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Snakes, Ladders, and Isolas of Localized Patterns
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What this work says
Outgoing claims: the papers this work cites, builds on, or argues with.
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Fast and Slow Waves in the FitzHugh–Nagumo Equation
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Exponential Dichotomies for Solitary-Wave Solutions of Semilinear Elliptic Equations on Infinite Cylinders
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Homoclinic period blow-up in reversible and conservative systems
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Instability and Stability of Rolls in the Swift-Hohenberg Equation
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Geometric singular perturbation theory for ordinary differential equations
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Front motion, metastability and subcritical bifurcations in hydrodynamics
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Stable localized solutions of arbitrary length for the quintic Swift-Hohenberg equation
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Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit
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Exchange lemmas 1: Deng's lemma
inferredcites →https://doi.org/10.1016/j.physd.2005.04.018openalex-snapshot
Homoclinic snaking near a heteroclinic cycle in reversible systems
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Bifurcation structure of dissipative solitons
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Exponential asymptotics of localised patterns and snaking bifurcation diagrams
inferredcites →https://doi.org/10.1016/j.physleta.2006.08.072openalex-snapshot
Snakes and ladders: Localized states in the Swift–Hohenberg equation
inferredcites →https://doi.org/10.1016/s0020-7683(01)00234-7openalex-snapshot
Spatially complex localisation in twisted elastic rods constrained to a cylinder
inferredcites →https://doi.org/10.1016/s0045-7825(98)00197-2openalex-snapshot
Homoclinic and heteroclinic orbits underlying the post-buckling of axially-compressed cylindrical shells
inferredcites →https://doi.org/10.1016/s0167-2789(00)00114-7openalex-snapshot
Absolute and convective instabilities of waves on unbounded and large bounded domains
inferredcites →https://doi.org/10.1016/s0167-2789(02)00350-0openalex-snapshot
Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation
inferredcites →https://doi.org/10.1016/s0167-2789(97)00310-2openalex-snapshot
Localized patterns for the quintic complex Swift-Hohenberg equation
inferredcites →https://doi.org/10.1016/s0167-2789(98)00172-9openalex-snapshot
Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations
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Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian–Hopf bifurcation
inferredcites →https://doi.org/10.1016/s0167-2789(99)00010-xopenalex-snapshot
A skeleton structure of self-replicating dynamics
inferredcites →https://doi.org/10.1016/s1874-575x(02)80039-xopenalex-snapshot
Stability of Travelling Waves
inferredcites →https://doi.org/10.1023/a:1008398006403openalex-snapshot
Cellular Buckling in Long Structures
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Homoclinic snaking: Structure and stability
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Existence and stability of standing hole solutions to complex Ginzburg-Landau equations
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Gluing unstable fronts and backs together can produce stable pulses
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Multi-pulse evolution and space-time chaos in dissipative systems
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Pattern selection in the generalized Swift-Hohenberg model
inferredcites →https://doi.org/10.1103/physreve.73.056211openalex-snapshot
Localized states in the generalized Swift-Hohenberg equation
inferredcites →https://doi.org/10.1103/physrevlett.84.3069openalex-snapshot
Stable Static Localized Structures in One Dimension
inferredcites →https://doi.org/10.1103/physrevlett.97.044502openalex-snapshot
Asymptotics of Large Bound States of Localized Structures
inferredcites →https://doi.org/10.1103/physrevlett.99.104503openalex-snapshot
Proposed Resolution of Theory-Experiment Discrepancy in Homoclinic Snaking
inferredcites →https://doi.org/10.1137/030600192openalex-snapshot
Defects in Oscillatory Media: Toward a Classification
inferredcites →https://doi.org/10.1137/050635390openalex-snapshot
Geometric Relations of Absolute and Essential Spectra of Wave Trains
inferredcites →https://doi.org/10.1137/06067794xopenalex-snapshot
Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking
inferredcites →https://doi.org/10.1137/070695800openalex-snapshot
Snaking of Multiple Homoclinic Orbits in Reversible Systems
inferredcites →https://doi.org/10.1137/070707622openalex-snapshot
Localized Hexagon Patterns of the Planar Swift–Hohenberg Equation
inferredcites →https://doi.org/10.1137/s0036141002418637openalex-snapshot
Bifurcations of Periodic Solutions Satisfying the Zero-Hamiltonian Constraint in Reversible Differential Equations
inferredcites →https://doi.org/10.1137/s1064827597321647openalex-snapshot
Computation of Homoclinic Orbits in Partial Differential Equations: An Application to Cylindrical Shell Buckling
inferredcites →https://doi.org/10.1209/epl/i1997-00249-7openalex-snapshot
Stable localized squares in pattern-forming nonequilibrium systems
inferredcites →https://doi.org/10.3934/proc.2009.2009.109openalex-snapshot
Multipulse states in the Swift-Hohenberg equation
What points at this work
Incoming claims: where this work appears in other papers’ reference lists and arguments.
