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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
References:
[1]

P. AntonelliJ.-C. Saut and C. Sparber, well-posedness, averagin of NLS with timeperiodic dispersion management, Adv. Diff. Eq., 18 (2013), 49-68. Google Scholar

[2]

A. BabinA. Ilyin and E. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math., 64 (2011), 591-648. doi: 10.1002/cpa.20356. Google Scholar

[3]

M. Barton-SmithA. Debussche and L. Di Menza, Numerical Study of two-dimensional stochastic NLS Equations,, Num. Meth. for PDE's, 21 (2005), 810-842. doi: 10.1002/num.20064. Google Scholar

[4]

C.-H. BruneauL. Di Menza and T. Lehner, Numerical resolution of some nonlinear Schrödinger equations in plasmas, Num. Meth. for PDE's, 15 (1999), 672-696. doi: 10.1002/(SICI)1098-2426(199911)15:6<672::AID-NUM5>3.3.CO;2-A. Google Scholar

[5]

X. CarvajalM. Panthee and M. Scialom, On the critical KDV equation with time-oscillating nonlinearity, Differential Integral Equations, 24 (2011), 541-567. Google Scholar

[6]

T. Cazenave and A. Haraux, Introduction aux problèmes, d'évolution semi-linéaires, Math. et Applications, Ellipses 1990. Google Scholar

[7]

T. Cazenave and M. Scialom, A Schrödinger equation with time-oscillating nonlinearity, Rev. Mat. Complut, 23 (2010), 321-339. doi: 10.1007/s13163-009-0018-7. Google Scholar

[8]

I. Damergi, Equation de Schrödinger non linéaire avecnon-linéarité oscillante, Ph. D Thesis, Ecole Polytechnique de Tunisie, (2014). Google Scholar

[9]

I. Damergi and O. Goubet, Blowup solutions to the nonlinear Schrodinger equation with oscillating nonlinearities, J. of Math. Anal., Appl., 352 (2009), 336-344. doi: 10.1016/j.jmaa.2008.07.079. Google Scholar

[10]

A. Debussche and L. Di Menza, Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Phys. D, 162 (2002), 131-154. doi: 10.1016/S0167-2789(01)00379-7. Google Scholar

[11]

L. Di Menza, Absorbing boundary conditions on a hypersurface for the linear Schrödinger equation, Appl. Math. Lett., 9 (1996), 55-59. doi: 10.1016/0893-9659(96)00051-1. Google Scholar

[12]

L. Di Menza, Numerical computations of solitons in optical systems, M2AN, 3 (2009), 173-208. doi: 10.1051/m2an:2008044. Google Scholar

[13]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equations, J. Math. Phy, 18 (1977), 1794-1797. doi: 10.1063/1.523491. Google Scholar

[14]

V. Konotop and P. Pacciani, On collapse in the nonlinear Schrödinger equation with time dependent nonlinearity. Application to Bose-Einstein condensates, arXiv: cond-mat/0504493 vl 19 apr 2005. doi: 10.1007/10928028_14. Google Scholar

[15]

M. Kunze, Infinitely many radial solutions of a variational problem related to dispersionmanaged optic fibers, Proceedings of the AMS, 131 (2003), 2181-2188. doi: 10.1090/S0002-9939-02-06780-1. Google Scholar

[16]

H. Liu and E. Tadmor, Rotation prevents finite time breakdown, Phys. D, 188 (2004), 262-276. doi: 10.1016/j.physd.2003.07.006. Google Scholar

[17]

G. D. Montesinos and V. M. Pérez-Garcia, Pérez-Garcia, Numerical studies of stabilized Townes solitons, Math. and Computers in Simulation, 69 (2005), 447-456. doi: 10.1016/j.matcom.2005.03.009. Google Scholar

[18]

M. Panthee and M. Scialom, On the supercritical KdV equation with time-oscillating nonlinearity, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1191-1212. doi: 10.1007/s00030-012-0204-z. Google Scholar

[19]

C. Sulem and P. L Sulem, The nonlinear Schröinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. Google Scholar

[20]

V. E. Zakharov, Collapse of Lungmuir waves, Soviet Phys. JETP, vol., 35 (1972), 908-914. Google Scholar

[21]

V. ZharnitskyE. GrenierCh. Jones and S. Turitsyn, Stabilizing effects of dispersion management, Physica D, 152-153 (2001), 794-817. doi: 10.1016/S0167-2789(01)00213-5. Google Scholar

show all references

References:
[1]

