Stabilizing blow up solutions to nonlinear schrÖdinger equations

Figure 1.  Profile of the solutions of (5) versus time for $ \Omega = 0 $, $ \Omega = 4 $ et $ \Omega = +\infty $

Figure 2.  Profile of the lifespan $ T_\Omega $ of the solution $ u^\Omega $ as a function of $ \Omega $

Figure 3.  Profiles of $ |u| $ versus $ x $ at final time computed with Dirichlet conditions (dot-dashed) and with transparent condition (continuous) starting from Gaussian data (dotted), linear case, amplitude $ q = 1 $, $ d = 2 $

Figure 4.  Profiles of $ |u| $ versus $ x $ and final time; computations performed with transparent conditions(up) and with Dirichlet condition (down) starting from the stationary state with mass excess $ \varepsilon = 0.2 $, one-dimensional case

Figure 5.  Profile of the solution amplitude at the origin versus time for $ t\in [80,100] $; computations performed with transparent conditions(continuous) and with Dirichlet condition (dashed) starting from the stationary state with mass excess $ \varepsilon = 0.2 $, one-dimensional case

Figure 6.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $ for different values of space and time steps. Convergence to the exact global solution $ e^{i\mu t} R_\mu(x) $ is observed

Figure 7.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $, $ \Omega = 0 $ (unperturbed case), $ 7 $, $ 8 $, $ 10 $, $ 40 $, Gaussian initial data with $ q = 4 $

Figure 8.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $, $ \Omega = 100 $, $ 200 $, $ 1000 $, $ 5000 $ and limit case, Gaussian initial data with $ q = 4 $

Figure 9.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $, $ \Omega = 0 $ (unperturbed case), $ 1.4 $, $ 1.43 $, $ 2 $, Gaussian initial data with $ q = 3 $

Figure 10.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $, $ \Omega = 10 $, $ 40 $, $ 100 $ and limit case, Gaussian initial data with $ q = 3 $

Figure 11.  Plot of $ |u(t, .)| $ at different times ($ t = 0 $, $ 1 $, $ 2 $, $ 3 $, $ 4 $), $ g_1 = 4 $

Figure 12.  Plot of $ |u(t, .)| $ at different times ($ t = 0 $, $ 1 $, $ 2 $, $ 3 $, $ 4 $), $ g_1 = 2 $

Figure 13.  Plot of $ |u(t, .)| $ at different times ($ t = 0 $, $ 1 $, $ 2 $, $ 3 $, $ 4 $), $ g_1 = 1 $

Figure 14.  Plot of $ |u(t, .)| $ at different times ($ t = 0 $, $ 1 $, $ 2 $, $ 3 $, $ 4 $), $ g_1 = \frac12 $

Figure 15.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $, $ \Omega = 0 $, $ 3 $, $ 5 $, $ 20 $, $ 40 $ and comparison with the linear case, Gaussian initial data

Figure 16.  Plot of the mean radius $ r_m(t) $ versus $ t $, $ \Omega = 0 $, $ 3 $, $ 5 $, $ 20 $, $ 40 $ and comparison with the linear case, Gaussian initial data

Figure 17.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $, $ \Omega = 0 $, $ 0.2 $, $ 0.5 $, $ 1 $, $ 20 $ and comparison with the linear case, soliton-like initial data

Figure 18.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $, $ \Omega = 20 $, $ 40 $, $ 80 $, soliton-like initial data

Figure 19.  Plot of the mean radius $ r_m(t) $ versus $ t $, $ \Omega = 20 $, $ 40 $, $ 80 $, soliton-like initial data

Figure 20.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $, $ \Omega = 40 $, $ 70 $, $ 100 $, $ 200 $ and comparison with the limit nonlinear case, soliton-like initial data

Figure 21.  Plot of the mean radius $ r_m(t) $ versus $ t $, $ \Omega = 40 $, $ 70 $, $ 100 $, $ 200 $ and comparison with the limit nonlinear case, soliton-like initial data

Figure 22.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $, $ \Omega = 20 $, $ 40 $, $ 80 $, Gaussian normalized Cauchy data

Figure 23.  Plot of the mean radius $ r_m(t) $ versus $ t $, $ \Omega = 20 $, $ 40 $, $ 80 $, Gaussian normalized Cauchy data

Figure 24.  Plot of $ \|u(t, .)\|_{L^\infty(\mathbb{R}^2) } $ versus $ t $ at small times, $ \Omega = 40 $, $ 80 $, $ 150 $, $ 300 $ and comparison with the limit nonlinear case, normalized Gaussian initial data

Figure 25.  Plot of the mean radius $ r_m(t) $ versus $ t $ at small times, $ \Omega = 40 $, $ 80 $, $ 150 $, $ 300 $ and comparison with the limit nonlinear case, normalized Gaussian initial data