, sparse coupling leads to snaking branches with many saddle-node bifurcations, while all-to-all coupling leads to branches with six saddle nodes, regardless of the size of the number of nodes in the graph.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49471335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Max Philipp Holl;Andrew J Archer;Svetlana V Gurevich;Edgar Knobloch;Lukas Ophaus;Uwe Thiele
The passive conserved Swift–Hohenberg equation (or phase-field-crystal [PFC] model) describes gradient dynamics of a single-order parameter field related to density. It provides a simple microscopic description of the thermodynamic transition between liquid and crystalline states. In addition to spatially extended periodic structures, the model describes a large variety of steady spatially localized structures. In appropriate bifurcation diagrams the corresponding solution branches exhibit characteristic slanted homoclinic snaking. In an active PFC model, encoding for instance the active motion of self-propelled colloidal particles, the gradient dynamics structure is broken by a coupling between density and an additional polarization field. Then, resting and traveling localized states are found with transitions characterized by parity-breaking drift bifurcations. Here, we briefly review the snaking behavior of localized states in passive and active PFC models before discussing the bifurcation behavior of localized states in systems of (i) two coupled passive PFC models with common gradient dynamics, (ii) two coupled passive PFC models where the coupling breaks the gradient dynamics structure and (iii) a passive PFC model coupled to an active PFC model.
{"title":"Localized states in passive and active phase-field-crystal models","authors":"Max Philipp Holl;Andrew J Archer;Svetlana V Gurevich;Edgar Knobloch;Lukas Ophaus;Uwe Thiele","doi":"10.1093/imamat/hxab025","DOIUrl":"10.1093/imamat/hxab025","url":null,"abstract":"The passive conserved Swift–Hohenberg equation (or phase-field-crystal [PFC] model) describes gradient dynamics of a single-order parameter field related to density. It provides a simple microscopic description of the thermodynamic transition between liquid and crystalline states. In addition to spatially extended periodic structures, the model describes a large variety of steady spatially localized structures. In appropriate bifurcation diagrams the corresponding solution branches exhibit characteristic slanted homoclinic snaking. In an active PFC model, encoding for instance the active motion of self-propelled colloidal particles, the gradient dynamics structure is broken by a coupling between density and an additional polarization field. Then, resting and traveling localized states are found with transitions characterized by parity-breaking drift bifurcations. Here, we briefly review the snaking behavior of localized states in passive and active PFC models before discussing the bifurcation behavior of localized states in systems of (i) two coupled passive PFC models with common gradient dynamics, (ii) two coupled passive PFC models where the coupling breaks the gradient dynamics structure and (iii) a passive PFC model coupled to an active PFC model.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42273427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organized by a codimension-three point at which the marginal stability curve has a quartic minimum. We develop a model equation to explore this situation, based on the Swift–Hohenberg equation; the model contains, amongst other things, snaking branches of patterns of one wavelength localized in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalized Ginzburg–Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift–Hohenberg equation with real coefficients. Localized solutions in this amplitude equation help interpret the localized patterns in the model. This work extends recent efforts to investigate snaking behaviour in pattern-forming systems where two different stable non-trivial patterns exist at the same parameter values.