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Stationary localised patterns without Turing instability
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Sufficient Conditions for Global Minimality of Metastable States in a Class of Non-convex Functionals: A Simple Approach Via Quadratic Lower Bounds
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Spatially Localized Structures in Lattice Dynamical Systems
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Isolas of 2-Pulse Solutions in Homoclinic Snaking Scenarios
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Isolas Versus Snaking of Localized Rolls
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The Maslov Index, Degenerate Crossings and the Stability of Pulse Solutions to the Swift-Hohenberg equation
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Localized Solutions of a Piecewise Linear Model of the Stationary Swift–Hohenberg Equation on the Line and on the Plane
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An Asymptotic Analysis of Spike Self-Replication and Spike Nucleation of Reaction-Diffusion Patterns on Growing 1-D Domains
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Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy
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Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs
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Numerical challenges for the understanding of localised solutions with different symmetries in non-local hyperbolic systems
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Isolas of localized structures and Raman–Kerr frequency combs in micro-structured resonators
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Anti-predator behavior: A mechanism to lose localized patterns for diffusive ratio-dependent predator–prey systems
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Numerical continuation for fractional PDEs: sharp teeth and bloated snakes
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Species coexistence by front pinning
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Lyapunov–Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies
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Non-conservative perturbations of homoclinic snaking scenarios
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Existence and stability of spatially localized patterns
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Stationary non-radial localized patterns in the planar Swift-Hohenberg PDE: Constructive proofs of existence
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Localised spatial structures in the Thomas model
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Bifurcation analysis of a modified Swift–Hohenberg equation
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Exponential asymptotics of localised patterns and snaking bifurcation diagrams
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Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation: A numerical study
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Defect-mediated snaking: A new growth mechanism for localized structures
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Bifurcation analysis to the Lugiato–Lefever equation in one space dimension
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Fronts and bumps in spatially extended Kuramoto networks
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On localised hotspots of an urban crime model
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Predicting the bifurcation structure of localized snaking patterns
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Snakes and ladders in an inhomogeneous neural field model
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Snaking branches of planar BCC fronts in the 3D Brusselator
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The Lefever–Lejeune nonlinear lattice: Convergence dynamics and the structure of equilibrium states
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Bistability, wave pinning and localisation in natural reaction–diffusion systems
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Dissecting the snake: Transition from localized patterns to spike solutions
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Traveling pulses with oscillatory tails, figure-eight-like stack of isolas, and dynamics in heterogeneous media
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On periodically modulated rolls in the generalized Swift–Hohenberg equation: Galerkin’ approximations
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Breathers and mixed oscillatory states near a Turing–Hopf instability in a two–component reaction–diffusion system
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From snaking to isolas: A one-active-site approximation in discrete optical cavities
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Homoclinic and Heteroclinic Bifurcations in Vector Fields
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Spontaneous formation of travelling localized structures and their asymptotic behaviour in binary fluid convection
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Travelling convectons in binary fluid convection
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Homoclinic snaking near the surface instability of a polarisable fluid
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Homoclinic snaking in plane Couette flow: bending, skewing and finite-size effects
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Modified snaking in plane Couette flow with wall-normal suction
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Isolas of multi-pulse solutions to lattice dynamical systems
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Convectons, anticonvectons and multiconvectons in binary fluid convection
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Characterising the disordered state of block copolymers: Bifurcations of localised states and self-replication dynamics
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Spikes and localised patterns for a novel Schnakenberg model in the semi-strong interaction regime
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Sequential buckling in fluid-filled cylindrical shells
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Spatially localized binary fluid convection in a porous medium
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Homoclinic snaking of localized states in doubly diffusive convection
