January  2021, 20(1): 215-242. doi: 10.3934/cpaa.2020264

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

* Corresponding author

Received  April 2020 Revised  August 2020 Published  January 2021 Early access  November 2020

Fund Project: The first author was supported in part by NSF grants DMS-1200455, DMS-1500106. The second author was supported in part by NSF grant DMS-1200455 (PI Justin Holmer)

We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,
$ \begin{equation} i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, \;\;\;\;\;\;(0.1)\end{equation} $
where
$ {\delta} = {\delta}(x) $
is the delta function supported at the origin. In the
$ L^2 $
supercritical setting
$ p>3 $
, we construct self-similar blow-up solutions belonging to the energy space
$ L_x^\infty \cap \dot H_x^1 $
. This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at
$ x = 0 $
imposed by the
$ \delta $
term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case
$ 0<p-3 \ll 1 $
using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.
Citation: Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure and Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
References:
[1]

R. AdamiG. Dell'AntonioR. 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. BuddShaohua 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 CacciapuotiDomenico FincoDiego 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. DyachenkoA. C. NewellA. 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. 

[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.

[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. LandmanG. C. PapanicolaouC. 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 MerlePierre 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. AdamiG. Dell'AntonioR. 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. BuddShaohua 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 CacciapuotiDomenico FincoDiego 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. DyachenkoA. C. NewellA. 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. 

[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.

[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. LandmanG. C. PapanicolaouC. 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 MerlePierre 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. 
Figure 1.  (left) plot of numerical solution of $ \sigma(h) $ versus $ h^{-1} $, (right) plot of numerical solution of $ \log \sigma(h) $ versus $ h^{-1} $ in solid black, together with $ h^{-1}\to \infty $ asymptotic $ \log \sigma(h) = \log 2 -\pi h^{-1} + \log h^{-1} $, from Theorem 1.4, in dashed red, showing good agreement for $ h^{-1}>1 $
Figure 2.  Graph of $ S(\omega) = \omega\sqrt{1-\omega^2}+\arcsin \omega $
Figure 3.  The case of $ 0<\omega<1 $. There are two relevant stationary points $ \zeta_\pm = 2\omega( \omega \pm i \sqrt{1-\omega^2}) $. The contour $ i\mathbb{R}_+ $ is deformed to the contour indicated in gray. It starts by going down on the imaginary axis until it picks up a descent curve corresponding stationary point $ \zeta_- $. At $ \zeta_- $, it follows the curve of ascent which is the curve of descent corresponding to $ \zeta_+ $. At $ \zeta_+ $, it follows the descent curve upward until it reaches the imaginary axis. The contour heat plot shows the value of $ \operatorname{Re} \phi(z) $. Along the chosen contour, it reaches its maximum at $ \zeta_+ $
Figure 4.  The case $ \omega>1 $. There is one relevant stationary point $ \zeta = 2\omega (\omega - \sqrt{\omega^2-1}) $ on the real axis. The contour $ i\mathbb{R}_+ $ is deformed to the contour illustrated in purple. Starting at $ 0 $, it proceeds down the negative imaginary axis until it links up with the descent curve corresponding to the stationary point $ \zeta $. Following this curve upward to the positive imaginary axis, it then transitions to the imaginary axis itself to continue to $ +i \infty $. Along this curve, the value of $ \operatorname{Re} \phi(z) $ reaches its maximum value of $ 0 $ at $ z = \zeta $
Figure 5.  The case $ \omega = 1 $. There is one relevant stationary point $ \zeta = 2 $. It is degenerate: $ \phi'(\zeta) = 0 $ and $ \phi''(\zeta) = 0 $ but $ \phi'''(\zeta)\neq 0 $. Thus there are three descent curves and three ascent curves emanating from the stationary point, alternating and separated from each other by $ 60 $ degrees. The desired contour follows two descent arcs into the stationary point, as indicated. The value of $ \operatorname{Re} \phi(z) $ reaches its maximum value of $ 0 $ at $ z = \zeta $
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