Fronts and patterns with a dynamic parameter ramp
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911006383407104 |
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| author | Avery, Montie Garcia-Lopez, Odalys Goh, Ryan Hosek, Benjamin Shade, Ethan |
| author_facet | Avery, Montie Garcia-Lopez, Odalys Goh, Ryan Hosek, Benjamin Shade, Ethan |
| contents | We examine the effect of a slowly-varying time-dependent parameter on invasion fronts for which an unstable homogeneous equilibrium is invaded by either another homogeneous state or a spatially periodic state. We first explain and motivate our approach by studying asymptotically constant invasion fronts in a scalar FKPP equation with time-dependent parameter which controls the stability of the trivial state. Following recent works in the area, we use a linearized analysis to derive formal predictions for front position and leading-edge spatial decay. We then use a comparison principle approach to establish a rigorous spreading result in the case of an unbounded temporal parameter. We then consider patterned-invasion in the complex Ginzburg-Landau equation with dynamic bifurcation parameter, a prototype for slow passage through a spatio-temporal Hopf instability. Linearized analysis once again gives front position and decay asymptotics, but also the selected spatial wavenumber at the leading edge. We then use a Burger's modulation analysis to predict the slowly-varying wavenumber in the wake of the front. Finally, in both equations, we used the recently developed concept of a space-time memory curve to characterize delayed invasion in the case where the parameter is initially stable before a subsequent slow passage through instability and invasion. We also provide preliminary results studying invasion in other prototypical pattern formation models modified with a dynamic parameter, as well as numerical results for delayed transition between pushed and pulled fronts in Nagumo's equation with dynamic parameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12142 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fronts and patterns with a dynamic parameter ramp Avery, Montie Garcia-Lopez, Odalys Goh, Ryan Hosek, Benjamin Shade, Ethan Pattern Formation and Solitons Analysis of PDEs Dynamical Systems 35B36, 35B32, 37L10, 35K57 We examine the effect of a slowly-varying time-dependent parameter on invasion fronts for which an unstable homogeneous equilibrium is invaded by either another homogeneous state or a spatially periodic state. We first explain and motivate our approach by studying asymptotically constant invasion fronts in a scalar FKPP equation with time-dependent parameter which controls the stability of the trivial state. Following recent works in the area, we use a linearized analysis to derive formal predictions for front position and leading-edge spatial decay. We then use a comparison principle approach to establish a rigorous spreading result in the case of an unbounded temporal parameter. We then consider patterned-invasion in the complex Ginzburg-Landau equation with dynamic bifurcation parameter, a prototype for slow passage through a spatio-temporal Hopf instability. Linearized analysis once again gives front position and decay asymptotics, but also the selected spatial wavenumber at the leading edge. We then use a Burger's modulation analysis to predict the slowly-varying wavenumber in the wake of the front. Finally, in both equations, we used the recently developed concept of a space-time memory curve to characterize delayed invasion in the case where the parameter is initially stable before a subsequent slow passage through instability and invasion. We also provide preliminary results studying invasion in other prototypical pattern formation models modified with a dynamic parameter, as well as numerical results for delayed transition between pushed and pulled fronts in Nagumo's equation with dynamic parameter. |
| title | Fronts and patterns with a dynamic parameter ramp |
| topic | Pattern Formation and Solitons Analysis of PDEs Dynamical Systems 35B36, 35B32, 37L10, 35K57 |
| url | https://arxiv.org/abs/2506.12142 |