The velocity diagram for traveling waves
In this Note, we consider traveling waves in a reaction-diffusion equation in dimension one. Motivated by the motion of dislocations in crystals, we introduce an additive parameter $\sigma $ in the reaction term, which may be interpreted as an exterior force applied on the crystal. Under certain nat...
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Académie des sciences
2023-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.433/ |
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| author | Al Haj, Mohammad Monneau, Régis |
| author_facet | Al Haj, Mohammad Monneau, Régis |
| author_sort | Al Haj, Mohammad |
| collection | DOAJ |
| description | In this Note, we consider traveling waves in a reaction-diffusion equation in dimension one. Motivated by the motion of dislocations in crystals, we introduce an additive parameter $\sigma $ in the reaction term, which may be interpreted as an exterior force applied on the crystal. Under certain natural assumptions and for every value of $\sigma \in [\sigma ^-,\sigma ^+]$, we show the existence of traveling waves $\phi $ of velocity $c$. The range $\sigma \in (\sigma ^-,\sigma ^+)$ corresponds to bistable cases with a unique velocity $c=c(\sigma )$. On the contrary, the case $\sigma =\sigma ^+$ is positively monostable with a branch of velocities $c\ge c^+$, while the case $\sigma =\sigma ^-$ is negatively monostable with a branch of velocities $c\le c^-$. This study gives rise to a natural connection between bistable cases and monostable cases in a single velocity diagram. We also give some qualitative properties of the velocity function $\sigma \mapsto c(\sigma )$. |
| format | Article |
| id | doaj-art-c6f3a28c30c04910b45537a6b56045dd |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-c6f3a28c30c04910b45537a6b56045dd2025-02-07T11:07:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-05-01361G477778210.5802/crmath.43310.5802/crmath.433The velocity diagram for traveling wavesAl Haj, Mohammad0Monneau, Régis1Lebanese University, Faculty of Science (section 5), Nabatiye 1700, LebanonCEREMADE, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France; CERMICS, Ecole des Ponts ParisTech, Université Paris-Est, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, FranceIn this Note, we consider traveling waves in a reaction-diffusion equation in dimension one. Motivated by the motion of dislocations in crystals, we introduce an additive parameter $\sigma $ in the reaction term, which may be interpreted as an exterior force applied on the crystal. Under certain natural assumptions and for every value of $\sigma \in [\sigma ^-,\sigma ^+]$, we show the existence of traveling waves $\phi $ of velocity $c$. The range $\sigma \in (\sigma ^-,\sigma ^+)$ corresponds to bistable cases with a unique velocity $c=c(\sigma )$. On the contrary, the case $\sigma =\sigma ^+$ is positively monostable with a branch of velocities $c\ge c^+$, while the case $\sigma =\sigma ^-$ is negatively monostable with a branch of velocities $c\le c^-$. This study gives rise to a natural connection between bistable cases and monostable cases in a single velocity diagram. We also give some qualitative properties of the velocity function $\sigma \mapsto c(\sigma )$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.433/ |
| spellingShingle | Al Haj, Mohammad Monneau, Régis The velocity diagram for traveling waves Comptes Rendus. Mathématique |
| title | The velocity diagram for traveling waves |
| title_full | The velocity diagram for traveling waves |
| title_fullStr | The velocity diagram for traveling waves |
| title_full_unstemmed | The velocity diagram for traveling waves |
| title_short | The velocity diagram for traveling waves |
| title_sort | velocity diagram for traveling waves |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.433/ |
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