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Convectons and secondary snaking in three-dimensional natural doubly diffusive convection
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Localized rotating convection with no-slip boundary conditions
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Bumps and oscillons in networks of spiking neurons
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Phase-field-crystal description of active crystallites: Elastic and inelastic collisions
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Complex localization mechanisms in networks of coupled oscillators: Two case studies
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Time-dependent localized patterns in a predator-prey model
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Nonreversible homoclinic snaking
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Snakes and isolas in non-reversible conservative systems
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Convectons in periodic and bounded domains
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Spatially localized structures in dissipative systems: open problems
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Lin's method for heteroclinic chains involving periodic orbits
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Exponential asymptotics of homoclinic snaking
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Non-reversible perturbations of homoclinic snaking scenarios
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Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disc
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Boundary crisis for degenerate singular cycles
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Beyond all order asymptotics for homoclinic snaking in a Schnakenberg system
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Localized patterns in planar bistable weakly coupled lattice systems
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Discontinuous shock solutions of the Whitham modulation equations as zero dispersion limits of traveling waves
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Approximate localised dihedral patterns near a turing instability
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Analysing transitions from a Turing instability to large periodic patterns in a reaction-diffusion system
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Homoclinic snaking of contact defects in reaction-diffusion equations
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Bifurcation behavior of the coupled SPG-AMOC system
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Stationary and oscillatory localized patterns in ratio-dependent predator–prey systems
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Snaking bifurcations of localized patterns on ring lattices
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Spatial localization beyond steady states in the neighbourhood of the Takens–Bogdanov bifurcation
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Origin, bifurcation structure and stability of localized states in Kerr dispersive optical cavities
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Snaking without subcriticality: grain boundaries as non-topological defects
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Localized patterns in a generalized Swift–Hohenberg equation with a quartic marginal stability curve
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Editorial to Homoclinic snaking at 21: in memory of Patrick Woods
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Unfolding of homoclinic and heteroclinic behaviour in a multiply-symmetric strut buckling problem
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Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments
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Localized auxin peaks in concentration-based transport models of the shoot apical meristem
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The emergence of a coherent structure for coherent structures: localized states in nonlinear systems
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Spatially localized structures in the Gray–Scott model
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Unified framework for localized patterns in reaction–diffusion systems; the Gray–Scott and Gierer–Meinhardt cases
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Dissipative soliton combs with spectral filtering
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Cavity solitons and localized patterns in a finite-size optical cavity
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Defectlike structures and localized patterns in the cubic-quintic-septic Swift-Hohenberg equation
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Origin of localized snakes-and-ladders solutions of plane Couette flow
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Two-dimensional localized states in an active phase-field-crystal model
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Transition fronts and their universality classes
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Discrete breathers in a mechanical metamaterial
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Breathers in lattices with alternating strain-hardening and strain-softening interactions
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Collisions of localized patterns in a nonvariational Swift-Hohenberg equation
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Homoclinic snaking in bounded domains
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Swift-Hohenberg equation with broken reflection symmetry
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Faceting and coarsening dynamics in the complex Swift-Hohenberg equation
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Variational approximations to homoclinic snaking
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Swift-Hohenberg equation with broken cubic-quintic nonlinearity
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Variational approximations to homoclinic snaking in continuous and discrete systems
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Homoclinic snaking near a codimension-two Turing-Hopf bifurcation point in the Brusselator model
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Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity
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Spatial localization in heterogeneous systems
