An amplitude equation for the conserved-Hopf bifurcation -- derivation, analysis and assessment

Abstract

We employ weakly nonlinear theory to derive an amplitude equation for the conserved-Hopf instability, i.e., a generic large-scale oscillatory instability for systems with two conservation laws. The resulting equation represents in the conserved case the equivalent of the complex Ginzburg-Landau equation obtained in the nonconserved case as amplitude equation for the standard Hopf bifurcation.Considering first the case of a relatively simple symmetric Cahn-Hilliard model with purely nonreciprocal coupling, we derive the nonlinear nonlocal amplitude equation with real coefficients and show that its bifurcation diagram and time evolution well agree with results for the full model. The solutions of the amplitude equation and their stability are analytically obtained thereby showing that in oscillatory phase separation the suppression of coarsening is universal. Second, we lift the two restrictions and obtain the amplitude equation in the generic case that has complex coefficients, that also shows very good agreement with the full model as exemplified for some transient dynamics that converges to traveling wave states.

References in corpus (20)

• The World of the Complex Ginzburg-Landau Equation
• Nonreciprocity as a generic route to traveling states
• A mass conserved reaction-diffusion system captures properties of cell polarity
• Scalar Active Mixtures: The Non-Reciprocal Cahn-Hilliard Model
• Self-organization principles of intracellular pattern formation
• Morphology changes in the evolution of liquid two-layer films
• Pattern formation with a conservation law
• Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity
• Phase-space geometry of mass-conserving reaction-diffusion dynamics
• Nonreciprocal pattern formation of conserved fields
• Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling
• On the transient and steady state of mass-conserved reaction diffusion systems
• Wavelength selection by interrupted coarsening in reaction-diffusion systems
• Nonreciprocal Cahn-Hilliard model emerges as a universal amplitude equation
• Localized states in coupled Cahn-Hilliard equations
• Nonreciprocity induces resonances in two-field Cahn-Hilliard model
• Stationary broken parity states in active matter models
• Why A Large Scale Mode Can Be Essential For Understanding Intracellular Actin Waves
• Mode instabilities and dynamic patterns in a colony of self-propelled surfactant particles covering a thin liquid layer
• Effervescent waves in a binary mixture with non-reciprocal couplings

Cited by in corpus (7)

• Coexistence of uniform and oscillatory states resulting from nonreciprocity and conservation laws
• Local Order Controls the Onset of Oscillations in the Nonreciprocal Ising Model
• Theory of Nonequilibrium Coexistence with Coupled Conserved and Nonconserved Order Parameters
• Motility-induced crystallization and rotating crystallites
• Higher-codimension points as organizing centers in nonreciprocal pattern-forming systems with O(2)-symmetry
• The effect of self-induced Marangoni flow on polar-nematic waves in active-matter systems
• Derivation and Analysis of Amplitude Equation for Generalized AMB+ in Presence of Chemical Reaction