Open Access
Issue |
Math. Model. Nat. Phenom.
Volume 17, 2022
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Article Number | 13 | |
Number of page(s) | 25 | |
DOI | https://doi.org/10.1051/mmnp/2022020 | |
Published online | 09 June 2022 |
- J. Alexander, R.A. Gardner and C.K.R.T. Jones, A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410 (1990) 167–212. [MathSciNet] [Google Scholar]
- E. Álvarez and R.G. Plaza, Existence and spectral instability of bounded spatially periodic traveling waves for scalar viscous balance laws. Quart. Appl. Math. 79 (2021) 493–544. [CrossRef] [MathSciNet] [Google Scholar]
- A.A. Andronov, Les cycles limites de Poincare et la theorie des oscillations auto—entretenues. C.R. Acad. Sci. Paris 189 (1929) 559–561. [Google Scholar]
- E. Bouin, V. Calvez and G. Nadin, Hyperbolic traveling waves driven by growth. Math. Models Methods Appl. Sci. 24 (2014) 1165–1195. [CrossRef] [MathSciNet] [Google Scholar]
- J.M. Burgers, A mathematical model illustrating the theory of turbulence, in Advances in Applied Mechanics, edited by R. von Mises and T. von Karman. Academic Press Inc., New York, N.Y. (1948) 171–199. [CrossRef] [Google Scholar]
- C. Cattaneo, Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3 (1949) 83–101. [MathSciNet] [Google Scholar]
- C. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée. C.R. Acad. Sci. Paris 247 (1958) 431–433. [MathSciNet] [Google Scholar]
- S. Chen and J. Duan, Instability of small-amplitude periodic waves from fold-Hopf bifurcation. Preprint arXiv:2012.07484. (2020). [Google Scholar]
- J. Crank, The mathematics of diffusion, Clarendon Press, Oxford, second ed. (1975). [Google Scholar]
- E.C.M. Crooks and C. Mascia, Front speeds in the vanishing diffusion limit for reaction-diffusion-convection equations. Differ. Integral Equ. 20 (2007) 499–514. [Google Scholar]
- C.M. Dafermos, Hyperbolic conservation laws in continuum physics. Vol. 325 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, fourth ed. (2016). [CrossRef] [Google Scholar]
- S.R. Dunbar and H.G. Othmer, On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks, in Nonlinear oscillations in biology and chemistry (Salt Lake City, Utah, 1985), H.G. Othmer, ed., vol. 66 of Lecture Notes in Biomath. Springer, Berlin (1986) 274–289. [CrossRef] [Google Scholar]
- N. Dunford and J.T. Schwartz, Linear operators. Part II: Spectral theory. Selfadjoint operators in Hilbert space. Wiley Classics Library, John Wiley & Sons Inc., New York (1988). [Google Scholar]
- S. Fedotov, Traveling waves in a reaction-diffusion system: diffusion with finite velocity and Kolmogorov-Petrovskii-Piskunov kinetics. Phys. Rev. E 58 (1998) 5143–5145. [CrossRef] [MathSciNet] [Google Scholar]
- A. Fick, On liquid diffusion. J. Membr. Sci. 100 (1995) 33–38. Reprint of the 1855 original. [CrossRef] [Google Scholar]
- R.A. Fisher, The wave of advance of advantageous genes. Ann. Eugen. 7 (1937) 355–369. [CrossRef] [Google Scholar]
- R.A. Gardner, On the structure of the spectra of periodic travelling waves. J. Math. Pures Appl. (9) 72 (1993) 415–439. [MathSciNet] [Google Scholar]
- H. Gomez, I. Colominas, F. Navarrina, J. Paris and M. Casteleiro, A hyperbolic theory for advection-diffusion problems: mathematical foundations and numerical modeling. Arch. Comput. Methods Eng. 17 (2010) 191–211. [CrossRef] [MathSciNet] [Google Scholar]
- M. Gorgone and F. Oliveri, Consistent approximate Q-conditional symmetries of PDEs: application to a hyperbolic reaction-diffusion-convection equation. Z. Angew. Math. Phys. 72 (2021) Paper No. 119, 25. [CrossRef] [Google Scholar]
- J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Vol. 42 of Applied Mathematical Sciences. Springer-Verlag, New York (1983). [CrossRef] [Google Scholar]
- K.P. Hadeler, Hyperbolic travelling fronts. Proc. Edinburgh Math. Soc. 31 (1988) 89–97. [CrossRef] [MathSciNet] [Google Scholar]
- K.P. Hadeler, Travelling fronts for correlated random walks. Canad. Appl. Math. Quart. 2 (1994) 27–43. [MathSciNet] [Google Scholar]
- K.P. Hadeler, Reaction telegraph equations and random walk systems, in Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), S.J. van Strien and S.M. Verduyn Lunel, eds., vol. 45 of Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, North-Holland, Amsterdam (1996) 133–161. Proceedings of the meeting held in Amsterdam, January 1995. [Google Scholar]
- K.P. Hadeler, Reaction transport systems in biological modelling, in Mathematics inspired by biology (Martina Franca, 1997), edited by V. Capasso and O. Diekmann. vol. 1714 of Lecture Notes in Math. Springer, Berlin (1999) 95–150. [CrossRef] [Google Scholar]
- J.K. Hale and H. Koçak, Dynamics and bifurcations, vol. 3 of Texts in Applied Mathematics. Springer-Verlag, New York (1991). [CrossRef] [Google Scholar]
