Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation

Hirokazu Ninomiya

doi:10.3934/dcds.2019084

Article Contents

2019, Volume 39, Issue 4: 2001-2019. Doi: 10.3934/dcds.2019084

This issue Previous Article Effect of quantified irreducibility on the computability of subshift entropy Next Article Global existence and optimal decay estimates of the compressible viscoelastic flows in

$ L^p $

critical spaces

Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation

Hirokazu Ninomiya^,

School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan

^* Corresponding author: H. Ninomiya
^* Corresponding author: H. Ninomiya

Received: January 31, 2018

Published: January 2019

The author was partially supported by JSPS KAKENHI Grant Numbers JP26287024, JP15K04963, JP16K13778, and JP16KT0022

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Abstract

In this paper, the propagation phenomena in the Allen-Cahn-Nagumo equation are considered. Especially, the relation between traveling wave solutions and entire solutions is discussed. Indeed, several types of one-dimensional entire solutions are constructed by composing one-dimensional traveling wave solutions. Combining planar traveling wave solutions provides several types of multi-dimensional traveling wave solutions. The relation between multi-dimensional traveling wave solutions and entire solutions suggests the existence of new traveling wave solutions and new entire solutions.

Keywords:

Traveling wave,
reaction-diffusion equations.

Mathematics Subject Classification: Primary: 35C07, 35K57; Secondary: 35B40, 35C08.

Citation:

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Figure 1. Profiles of supersolutions for one-dimensional entire solutions when $ -t $ is large, where $ f(u) = u(1-u)(u-1/12) $

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Figure 2. Profiles of supersolutions for one-dimensional entire solutions when $ -t $ is large, where $ f(u) = u(1-u)(u-1/12) $

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Figure 3. Contours of $ V_1^- $

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Figure 4. Zipping wave solution when $ f(u) = u(1-u)(u-1/2) $

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Figure 5. Example of $ \Gamma_0 $ and the equidistant curves from $ K_0 $ in $ \mathbb{R} ^2 $

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