{"title":"Localized patterns in a generalized Swift–Hohenberg equation with a quartic marginal stability curve","authors":"David C Bentley;Alastair M Rucklidge","doi":"10.1093/imamat/hxab035","DOIUrl":"https://doi.org/10.1093/imamat/hxab035","url":null,"abstract":"In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organized by a codimension-three point at which the marginal stability curve has a quartic minimum. We develop a model equation to explore this situation, based on the Swift–Hohenberg equation; the model contains, amongst other things, snaking branches of patterns of one wavelength localized in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalized Ginzburg–Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift–Hohenberg equation with real coefficients. Localized solutions in this amplitude equation help interpret the localized patterns in the model. This work extends recent efforts to investigate snaking behaviour in pattern-forming systems where two different stable non-trivial patterns exist at the same parameter values.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50424517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Priya Subramanian;Andrew J Archer;Edgar Knobloch;Alastair M Rucklidge
Non-topological defects in spatial patterns such as grain boundaries in crystalline materials arise from local variations of the pattern properties such as amplitude, wavelength and orientation. Such non-topological defects may be treated as spatially localized structures, i.e. as fronts connecting distinct periodic states. Using the two-dimensional quadratic-cubic Swift–Hohenberg equation, we obtain fully nonlinear equilibria containing grain boundaries that separate a patch of hexagons with one orientation (the grain) from an identical hexagonal state with a different orientation (the background). These grain boundaries take the form of closed curves with multiple penta-hepta defects that arise from local orientation mismatches between the two competing hexagonal structures. Multiple isolas occurring robustly over a wide range of parameters are obtained even in the absence of a unique Maxwell point, underlining the importance of retaining pinning when analysing patterns with defects, an effect omitted from the commonly used amplitude-phase description. Similar results are obtained for quasiperiodic structures in a two-scale phase-field model.
{"title":"Snaking without subcriticality: grain boundaries as non-topological defects","authors":"Priya Subramanian;Andrew J Archer;Edgar Knobloch;Alastair M Rucklidge","doi":"10.1093/imamat/hxab032","DOIUrl":"10.1093/imamat/hxab032","url":null,"abstract":"Non-topological defects in spatial patterns such as grain boundaries in crystalline materials arise from local variations of the pattern properties such as amplitude, wavelength and orientation. Such non-topological defects may be treated as spatially localized structures, i.e. as fronts connecting distinct periodic states. Using the two-dimensional quadratic-cubic Swift–Hohenberg equation, we obtain fully nonlinear equilibria containing grain boundaries that separate a patch of hexagons with one orientation (the grain) from an identical hexagonal state with a different orientation (the background). These grain boundaries take the form of closed curves with multiple penta-hepta defects that arise from local orientation mismatches between the two competing hexagonal structures. Multiple isolas occurring robustly over a wide range of parameters are obtained even in the absence of a unique Maxwell point, underlining the importance of retaining pinning when analysing patterns with defects, an effect omitted from the commonly used amplitude-phase description. Similar results are obtained for quasiperiodic structures in a two-scale phase-field model.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41636949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper uncovers new manifestations of the homoclinic snaking mechanism in the post-buckling regime of a pressurized thin cylindrical shell under axial load. These new forms tend to propagate either wholly or partially in a direction that is orthogonal to the direction of the applied load and so, unlike earlier forms in Woods & Champneys (1999, Heteroclinic tangles in the unfolding of a degenerate Hamiltonian Hopf bifurcation. Phys. D, 129, 147–170), are fundamentally 2D in nature. The main effect of internal pressurization on the snaking mechanism is firstly to transition the circumferential multiplication of buckles from a one-tier pattern to a three-tier pattern. Secondly, internal pressurization can induce oblique snaking, whereby the sequential multiplication of buckles occurs in a helical pattern across the cylinder domain. For low levels of internal pressure, the single dimple remains—as in the unpressurized case—the unstable edge state that forms the smallest energy barrier around the stable pre-buckling equilibrium. For greater levels of pressure, the edge state changes to a single dimple surrounded by four smaller dimples. By tracing the limit point that denotes the onset of these edge states in the parameter space of internal pressure and axial load, we justify and validate the empirically derived design guideline for buckling of pressurized cylinders proposed by Fung & Sechler (1957, Buckling of thin-walled circular cylinders under axial compression and internal pressure. J. Aeronaut. Sci., 24, 351–356).
