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May  2017, 16(3): 1059-1082. doi: 10.3934/cpaa.2017051

## Stabilizing blow up solutions to nonlinear schrÖdinger equations

 1 LMR EA 4535, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP1039,51687 Reims cedex, France 2 LUTH CNRS UMR 8102, Observatoire de Paris, 5 place Jules Janssen 92195 Meudon cedex, France 3 LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex, France

Received  July 2016 Revised  December 2016 Published  February 2017

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.

Citation: Laurent Di Menza, Olivier Goubet. Stabilizing blow up solutions to nonlinear schrÖdinger equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1059-1082. doi: 10.3934/cpaa.2017051
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##### References:
Profile of the solutions of (5) versus time for $\Omega = 0$, $\Omega = 4$ et $\Omega = +\infty$
Profile of the lifespan $T_\Omega$ of the solution $u^\Omega$ as a function of $\Omega$
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 $q = 1$, $d = 2$
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
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
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
Plot of $\|u(t, .)\|_{L^\infty(\mathbb{R}^2) }$ versus $t$, $\Omega = 0$ (unperturbed case), $7$, $8$, $10$, $40$, Gaussian initial data with $q = 4$
Plot of $\|u(t, .)\|_{L^\infty(\mathbb{R}^2) }$ versus $t$, $\Omega = 100$, $200$, $1000$, $5000$ and limit case, Gaussian initial data with $q = 4$
Plot of $\|u(t, .)\|_{L^\infty(\mathbb{R}^2) }$ versus $t$, $\Omega = 0$ (unperturbed case), $1.4$, $1.43$, $2$, Gaussian initial data with $q = 3$
Plot of $\|u(t, .)\|_{L^\infty(\mathbb{R}^2) }$ versus $t$, $\Omega = 10$, $40$, $100$ and limit case, Gaussian initial data with $q = 3$
Plot of $|u(t, .)|$ at different times ($t = 0$, $1$, $2$, $3$, $4$), $g_1 = 4$
Plot of $|u(t, .)|$ at different times ($t = 0$, $1$, $2$, $3$, $4$), $g_1 = 2$
Plot of $|u(t, .)|$ at different times ($t = 0$, $1$, $2$, $3$, $4$), $g_1 = 1$
Plot of $|u(t, .)|$ at different times ($t = 0$, $1$, $2$, $3$, $4$), $g_1 = \frac12$
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
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
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
Plot of $\|u(t, .)\|_{L^\infty(\mathbb{R}^2) }$ versus $t$, $\Omega = 20$, $40$, $80$, soliton-like initial data
Plot of the mean radius $r_m(t)$ versus $t$, $\Omega = 20$, $40$, $80$, soliton-like initial data
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
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
Plot of $\|u(t, .)\|_{L^\infty(\mathbb{R}^2) }$ versus $t$, $\Omega = 20$, $40$, $80$, Gaussian normalized Cauchy data
Plot of the mean radius $r_m(t)$ versus $t$, $\Omega = 20$, $40$, $80$, Gaussian normalized Cauchy data
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
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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