Standing and travelling waves in a parabolic-hyperbolic system

Figure 1.  Dependency of the wave velocity of segregated travelling wave solutions on $ \alpha $. The horizontal and the vertical axes are respectively the wave velocity $ \overline c $ and the value $ \alpha $. The parameter values are $ \gamma = 1 $ and $ k = 2 $

Figure 2.  Typical profiles of segregated TWs. The parameter values are $ \gamma = 1 $ and $ k = 2 $, the same as the ones in Figure 1. The solid and the dashed curves respectively denote $ U $ and $ V $. (a) $ \overline{c} = 0.2238529804 $ (b) $ \overline{c} = 0 $ (c) $ \overline{c} = -0.2895584564 $

Figure 3.  Profiles of overlapping TWs. The solid and the dashed curves respectively denote $ U $ and $ V $ and the gray curve denotes $ U+V $. The parameter values are $ \alpha = 1 $, $ k = 2 $ and $ \gamma = 1 $. For this parameter setting, the velocity of segregated TW is $ \overline{c} = 0.4094908611 $

Figure 4.  Profiles of overlapping TWs. The solid and the dashed curves respectively denote $ U $ and $ V $ and the gray curve denotes $ U+V $. The parameter values are $ \alpha = 4 $, $ k = 2 $ and $ \gamma = 1 $. For this parameter setting, the velocity of segregated TW is $ \overline{c} = -0.2895584564 $. See also Figure 2(c)

Figure 5.  Profiles of overlapping TWs. The solid and the dashed curves respectively denote $ U $ and $ V $ and the gray curve denotes $ U+V $. The parameter values are $ \alpha = 4 $, $ k = 2 $, $ \gamma = 0.4 $ and $ c = 1 $

Figure 6.  Profiles of $ t_\gamma $ in cases (ⅰ), (ⅱ) and (ⅲ) of Lemma 2.1

Figure 7.  Profiles of $ \Omega_1 $ and $ \Omega_2 $ in cases (ii) and (iii) of Lemma 2.2

Figure 8.  The case $ \lambda_1<\lambda_2 $ and $ \gamma>k(\alpha-1)/(k-1) $: $ (\gamma, c) = (10, 1) $

Figure 9.  The case $ \lambda_1>\lambda_2 $ and $ \gamma>k(\alpha-1)/(k-1) $: $ (\gamma, c) = (9, 2) $

Figure 10.  The case $ \lambda_1<\lambda_2 $ and $ 0<\gamma<k(\alpha-1)/(k-1) $: $ (\gamma, c) = (2, 1) $ ($ k = 4 $ and $ \alpha = 10 $)

Figure 11.  The case $ \lambda_1>\lambda_2 $ and $ 0<\gamma<k(\alpha-1)/(k-1) $: $ (\gamma, c) = (3, 1) $ ($ k = 4 $ and $ \alpha = 10 $)

Figure 12.  The case $ \Lambda_1<\Lambda_2 $ and $ \gamma>\alpha(k-1)/(\alpha-1) $: $ (\gamma, c) = (10, 1) $

Figure 13.  The case $ \Lambda_1>\Lambda_2 $ and $ \gamma>\alpha(k-1)/(\alpha-1) $: $ (\gamma, c) = (25, 0.5) $ ($ k = 8 $ and $ \alpha = 20 $)

Figure 14.  The case $ \Lambda_1<\Lambda_2 $ and $ 0<\gamma<\alpha(k-1)/(\alpha-1) $: $ (\gamma, c) = (0.00066, 0.02475) $ ($ k = 1.0004 $ and $ \alpha = 2.5020 $)

Figure 15.  The case $ \Lambda_1>\Lambda_2 $ and $ 0<\gamma<\alpha(k-1)/(\alpha-1) $: $ (\gamma, c) = (3, 1) $ ($ k = 4 $ and $ \alpha = 10 $)

Figure 16.  Global portrait of the system (Ⅰ) $ (\gamma, c) = (10, 1) $