{"title":"Localization and snaking in axially compressed and internally pressurized thin cylindrical shells","authors":"Rainer M J Groh;Giles W Hunt","doi":"10.1093/imamat/hxab024","DOIUrl":"10.1093/imamat/hxab024","url":null,"abstract":"This paper uncovers new manifestations of the homoclinic snaking mechanism in the post-buckling regime of a pressurized thin cylindrical shell under axial load. These new forms tend to propagate either wholly or partially in a direction that is orthogonal to the direction of the applied load and so, unlike earlier forms in Woods & Champneys (1999, Heteroclinic tangles in the unfolding of a degenerate Hamiltonian Hopf bifurcation. Phys. D, 129, 147–170), are fundamentally 2D in nature. The main effect of internal pressurization on the snaking mechanism is firstly to transition the circumferential multiplication of buckles from a one-tier pattern to a three-tier pattern. Secondly, internal pressurization can induce oblique snaking, whereby the sequential multiplication of buckles occurs in a helical pattern across the cylinder domain. For low levels of internal pressure, the single dimple remains—as in the unpressurized case—the unstable edge state that forms the smallest energy barrier around the stable pre-buckling equilibrium. For greater levels of pressure, the edge state changes to a single dimple surrounded by four smaller dimples. By tracing the limit point that denotes the onset of these edge states in the parameter space of internal pressure and axial load, we justify and validate the empirically derived design guideline for buckling of pressurized cylinders proposed by Fung & Sechler (1957, Buckling of thin-walled circular cylinders under axial compression and internal pressure. J. Aeronaut. Sci., 24, 351–356).","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47357326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Haifaa Alrihieli;Alastair M Rucklidge;Priya Subramanian
Double-zero eigenvalues at a Takens–Bogdanov (TB) bifurcation occur in many physical systems such as double-diffusive convection, binary convection and magnetoconvection. Analysis of the associated normal form, in 1D with periodic boundary condition, shows the existence of steady patterns, standing waves, modulated waves (MW) and travelling waves, and describes the transitions and bifurcations between these states. Values of coefficients of the terms in the normal form classify all possible different bifurcation scenarios in the neighbourhood of the TB bifurcation (Dangelmayr, G. & Knobloch, E. (1987) The Takens–Bogdanov bifurcation with O(2)-symmetry. Phil. Trans. R. Soc. Lond. A, 322, 243-279). In this work we develop a new and simple pattern-forming partial differential equation (PDE) model, based on the Swift–Hohenberg equation, adapted to have the TB normal form at onset. This model allows us to explore the dynamics in a wide range of bifurcation scenarios, including in domains much wider than the lengthscale of the pattern. We identify two bifurcation scenarios in which coexistence between different types of solutions is indicated from the analysis of the normal form equation. In these scenarios, we look for spatially localized solutions by examining pattern formation in wide domains. We are able to recover two types of localized states, that of a localized steady state (LSS) in the background of the trivial state (TS) and that of a spatially localized travelling wave (LTW) in the background of the TS, which have previously been observed in other systems. Additionally, we identify two new types of spatially localized states: that of a LSS in a MW background and that of a LTW in a steady state (SS) background. The PDE model is easy to solve numerically in large domains and so will allow further investigation of pattern formation with a TB bifurcation in one or more dimensions and the exploration of a range of background and foreground pattern combinations beyond SSs.