- J. Harterich, Viscous profiles of traveling waves in scalar balance laws: the canard case. Methods Appl. Anal. 10 (2003) 97–117. [CrossRef] [MathSciNet] [Google Scholar]
- J. Harterich and K. Sakamoto, Front motion in viscous conservation laws with stiff source terms. Adv. Differ. Equ. 11 (2006) 721–750. [Google Scholar]
- L. Herrmann, Hyperbolic diffusion equation. AIP Conf. Proc. 1504 (2012) 1337–1340. [CrossRef] [Google Scholar]
- T. Hillen, Invariance principles for hyperbolic random walk systems. J. Math. Anal. Appl. 210 (1997) 360–374. [CrossRef] [MathSciNet] [Google Scholar]
- T. Hillen, Qualitative analysis of semilinear Cattaneo equations. Math. Models Methods Appl. Sci. 8 (1998) 507–519. [CrossRef] [Google Scholar]
- P.D. Hislop and I.M. Sigal, Introduction to spectral theory. With applications to Schrödinger operators. Vol. 113 of Applied Mathematical Sciences. Springer-Verlag, New York (1996). [CrossRef] [Google Scholar]
- E.E. Holmes, Is diffusion too simple? Comparisons with a telegraph model of dispersal. Am,. Nat. 142 (1993) 779–796. [CrossRef] [PubMed] [Google Scholar]
- E. Hopf, Abzweigung einer periodischen Losung von einer stationaren Losung eines Differentialsystems. Ber. Math.-Phys. Kl. Sachs. Akad. Wiss. Leipzig 94 (1942) 1–22. [Google Scholar]
- C.K.R.T. Jones, R. Marangell, P.D. Miller and R.G. Plaza, Spectral and modulational stability of periodic wavetrains for the nonlinear Klein-Gordon equation. J. Differ. Equ. 257 (2014) 4632–4703. [CrossRef] [Google Scholar]
- D.D. Joseph and L. Preziosi, Heat waves. Rev. Modern Phys. 61 (1989) 41–73. [CrossRef] [MathSciNet] [Google Scholar]
- M. Kac, A stochastic model related to the telegrapher’s equation. Rocky Mountain J. Math. 4 (1974) 497–509. [MathSciNet] [Google Scholar]
- T. Kapitula and K. Promislow, Spectral and dynamical stability of nonlinear waves. Vol. 185 of Applied Mathematical Sciences. Springer-Verlag, New York (2013). [CrossRef] [Google Scholar]
- T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics. Springer-Verlag, New York, Second ed. (1980). [Google Scholar]
- R. Kollar, B. Deconinck and O. Trichtchenko, Direct characterization of spectral stability of small-amplitude periodic waves in scalar Hamiltonian problems via dispersion relation. SIAM J. Math. Anal. 51 (2019) 3145–3169. [CrossRef] [MathSciNet] [Google Scholar]
- A.N. Kolmogorov, I. Petrovsky and N. Piskunov, Etude de l'equation de la diffusion avec croissance de la quantite de matiere et son applicationa un probleme biologique. Mosc. Univ. Bull. Math. 1 (1937) 1–25. [Google Scholar]
- Y.A. Kuznetsov, Elements of applied bifurcation theory, vol. 112 of Applied Mathematical Sciences. Springer-Verlag, New York, second ed. (1998). [Google Scholar]
- C. Lattanzio, C. Mascia, R.G. Plaza and C. Simeoni, Analytical and numerical investigation of traveling waves for the Allen- Cahn model with relaxation. Math. Models Methods Appl. Sci. 26 (2016) 931–985. [CrossRef] [MathSciNet] [Google Scholar]
- C. Lattanzio, C. Mascia, R.G. Plaza and C. Simeoni, Kinetic schemes for assessing stability of traveling fronts for the Allen- Cahn equation with relaxation. Appl. Numer. Math. 141 (2019) 234–247. [CrossRef] [MathSciNet] [Google Scholar]
- P.D. Lax, Hyperbolic systems of conservation laws II. Comm.. Pure Appl. Math. 10 (1957) 537–566. [CrossRef] [MathSciNet] [Google Scholar]
- P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, no. 11 in CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1973). [Google Scholar]
- T.-P. Liu, Hyperbolic conservation laws with relaxation. Comm.. Math. Phys. 108 (1987) 153–175. [CrossRef] [MathSciNet] [Google Scholar]
- J.E. Marsden and M. McCracken, The Hopf bifurcation and its applications. Vol. 19 of Applied Mathematical Sciences. Springer-Verlag, New York (1976). [CrossRef] [Google Scholar]
- J.C. Maxwell, On the dynamical theory of gases. Trans. Royal Soc. London 157 (1867) 49–88. [CrossRef] [Google Scholar]
- B. Sandstede, Stability of travelling waves, in Vol. 2 of Handbook of dynamical systems, edited by B. Fiedler. North-Holland, Amsterdam (2002) 983–1055. [Google Scholar]
- S.H. Strogatz, Nonlinear dynamics and chaos. With applications to physics, biology, chemistry, and engineering. Westview Press, Boulder, CO, second ed. (2015). [Google Scholar]
- G.F. Verduzco, The first Lyapunov coefficient for a class of systems. 16th IFAC World Congress. IFAC Proc. Vols. 38 (2005) 1205–1209. [Google Scholar]
- P.-F. Verhulst, Notice sur la loi que la population suit dans son accroissement. Corr. Math. Phys. 10 (1838) 113–121. [Google Scholar]
- P. Vernotte, Les paradoxes de la theorie continue de l'equation de la chaleur. C.R. Acad. Sci. Paris 246 (1958) 3154–3155. [MathSciNet] [Google Scholar]
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