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Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles
Brown University, Department of Mathematics, Box 1917,151 Thayer St. Providence, RI 02912, USA |
$ \begin{equation} i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, \;\;\;\;\;\;(0.1)\end{equation} $ |
$ {\delta} = {\delta}(x) $ |
$ L^2 $ |
$ p>3 $ |
$ L_x^\infty \cap \dot H_x^1 $ |
$ x = 0 $ |
$ \delta $ |
$ 0<p-3 \ll 1 $ |
References:
[1] |
R. Adami, G. Dell'Antonio, R. Figari and A. Teta,
Blow-up solutions for the Schrödinger equation in dimension three with a concentrated nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 121-137.
doi: 10.1016/S0294-1449(03)00035-0. |
[2] |
Carl M. Bender and Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4757-3069-2. |
[3] |
Chris J. Budd, Shaohua Chen and Robert D. Russell,
New self-similar solutions of the nonlinear Schrödinger equation with moving mesh computations, J. Comput. Phys., 152 (1999), 756-789.
doi: 10.1006/jcph.1999.6262. |
[4] |
Claudio Cacciapuoti, Domenico Finco, Diego Noja and Alessandro Teta,
The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit, Lett. Math. Phys., 104 (2014), 1557-1570.
doi: 10.1007/s11005-014-0725-y. |
[5] |
S. Dyachenko, A. C. Newell, A. Pushkarev and V. E. Zakharov,
Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation, Phys. D, 57 (1992), 96-160.
doi: 10.1016/0167-2789(92)90090-A. |
[6] |
Gadi Fibich, The Nonlinear Schrödinger Equation, Springer, Cham, 2015.
doi: 10.1007/978-3-319-12748-4. |
[7] |
G. M. Fraǐman, Asymptotic stability of manifold of self-similar solutions in self-focusing, Zh. Èksper. Teoret. Fiz., 88 (1985), 390-400. Google Scholar |
[8] |
Justin Holmer and Chang Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory, J. Math. Anal. Appl., 483 (2020), 20 pp.
doi: 10.1016/j.jmaa.2019.123522. |
[9] |
Justin Holmer and Chang Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity III: $L^2$-critical log log blowup, in preparation. Google Scholar |
[10] |
Russell Johnson and Xing Bin Pan,
On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 763-782.
doi: 10.1017/S030821050003095X. |
[11] |
Nancy Kopell and Michael Landman,
Spatial structure of the focusing singularity of the nonlinear Schrödinger equation: a geometrical analysis, SIAM J. Appl. Math., 55 (1995), 1297-1323.
doi: 10.1137/S0036139994262386. |
[12] |
M. J. Landman, G. C. Papanicolaou, C. Sulem and P. L. Sulem,
Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension, Phys. Rev. A, 38 (1988), 3837-3843.
doi: 10.1103/PhysRevA.38.3837. |
[13] |
F. Merle and P. Raphael,
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal., 13 (2003), 591-642.
doi: 10.1007/s00039-003-0424-9. |
[14] |
Frank Merle and Pierre Raphael,
On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672.
doi: 10.1007/s00222-003-0346-z. |
[15] |
Frank Merle and Pierre Raphael,
The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. Math., 161 (2005), 157-222.
doi: 10.4007/annals.2005.161.157. |
[16] |
Frank Merle and Pierre Raphael,
Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Commun. Math. Phys., 253 (2005), 675-704.
doi: 10.1007/s00220-004-1198-0. |
[17] |
Frank Merle and Pierre Raphael,
On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation, J. Am. Math. Soc., 19 (2006), 37-90.
doi: 10.1090/S0894-0347-05-00499-6. |
[18] |
Frank Merle, Pierre Raphaël and Jeremie Szeftel,
Stable self-similar blow-up dynamics for slightly $L^2$ super-critical NLS equations, Geom. Funct. Anal., 20 (2010), 1028-1071.
doi: 10.1007/s00039-010-0081-8. |
[19] |
Diego Noja and An drea Posilicano,
Wave equations with concentrated nonlinearities, J. Phys. A., 38 (2005), 5011-5022.
doi: 10.1088/0305-4470/38/22/022. |
[20] |
Galina Perelman,
On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, 2 (2001), 605-673.
doi: 10.1007/PL00001048. |
[21] |
Pierre Raphael,
Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann., 331 (2005), 577-609.
doi: 10.1007/s00208-004-0596-0. |
[22] |
S. Yu. Slavyanov, Asymptotic Solutions of The One-dimensional Schrödinger Equation, American Mathematical Society, Providence, RI, 1996.
doi: 10.1090/mmono/151. |
[23] |
C. Sulem and P. L. Sulem,
Focusing nonlinear Schr${\rm{\ddot d}}$inger equation and wave-packet collapse, Nonlinear Anal., 30 (1997), 833-844.
doi: 10.1016/S0362-546X(96)00168-X. |
[24] |
Catherine Sulem and Pierre-Louis Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, New York, 1999. |
[25] |
Elias M. Stein and Rami Shakarchi, Complex Analysis, Princeton University Press, Princeton, NJ, 2003.
