A variational approach to three-phase traveling waves for a gradient system

Chiun-Chuan Chen; Hung-Yu Chien; Chih-Chiang Huang

doi:10.3934/dcds.2021055

Article Contents

2021, Volume 41, Issue 10: 4737-4765. Doi: 10.3934/dcds.2021055

This issue Previous Article Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials Next Article On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system

A variational approach to three-phase traveling waves for a gradient system

Department of Mathematics, National Taiwan University, Taipei, 10617, Taiwan

^* Corresponding author: Hung-Yu Chien, andy1010@gmail.com
^* Corresponding author: Hung-Yu Chien, andy1010@gmail.com

Received: February 2020

Revised: January 2021

Early access: March 2021

Published: October 2021

C.-C. Chen and H.-Y. Chien are supported by the grants 108-2115-M-002-011-MY3 of MOST, Taiwan. C.-C. Huang is supported by the grants 109-2811-M-002-623 and 108-2115-M-002-012-MY2, MOST

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Abstract

In this paper, we use a variational approach to study traveling wave solutions of a gradient system in an infinite strip. As the even-symmetric potential of the system has three local minima, we prove the existence of a traveling wave that propagates from one phase to the other two phases, where these phases corresponds to the three local minima of the potential. To control the asymptotic behavior of the wave at minus infinity, we successfully find a certain convexity condition on the potential, which guarantees the convergence of the wave to a constant state but not to a one-dimensional homoclinic solution or other equilibria. In addition, a non-trivial steady state in $ \mathbb R^2 $ is established by taking a limit of the traveling wave solutions in the strip as the width of the strip tends to infinity.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35C07, 35A15.

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Figure 1. The union of gray regions is $ \{W(u)\le\epsilon\} $. The green curves are geodesic balls. And the red circle is $ \{ \mathbf p\ |\ | \mathbf p - \mathbf a| = \delta\} $

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Figure 2. The portrait of $ \partial_u\widetilde W = 0 $, $ \det( {{\rm{Hess\; }}}\widetilde W) = 0 $, $ \partial_{uu}W = 0 $ and $ | \mathbf p- \mathbf a| = 0.2 $

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