Takens-Bogdanov (TB)分岔的双零特征值存在于许多物理系统中，如双扩散对流、二元对流和磁对流。在一维周期边界条件下，对相关的范式进行分析，证明了稳态模式、驻波、调制波和行波的存在，并描述了这些状态之间的转换和分岔。正规形式项的系数值对TB分岔的邻域中所有可能的不同分岔情况进行分类(Dangelmayr, G. & Knobloch, E. (1987) O(2)-对称的Takens-Bogdanov分岔。菲尔。反式。r . Soc。Lond。A, 322, 243-279)。在这项工作中，我们开发了一个新的和简单的模式形成的偏微分方程(PDE)模型，基于Swift-Hohenberg方程，适应于在开始时具有TB标准形式。这个模型允许我们在大范围的分岔场景中探索动态，包括在比模式的长度尺度更宽的领域。通过对范式方程的分析，我们确定了两种不同类型解共存的分岔情形。在这些情况下，我们通过检查广泛领域的模式形成来寻找空间局部化的解决方案。我们能够恢复两种类型的局域状态，一种是在平凡状态(TS)背景下的局域稳态(LSS)，另一种是在TS背景下的空间局域行波(LTW)，这在其他系统中已经被观察到。此外，我们还确定了两种新的空间局域化状态:MW背景下的LSS和稳态背景下的LTW。PDE模型很容易在大范围内进行数值求解，因此将允许在一个或多个维度上进一步研究具有TB分支的模式形成，并探索超出SSs的一系列背景和前景模式组合。
{"title":"Spatial localization beyond steady states in the neighbourhood of the Takens–Bogdanov bifurcation","authors":"Haifaa Alrihieli;Alastair M Rucklidge;Priya Subramanian","doi":"10.1093/imamat/hxab030","DOIUrl":"10.1093/imamat/hxab030","url":null,"abstract":"Double-zero eigenvalues at a Takens–Bogdanov (TB) bifurcation occur in many physical systems such as double-diffusive convection, binary convection and magnetoconvection. Analysis of the associated normal form, in 1D with periodic boundary condition, shows the existence of steady patterns, standing waves, modulated waves (MW) and travelling waves, and describes the transitions and bifurcations between these states. Values of coefficients of the terms in the normal form classify all possible different bifurcation scenarios in the neighbourhood of the TB bifurcation (Dangelmayr, G. & Knobloch, E. (1987) The Takens–Bogdanov bifurcation with O(2)-symmetry. Phil. Trans. R. Soc. Lond. A, 322, 243-279). In this work we develop a new and simple pattern-forming partial differential equation (PDE) model, based on the Swift–Hohenberg equation, adapted to have the TB normal form at onset. This model allows us to explore the dynamics in a wide range of bifurcation scenarios, including in domains much wider than the lengthscale of the pattern. We identify two bifurcation scenarios in which coexistence between different types of solutions is indicated from the analysis of the normal form equation. In these scenarios, we look for spatially localized solutions by examining pattern formation in wide domains. We are able to recover two types of localized states, that of a localized steady state (LSS) in the background of the trivial state (TS) and that of a spatially localized travelling wave (LTW) in the background of the TS, which have previously been observed in other systems. Additionally, we identify two new types of spatially localized states: that of a LSS in a MW background and that of a LTW in a steady state (SS) background. The PDE model is easy to solve numerically in large domains and so will allow further investigation of pattern formation with a TB bifurcation in one or more dimensions and the exploration of a range of background and foreground pattern combinations beyond SSs.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48680311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An activator–inhibitor–substrate model of side branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side branching. The model consists of four coupled reaction–diffusion equations, and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one spatial dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov–Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with $N$ identical equidistant peaks, $N=1,2,dots ,$ , together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the $N=1$ state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D.
{"title":"Stationary peaks in a multivariable reaction–diffusion system: foliated snaking due to subcritical Turing instability","authors":"Edgar Knobloch;Arik Yochelis","doi":"10.1093/imamat/hxab029","DOIUrl":"10.1093/imamat/hxab029","url":null,"abstract":"An activator–inhibitor–substrate model of side branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side branching. The model consists of four coupled reaction–diffusion equations, and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one spatial dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov–Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with \u0000<tex>$N$</tex>\u0000 identical equidistant peaks, \u0000<tex>$N=1,2,dots ,$</tex>\u0000, together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the \u0000<tex>$N=1$</tex>\u0000 state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47707535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we revisit some general results on the dynamics of circular fronts between homogeneous states and the formation of localized structures in two dimensions (2D). We show how the bifurcation diagram of axisymmetric structures localized in radius fits within the framework of collapsed homoclinic snaking. In 2D, owing to curvature effects, the collapse of the snaking structure follows a different scaling that is determined by the so-called nucleation radius. Moreover, in the case of fronts between two symmetry-related states, the precise point in parameter space to which radial snaking collapses is not a ‘Maxwell’ point but is determined by the curvature-driven dynamics only. In this case, the snaking collapses to a ‘zero surface tension’ point. Near this point, the breaking of symmetry between the homogeneous states tilts the snaking diagram. A different scaling law is found for the collapse of the snaking curve in each case. Curvature effects on axisymmetric localized states with internal structure are also discussed, as are cellular structures separated from a homogeneous state by a circular front. While some of these results are well understood in terms of curvature-driven dynamics and front interactions, a proper mathematical description in terms of homoclinic trajectories in a radial spatial dynamics description is lacking.