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Damped-driven granular chains: An ideal playground for dark breathers and multibreathers
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Damping filter method for obtaining spatially localized solutions
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Subcritical Turing bifurcation and the morphogenesis of localized patterns
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Rodlike localized structure in isotropic pattern-forming systems
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Homoclinic snaking in the discrete Swift-Hohenberg equation
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Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion
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Tristability between stripes, up-hexagons, and down-hexagons and snaking bifurcation branches of spatial connections between up- and down-hexagons
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Resting and traveling localized states in an active phase-field-crystal model
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Slanted snaking of localized Faraday waves
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Influence of Boundaries on Localized Patterns
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Snakes and Ladders: Localized Solutions of Plane Couette Flow
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Analytical Results for Front Pinning between an Hexagonal Pattern and a Uniform State in Pattern-Formation Systems
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Dissipative localized states and breathers in phase-mismatched singly resonant optical parametric oscillators: Bifurcation structure and stability
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Localized Hexagon Patterns of the Planar Swift–Hohenberg Equation
inferred← citeshttps://doi.org/10.1137/080724344openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/080734297openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/080734923openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/090762865openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/100782747openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/110843976openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/110855429openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/120893008openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/130918484openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/130918721openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/140966897openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/15m1012554openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/16m1093847openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/18m1218728openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/18m1225653openalex-snapshot
Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift--Hohenberg Equation
inferred← citeshttps://doi.org/10.1137/20m1369622openalex-snapshot
Hexagon Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift--Hohenberg Equation
inferred← citeshttps://doi.org/10.1137/22m1509734openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/22m1514234openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/24m1644328openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/24m1695245openalex-snapshot
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inferred← citeshttps://doi.org/10.1137/25m173973xopenalex-snapshot
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inferred← citeshttps://doi.org/10.1137/25m1781851openalex-snapshot
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inferred← citeshttps://doi.org/10.1140/epjst/e2014-02087-0openalex-snapshot
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inferred← citeshttps://doi.org/10.1140/epjst/e2016-60006-8openalex-snapshot
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inferred← citeshttps://doi.org/10.1142/s0218127415300013openalex-snapshot
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inferred← citeshttps://doi.org/10.1146/annurev-conmatphys-031214-014514openalex-snapshot
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inferred← citeshttps://doi.org/10.13140/rg.2.1.1138.7368openalex-snapshot
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inferred← citeshttps://doi.org/10.1364/ol.463061openalex-snapshot
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inferred← citeshttps://doi.org/10.3934/dcdss.2011.4.1019openalex-snapshot
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inferred← citeshttps://doi.org/10.3934/dcdss.2011.4.1213openalex-snapshot
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inferred← citeshttps://doi.org/10.3934/dcdss.2011.4.1341openalex-snapshot
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inferred← citeshttps://doi.org/10.3934/dcdss.2012.5.939openalex-snapshot
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inferred← citeshttps://doi.org/10.3934/dcdss.2016068openalex-snapshot
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inferred← citeshttps://doi.org/10.3934/dcdss.2022120openalex-snapshot
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inferred← citeshttps://doi.org/10.3934/proc.2009.2009.109openalex-snapshot
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inferred← citeshttps://doi.org/10.3934/proc.2011.2011.1244openalex-snapshot
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inferred← citeshttps://doi.org/10.4208/aam.v33.n1.p6openalex-snapshot
Localized Patterns of the Swift-Hohenberg Equation with a Dissipative Term
inferred← citeshttps://doi.org/10.4208/aam.v34.n1.p94openalex-snapshot
Localized Patterns of the Cubic-Quintic Swift-Hohenberg Equations with Two Symmetry-Breaking Terms
inferred← citeshttps://doi.org/10.48550/arxiv.0904.3387openalex-snapshot
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inferred← citeshttps://doi.org/10.48550/arxiv.1011.6546openalex-snapshot
Nonreversible Homoclinic Snaking
inferred← citeshttps://doi.org/10.48550/arxiv.1108.1984openalex-snapshot
Localised states in an extended Swift-Hohenberg equation
inferred← citeshttps://doi.org/10.48550/arxiv.1609.06108openalex-snapshot
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inferred← citeshttps://doi.org/10.48550/arxiv.1809.07847openalex-snapshot
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inferred← citeshttps://doi.org/10.48550/arxiv.1908.05211openalex-snapshot
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Localized Structures in the Multi-dimensional Swift-Hohenberg Equation
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