Abstract
The existence of travelling waves for a coupled system of hyperbolic/parabolic equations is established in the case of a finite number of velocities in the kinetic equation. This finds application in collective motion of chemotactic bacteria. The analysis builds on the previous work by the first author (arXiv:1607.00429) in the case of a continuum of velocities. Here, the proof is specific to the discrete setting, based on the decomposition of the population density in special Case’s modes. Some counter-intuitive results are discussed numerically, including the co-existence of several travelling waves for some sets of parameters, as well as the possible non-existence of travelling waves.
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Notes
- 1.
- 2.
Notice that these weights are signed.
- 3.
Moreover, we have no argument so far to determine the signs of the coefficients (a j , b j ), if they have any.
- 4.
This is clearly the case due to integrability over \(\mathbb{R}\).
- 5.
Notice that the value z = 0 is not an issue here. In fact, Lemma 7 includes the values z = 0+ and z = 0− which must be distinguished from each other because I is not continuous at z = 0.
- 6.
The restriction c > c ⋆ is due to confinement on the right hand side, whereas the condition c > 0 is due to the arbitrary choice of the direction of propagation of the wave (here, left to right), which influences itself the monotonicity of N (here, increasing). It is an arbitrary choice, of course, since the problem is symmetric.
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Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 639638). M.T. has benefited from the PICS Project CNR-CNRS 2015-2017 Modèles mathématiques et simulations numériques pour le mouvement de cellules.
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Calvez, V., Gosse, L., Twarogowska, M. (2017). Concentration Waves of Chemotactic Bacteria: The Discrete Velocity Case. In: Gosse, L., Natalini, R. (eds) Innovative Algorithms and Analysis. Springer INdAM Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-49262-9_3
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