{"title":"Curvature effects and radial homoclinic snaking","authors":"Damià Gomila;Edgar Knobloch","doi":"10.1093/imamat/hxab028","DOIUrl":"10.1093/imamat/hxab028","url":null,"abstract":"In this work, we revisit some general results on the dynamics of circular fronts between homogeneous states and the formation of localized structures in two dimensions (2D). We show how the bifurcation diagram of axisymmetric structures localized in radius fits within the framework of collapsed homoclinic snaking. In 2D, owing to curvature effects, the collapse of the snaking structure follows a different scaling that is determined by the so-called nucleation radius. Moreover, in the case of fronts between two symmetry-related states, the precise point in parameter space to which radial snaking collapses is not a ‘Maxwell’ point but is determined by the curvature-driven dynamics only. In this case, the snaking collapses to a ‘zero surface tension’ point. Near this point, the breaking of symmetry between the homogeneous states tilts the snaking diagram. A different scaling law is found for the collapse of the snaking curve in each case. Curvature effects on axisymmetric localized states with internal structure are also discussed, as are cellular structures separated from a homogeneous state by a circular front. While some of these results are well understood in terms of curvature-driven dynamics and front interactions, a proper mathematical description in terms of homoclinic trajectories in a radial spatial dynamics description is lacking.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41963536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Localized coherent structures can form in externally driven dispersive optical cavities with a Kerr-type non-linearity. Such systems are described by the Lugiato–Lefever (LL) equation, which supports a large variety of dynamical states. Here, we review our current knowledge of the formation, stability and bifurcation structure of localized structures in the one-dimensional LL equation. We do so by focusing on two main regimes of operation: anomalous and normal second-order dispersion. In the anomalous regime, localized patterns are organized in a homoclinic snaking scenario, which is eventually destroyed, leading to a foliated snaking bifurcation structure. In the normal regime, localized structures undergo a different type of bifurcation structure, known as collapsed snaking. The effects of third-order dispersion and various dynamical regimes are also described.
{"title":"Origin, bifurcation structure and stability of localized states in Kerr dispersive optical cavities","authors":"P Parra-Rivas;E Knobloch;L Gelens;D Gomila","doi":"10.1093/imamat/hxab031","DOIUrl":"10.1093/imamat/hxab031","url":null,"abstract":"Localized coherent structures can form in externally driven dispersive optical cavities with a Kerr-type non-linearity. Such systems are described by the Lugiato–Lefever (LL) equation, which supports a large variety of dynamical states. Here, we review our current knowledge of the formation, stability and bifurcation structure of localized structures in the one-dimensional LL equation. We do so by focusing on two main regimes of operation: anomalous and normal second-order dispersion. In the anomalous regime, localized patterns are organized in a homoclinic snaking scenario, which is eventually destroyed, leading to a foliated snaking bifurcation structure. In the normal regime, localized structures undergo a different type of bifurcation structure, known as collapsed snaking. The effects of third-order dispersion and various dynamical regimes are also described.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45011913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This editorial serves as an extended introduction to the Special Issue. It gives the context to homoclinic snaking, especially the contribution of Patrick Woods. A very brief summary of more recent developments serves as a motivation to each paper that follows.
{"title":"Editorial to Homoclinic snaking at 21: in memory of Patrick Woods","authors":"Alan Champneys","doi":"10.1093/imamat/hxab041","DOIUrl":"10.1093/imamat/hxab041","url":null,"abstract":"This editorial serves as an extended introduction to the Special Issue. It gives the context to homoclinic snaking, especially the contribution of Patrick Woods. A very brief summary of more recent developments serves as a motivation to each paper that follows.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42448854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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