Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations
In this paper we study the existence of one-dimensional travelling wave solutions $u(x,t)=\phi(x-ct)$ for the non-linear degenerate (at u=0) reaction-diffusion equation $u_t=[D(u)u_x]_x+g(u)$ where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing...
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| Format: | Journal article |
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1997
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| _version_ | 1826269030531465216 |
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| author | Sánchez-Garduño, F Maini, P |
| author_facet | Sánchez-Garduño, F Maini, P |
| author_sort | Sánchez-Garduño, F |
| collection | OXFORD |
| description | In this paper we study the existence of one-dimensional travelling wave solutions $u(x,t)=\phi(x-ct)$ for the non-linear degenerate (at u=0) reaction-diffusion equation $u_t=[D(u)u_x]_x+g(u)$ where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c=0,2. The existence of a unique value $c^{*}>0$ of c for which $\phi(x-c^{*}t)$ is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for $c \neq c^{*}$. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation. |
| first_indexed | 2024-03-06T21:18:41Z |
| format | Journal article |
| id | oxford-uuid:40b39b06-774e-41a2-b9ad-a0efa7d3a88e |
| institution | University of Oxford |
| last_indexed | 2024-03-06T21:18:41Z |
| publishDate | 1997 |
| record_format | dspace |
| spelling | oxford-uuid:40b39b06-774e-41a2-b9ad-a0efa7d3a88e2022-03-26T14:39:19ZTravelling wave phenomena in non-linear diffusion degenerate Nagumo equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:40b39b06-774e-41a2-b9ad-a0efa7d3a88eMathematical Institute - ePrints1997Sánchez-Garduño, FMaini, PIn this paper we study the existence of one-dimensional travelling wave solutions $u(x,t)=\phi(x-ct)$ for the non-linear degenerate (at u=0) reaction-diffusion equation $u_t=[D(u)u_x]_x+g(u)$ where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c=0,2. The existence of a unique value $c^{*}>0$ of c for which $\phi(x-c^{*}t)$ is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for $c \neq c^{*}$. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation. |
| spellingShingle | Sánchez-Garduño, F Maini, P Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations |
| title | Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations |
| title_full | Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations |
| title_fullStr | Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations |
| title_full_unstemmed | Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations |
| title_short | Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations |
| title_sort | travelling wave phenomena in non linear diffusion degenerate nagumo equations |
| work_keys_str_mv | AT sanchezgardunof travellingwavephenomenainnonlineardiffusiondegeneratenagumoequations AT mainip travellingwavephenomenainnonlineardiffusiondegeneratenagumoequations |