Existence and Stability of Periodic Planar Standing Waves in Phase-Transitional Elasticity with Strain-Gradient Effects II: Examples

Abstract

In a companion paper, we investigated phase-transitional elasticity models with strain-gradient effect, and established the existence of non-constant planar periodic standing waves in these models by variational methods. The variational methods enable us to deal with existence of periodic waves no matter the unknowns are scalar or not. Here, we list specific phase-transitional models with strain-gradient effects and list conditions that guarantee the existence of non-constant periodic waves. Also, when the unknowns are scalars, we do a phase-plane analysis, and compare the results obtained by phase-plane analysis with those obtained by our general variational methods. Finally, we briefly discuss relations between spectral and nonlinear stabilities by using a change of variables introduced by Kotschote to transform our system to a strictly parabolic system to which general results of Johnson–Zumbrun and Howard-Zumbrun apply.