![]() |
show all references
References:
[1] |
R. Adami, G. Dell'Antonio, R. Figari and A. Teta,
Blow-up solutions for the Schrödinger equation in dimension three with a concentrated nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 121-137.
doi: 10.1016/S0294-1449(03)00035-0. |
[2] |
Carl M. Bender and Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4757-3069-2. |
[3] |
Chris J. Budd, Shaohua Chen and Robert D. Russell,
New self-similar solutions of the nonlinear Schrödinger equation with moving mesh computations, J. Comput. Phys., 152 (1999), 756-789.
doi: 10.1006/jcph.1999.6262. |
[4] |
Claudio Cacciapuoti, Domenico Finco, Diego Noja and Alessandro Teta,
The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit, Lett. Math. Phys., 104 (2014), 1557-1570.
doi: 10.1007/s11005-014-0725-y. |
[5] |
S. Dyachenko, A. C. Newell, A. Pushkarev and V. E. Zakharov,
Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation, Phys. D, 57 (1992), 96-160.
doi: 10.1016/0167-2789(92)90090-A. |
[6] |
Gadi Fibich, The Nonlinear Schrödinger Equation, Springer, Cham, 2015.
doi: 10.1007/978-3-319-12748-4. |
[7] |
G. M. Fraǐman, Asymptotic stability of manifold of self-similar solutions in self-focusing, Zh. Èksper. Teoret. Fiz., 88 (1985), 390-400. Google Scholar |
[8] |
Justin Holmer and Chang Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory, J. Math. Anal. Appl., 483 (2020), 20 pp.
doi: 10.1016/j.jmaa.2019.123522. |
[9] |
Justin Holmer and Chang Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity III: $L^2$-critical log log blowup, in preparation. Google Scholar |
[10] |
Russell Johnson and Xing Bin Pan,
On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 763-782.
doi: 10.1017/S030821050003095X. |
[11] |
Nancy Kopell and Michael Landman,
Spatial structure of the focusing singularity of the nonlinear Schrödinger equation: a geometrical analysis, SIAM J. Appl. Math., 55 (1995), 1297-1323.
doi: 10.1137/S0036139994262386. |
[12] |
M. J. Landman, G. C. Papanicolaou, C. Sulem and P. L. Sulem,
Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension, Phys. Rev. A, 38 (1988), 3837-3843.
doi: 10.1103/PhysRevA.38.3837. |
[13] |
F. Merle and P. Raphael,
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal., 13 (2003), 591-642.
doi: 10.1007/s00039-003-0424-9. |
[14] |
Frank Merle and Pierre Raphael,
On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672.
doi: 10.1007/s00222-003-0346-z. |
[15] |
Frank Merle and Pierre Raphael,
The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. Math., 161 (2005), 157-222.
doi: 10.4007/annals.2005.161.157. |
[16] |
Frank Merle and Pierre Raphael,
Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Commun. Math. Phys., 253 (2005), 675-704.
doi: 10.1007/s00220-004-1198-0. |
[17] |
Frank Merle and Pierre Raphael,
On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation, J. Am. Math. Soc., 19 (2006), 37-90.
doi: 10.1090/S0894-0347-05-00499-6. |
[18] |
Frank Merle, Pierre Raphaël and Jeremie Szeftel,
Stable self-similar blow-up dynamics for slightly $L^2$ super-critical NLS equations, Geom. Funct. Anal., 20 (2010), 1028-1071.
doi: 10.1007/s00039-010-0081-8. |
[19] |
Diego Noja and An drea Posilicano,
Wave equations with concentrated nonlinearities, J. Phys. A., 38 (2005), 5011-5022.
doi: 10.1088/0305-4470/38/22/022. |
[20] |
Galina Perelman,
On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, 2 (2001), 605-673.
doi: 10.1007/PL00001048. |
[21] |
Pierre Raphael,
Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann., 331 (2005), 577-609.
doi: 10.1007/s00208-004-0596-0. |
[22] |
S. Yu. Slavyanov, Asymptotic Solutions of The One-dimensional Schrödinger Equation, American Mathematical Society, Providence, RI, 1996.
doi: 10.1090/mmono/151. |
[23] |
C. Sulem and P. L. Sulem,
Focusing nonlinear Schr${\rm{\ddot d}}$inger equation and wave-packet collapse, Nonlinear Anal., 30 (1997), 833-844.
doi: 10.1016/S0362-546X(96)00168-X. |
[24] |
Catherine Sulem and Pierre-Louis Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, New York, 1999. |
[25] |
Elias M. Stein and Rami Shakarchi, Complex Analysis, Princeton University Press, Princeton, NJ, 2003.
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