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Nonlinear waves in a simple model of high-grade glioma

Arturo Álvarez-Arenas

Arturo Álvarez-Arenas

Department of Mathematics and Mathematical Oncology Laboratory, University of Castilla-La ManchaCiudad Real, Spain
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Álvarez-Arenas, Arturo
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Juan Belmonte-Beitia

Juan Belmonte-Beitia

Department of Mathematics and Mathematical Oncology Laboratory, University of Castilla-La ManchaCiudad Real, Spain
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Belmonte-Beitia, Juan
 and 
Gabriel F. Calvo

Gabriel F. Calvo

Department of Mathematics and Mathematical Oncology Laboratory, University of Castilla-La ManchaCiudad Real, Spain
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Calvo, Gabriel F.
  
Oct 01, 2016
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Published Online: Oct 01, 2016
Page range: 405 - 422
Received: May 15, 2016
Accepted: Oct 01, 2016
DOI: https://doi.org/10.21042/AMNS.2016.2.00035
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© 2016 Arturo Álvarez-Arenas, Juan Belmonte-Beitia, and Gabriel F. Calvo, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

(a) Heteroclinic orbits of Eq. (10) for β = 0.1. (b) Kink solution for β = 0.1 corresponding to the marked heteroclinic orbit in (a). (c) The bright soliton solution 𝓤 corresponds to the kink obtained in (b)
(a) Heteroclinic orbits of Eq. (10) for β = 0.1. (b) Kink solution for β = 0.1 corresponding to the marked heteroclinic orbit in (a). (c) The bright soliton solution 𝓤 corresponds to the kink obtained in (b)

Fig. 2

Bright solitary waves for β = 0.3, ϕ0 = 0.5 and c = 2.5. (a) Profiles from Eq. (45) (solid curve) and the explicit solution given by Eq. (46) (dashed curve). (b) Homoclinic orbits from Eq. (45) (solid curve) and Eq. (46) (dashed curve).
Bright solitary waves for β = 0.3, ϕ0 = 0.5 and c = 2.5. (a) Profiles from Eq. (45) (solid curve) and the explicit solution given by Eq. (46) (dashed curve). (b) Homoclinic orbits from Eq. (45) (solid curve) and Eq. (46) (dashed curve).

Fig. 3

Characteristic curves of the minimum speed c0 for positive bright solitary solutions, in terms of (a) β and (b) ϕ0, calculated from Eq. (53).
Characteristic curves of the minimum speed c0 for positive bright solitary solutions, in terms of (a) β and (b) ϕ0, calculated from Eq. (53).

Fig. 4

Plot of the depth of resection b versus the recurrence time t for different values of the proliferation rate: ρ = 1/5 (blue solid line), ρ = 1/4 (green dashed line) and ρ = 1/3 (red dotted-dashed line).
Plot of the depth of resection b versus the recurrence time t for different values of the proliferation rate: ρ = 1/5 (blue solid line), ρ = 1/4 (green dashed line) and ρ = 1/3 (red dotted-dashed line).
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