P. AntonelliJ.-C. Saut and C. Sparber, well-posedness, averagin of NLS with timeperiodic dispersion management, Adv. Diff. Eq., 18 (2013), 49-68. Google Scholar

[2]

A. BabinA. Ilyin and E. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math., 64 (2011), 591-648. doi: 10.1002/cpa.20356. Google Scholar

[3]

M. Barton-SmithA. Debussche and L. Di Menza, Numerical Study of two-dimensional stochastic NLS Equations,, Num. Meth. for PDE's, 21 (2005), 810-842. doi: 10.1002/num.20064. Google Scholar

[4]

C.-H. BruneauL. Di Menza and T. Lehner, Numerical resolution of some nonlinear Schrödinger equations in plasmas, Num. Meth. for PDE's, 15 (1999), 672-696. doi: 10.1002/(SICI)1098-2426(199911)15:6<672::AID-NUM5>3.3.CO;2-A. Google Scholar

[5]

X. CarvajalM. Panthee and M. Scialom, On the critical KDV equation with time-oscillating nonlinearity, Differential Integral Equations, 24 (2011), 541-567. Google Scholar

[6]

T. Cazenave and A. Haraux, Introduction aux problèmes, d'évolution semi-linéaires, Math. et Applications, Ellipses 1990. Google Scholar

[7]

T. Cazenave and M. Scialom, A Schrödinger equation with time-oscillating nonlinearity, Rev. Mat. Complut, 23 (2010), 321-339. doi: 10.1007/s13163-009-0018-7. Google Scholar

[8]

I. Damergi, Equation de Schrödinger non linéaire avecnon-linéarité oscillante, Ph. D Thesis, Ecole Polytechnique de Tunisie, (2014). Google Scholar

[9]

I. Damergi and O. Goubet, Blowup solutions to the nonlinear Schrodinger equation with oscillating nonlinearities, J. of Math. Anal., Appl., 352 (2009), 336-344. doi: 10.1016/j.jmaa.2008.07.079. Google Scholar

[10]

A. Debussche and L. Di Menza, Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Phys. D, 162 (2002), 131-154. doi: 10.1016/S0167-2789(01)00379-7. Google Scholar

[11]

L. Di Menza, Absorbing boundary conditions on a hypersurface for the linear Schrödinger equation, Appl. Math. Lett., 9 (1996), 55-59. doi: 10.1016/0893-9659(96)00051-1. Google Scholar

[12]

L. Di Menza, Numerical computations of solitons in optical systems, M2AN, 3 (2009), 173-208. doi: 10.1051/m2an:2008044. Google Scholar

[13]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equations, J. Math. Phy, 18 (1977), 1794-1797. doi: 10.1063/1.523491. Google Scholar

[14]

V. Konotop and P. Pacciani, On collapse in the nonlinear Schrödinger equation with time dependent nonlinearity. Application to Bose-Einstein condensates, arXiv: cond-mat/0504493 vl 19 apr 2005. doi: 10.1007/10928028_14. Google Scholar

[15]

M. Kunze, Infinitely many radial solutions of a variational problem related to dispersionmanaged optic fibers, Proceedings of the AMS, 131 (2003), 2181-2188. doi: 10.1090/S0002-9939-02-06780-1. Google Scholar

[16]

H. Liu and E. Tadmor, Rotation prevents finite time breakdown, Phys. D, 188 (2004), 262-276. doi: 10.1016/j.physd.2003.07.006. Google Scholar

[17]

G. D. Montesinos and V. M. Pérez-Garcia, Pérez-Garcia, Numerical studies of stabilized Townes solitons, Math. and Computers in Simulation, 69 (2005), 447-456. doi: 10.1016/j.matcom.2005.03.009. Google Scholar

[18]

M. Panthee and M. Scialom, On the supercritical KdV equation with time-oscillating nonlinearity, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1191-1212. doi: 10.1007/s00030-012-0204-z. Google Scholar

[19]

C. Sulem and P. L Sulem, The nonlinear Schröinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. Google Scholar

[20]

V. E. Zakharov, Collapse of Lungmuir waves, Soviet Phys. JETP, vol., 35 (1972), 908-914. Google Scholar

[21]

V. ZharnitskyE. GrenierCh. Jones and S. Turitsyn, Stabilizing effects of dispersion management, Physica D, 152-153 (2001), 794-817. doi: 10.1016/S0167-2789(01)00213-5. Google Scholar

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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