A floer homology approach to traveling waves in reaction-diffusion equations on cylinders

Bente Bakker; Jan Bouwe Van Den Berg; Rob Vandervorst

doi:10.1137/17M1162019

A floer homology approach to traveling waves in reaction-diffusion equations on cylinders

Bente Bakker^*, Jan Bouwe Van Den Berg, Rob Vandervorst

^*Corresponding author for this work

Mathematics

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Abstract

Traveling waves form a prominent feature in the dynamics of scalar reaction-diffusion equations on unbounded cylinders. The traveling waves can be identified with the bounded solutions of the (Formula Prtsented) Ω ⊂ R d is a bounded domain, Δ is the Laplacian on Ω, and B denotes Dirichlet, Neumann, or periodic boundary data. We develop a new homological invariant for the dynamics of the bounded solutions of the above elliptic PDE. Restrictions on the nonlinearity f are kept to a minimum; for instance, any nonlinearity exhibiting polynomial growth in u can be considered. In particular, the set of bounded solutions of the traveling wave PDE may not be uniformly bounded. Despite this, the homology is invariant under lower order (but not necessarily small) perturbations of the nonlinearity f, thus making the homology amenable for computation. Using the new invariant we derive lower bounds on the number of bounded solutions of our PDE, thus obtaining existence and multiplicity results for traveling wave solutions of reaction-diffusion equations on unbounded cylinders.

Original language	English
Pages (from-to)	2634-2706
Number of pages	73
Journal	SIAM Journal on Applied Dynamical Systems
Volume	17
Issue number	4
Early online date	15 Nov 2018
DOIs	https://doi.org/10.1137/17M1162019
Publication status	Published - Dec 2018

Funding

\ast Received by the editors December 18, 2017; accepted for publication (in revised form) July 26, 2018; published electronically November 15, 2018. http://www.siam.org/journals/siads/17-4/M116201.html Funding: This research was supported by NWO VICI grant 639.033.109 and NWO TOP grant 613.001.351. \dagger Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands ([email protected], [email protected], [email protected]).

Funders	Funder number
Nederlandse Organisatie voor Wetenschappelijk Onderzoek	639.033.109, 613.001.351

Keywords

Floer homology
gradient-like structure
infinite dimensional dynamics
nonlocal equation
reaction-diffusion equations
strongly indefinite equation
traveling waves

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This output contributes to the following UN Sustainable Development Goals (SDGs)

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A floer homology approach to traveling waves in reactiondiffusion equations on cylindersFinal published version, 971 KBLicence: Article 25fa Dutch Copyright Act

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title = "A floer homology approach to traveling waves in reaction-diffusion equations on cylinders",

abstract = "Traveling waves form a prominent feature in the dynamics of scalar reaction-diffusion equations on unbounded cylinders. The traveling waves can be identified with the bounded solutions of the (Formula Prtsented) Ω ⊂ R d is a bounded domain, Δ is the Laplacian on Ω, and B denotes Dirichlet, Neumann, or periodic boundary data. We develop a new homological invariant for the dynamics of the bounded solutions of the above elliptic PDE. Restrictions on the nonlinearity f are kept to a minimum; for instance, any nonlinearity exhibiting polynomial growth in u can be considered. In particular, the set of bounded solutions of the traveling wave PDE may not be uniformly bounded. Despite this, the homology is invariant under lower order (but not necessarily small) perturbations of the nonlinearity f, thus making the homology amenable for computation. Using the new invariant we derive lower bounds on the number of bounded solutions of our PDE, thus obtaining existence and multiplicity results for traveling wave solutions of reaction-diffusion equations on unbounded cylinders.",

keywords = "Floer homology, gradient-like structure, infinite dimensional dynamics, nonlocal equation, reaction-diffusion equations, strongly indefinite equation, traveling waves",

author = "Bente Bakker and {Van Den Berg}, {Jan Bouwe} and Rob Vandervorst",

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AU - Van Den Berg, Jan Bouwe

AU - Vandervorst, Rob

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N2 - Traveling waves form a prominent feature in the dynamics of scalar reaction-diffusion equations on unbounded cylinders. The traveling waves can be identified with the bounded solutions of the (Formula Prtsented) Ω ⊂ R d is a bounded domain, Δ is the Laplacian on Ω, and B denotes Dirichlet, Neumann, or periodic boundary data. We develop a new homological invariant for the dynamics of the bounded solutions of the above elliptic PDE. Restrictions on the nonlinearity f are kept to a minimum; for instance, any nonlinearity exhibiting polynomial growth in u can be considered. In particular, the set of bounded solutions of the traveling wave PDE may not be uniformly bounded. Despite this, the homology is invariant under lower order (but not necessarily small) perturbations of the nonlinearity f, thus making the homology amenable for computation. Using the new invariant we derive lower bounds on the number of bounded solutions of our PDE, thus obtaining existence and multiplicity results for traveling wave solutions of reaction-diffusion equations on unbounded cylinders.

AB - Traveling waves form a prominent feature in the dynamics of scalar reaction-diffusion equations on unbounded cylinders. The traveling waves can be identified with the bounded solutions of the (Formula Prtsented) Ω ⊂ R d is a bounded domain, Δ is the Laplacian on Ω, and B denotes Dirichlet, Neumann, or periodic boundary data. We develop a new homological invariant for the dynamics of the bounded solutions of the above elliptic PDE. Restrictions on the nonlinearity f are kept to a minimum; for instance, any nonlinearity exhibiting polynomial growth in u can be considered. In particular, the set of bounded solutions of the traveling wave PDE may not be uniformly bounded. Despite this, the homology is invariant under lower order (but not necessarily small) perturbations of the nonlinearity f, thus making the homology amenable for computation. Using the new invariant we derive lower bounds on the number of bounded solutions of our PDE, thus obtaining existence and multiplicity results for traveling wave solutions of reaction-diffusion equations on unbounded cylinders.

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KW - gradient-like structure

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A floer homology approach to traveling waves in reaction-diffusion equations on cylinders

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