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

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 R2 is established by taking a limit of the traveling wave solutions in the strip as the width of the strip tends to infinity. ? 2021 American Institute of Mathematical Sciences. All rights reserved.