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Applied Mathematics and Nonlinear Sciences
Volume 1 (2016): Issue 2 (July 2016)
Open Access
Nonlinear waves in a simple model of high-grade glioma
Arturo Álvarez-Arenas
Arturo Álvarez-Arenas
,
Juan Belmonte-Beitia
Juan Belmonte-Beitia
and
Gabriel F. Calvo
Gabriel F. Calvo
| Oct 01, 2016
Applied Mathematics and Nonlinear Sciences
Volume 1 (2016): Issue 2 (July 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
Keywords
Solitary waves
,
bright solitons
,
dark solitons
,
mathematical oncology
© 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)
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).
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).
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).