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Stabilizing blow up solutions to nonlinear schrÖdinger equations

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  • In this paper, we consider the critical nonlinear Schrödinger equations in ${\mathbb{R}^2}$ with an oscillating nonlinearity, in a radial geometry. We numerically investigate the influence of the oscillations on the time of existence for the corresponding solution, on the spirit of the recent result of Cazenave and Scialom. It can be observed that the solution converges to the solution of a limit equation obtained with the weak limit of the oscillatory term, starting either with Gaussian data as well as standing waves solutions.

    Mathematics Subject Classification: 35Q55, 35B44.

    Citation:

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  • 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

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