American Institute of Mathematical Sciences

Advanced
January  2019, 18(1): 539-558. doi: 10.3934/cpaa.2019027

Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities

Türker Özsarı

Department of Mathematics, Izmir Institute of Technology, Urla, Izmir 35430, Turkey

Received  December 2017 Revised  March 2018 Published  August 2018

Fund Project: The author is supported by Izmir Institute of Technology's BAP Grant 2015IYTE43.

The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly supported or rapidly decaying weight function in virial identities - an idea borrowed from [18]. At the end of the paper, a numerical example satisfying the theory is provided.

Keywords: Blow-up, nonlinear Schrödinger equations, oscillating nonlinearities, infinite momentum, nonlinear boundary conditions.
Mathematics Subject Classification: Primary: 35B44, 35Q41, 35Q55; Secondary: 35B45.
Citation: Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027
References:
[1]

F. K. Abdullaev and M. Salerno, Gap-Townes solitons and localized excitations in low-dimensional Bose-Einstein condensates in optical lattices, Phys. Rev. A, 72 (2005), 033617.   Google Scholar

[2]

A. S. Ackleh and K. Deng, On the critical exponent for the Schrödinger equation with a nonlinear boundary condition, Differential Integral Equations, 17 (2004), 1293-1307.   Google Scholar

[3]

G. L. Alfimov, V. V. Konotop and P. Pacciani, Stationary localized modes of the quintic nonlinear Schrodinger equation with a periodic potential, Phys. Rev. A, 75 (2007), 023624.   Google Scholar

[4]

R. Balakrishan, Soliton propagation in nonuniform media, Phys. Rev. A, 32 (1985), 1144-1149.   Google Scholar

[5]

A. Batal and T. Özsarı, Nonlinear Schrödinger equation on the half-line with nonlinear boundary condition, Electron. J. Differential Equations, Paper No. 222 (2016), 20 pp.  Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[7]

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

[8]

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

[9]

A. V. Gurevich, Nonlinear Phonomena in the Ionosphere, Berlin: Springer, 1978. Google Scholar

[10]

J. Holmer and C. Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity Ⅰ: Basic theory, preprint, arXiv: 1510.03491. Google Scholar

[11]

V. K. Kalantarov and T. Özsarı, Qualitative properties of solutions for nonlinear Schrödinger equations with nonlinear boundary conditions on the half-line, J. Math. Phys., 18 (2016), 021511.  doi: 10.1063/1.4941459.  Google Scholar

[12]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.   Google Scholar

[13]

E. B. Kolomeisky, T. J. Newman and J. P. Straley, Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation, Phys. Rev. Lett., 85 (2000), 1146.   Google Scholar

[14]

E. H. Lieb, R. Seiringer and J. Yngvason, One-dimensional bosons in three-dimensional traps, Phys. Rev. Lett., 91 (2003), 150401.   Google Scholar

[15]

B. A. Malomed, Nonlinear Schrödinger equations, in Scott Alwyn, Encyclopedia of Nonlinear Science, New York: Routledge, (2005), 639–643.  Google Scholar

[16]

M. I. Molina and C. A. Bustamante, The attractive nonlinear delta-function potential, preprint, arXiv: physics/0102053. Google Scholar

[17]

T. Ogawa and Y. Tsutsumi, Blow-up of $ H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[18]

T. Ogawa and Y. Tsutsumi, Blow-up of $ H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.  doi: 10.2307/2048340.  Google Scholar

[19]

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch and I. Bloch, Tonks-Girardeau gas of ultracold atoms in an optical lattice, Nature, 249 (2004), 277-281.   Google Scholar

[20]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, 116. The Clarendon Press, Oxford University Press, Oxford, 2003.  Google Scholar

[21]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Series in Mathematical Sciences, Volume 139, Springer-Verlag, 1999.  Google Scholar

[22]

P. Yeh, Optical Waves in Layered Media, New York: Wiley, 1988. Google Scholar

[23]

V. E. Zakharov and A. B. Shabat, Exact theory of two dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69.   Google Scholar

[24]

J. Zhang and S. Zhu, Blow-up profile to solutions of NLS with oscillating nonlinearities, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 219-234.  doi: 10.1007/s00030-011-0125-2.  Google Scholar

show all references

References:
[1]

F. K. Abdullaev and M. Salerno, Gap-Townes solitons and localized excitations in low-dimensional Bose-Einstein condensates in optical lattices, Phys. Rev. A, 72 (2005), 033617.   Google Scholar

[2]

A. S. Ackleh and K. Deng, On the critical exponent for the Schrödinger equation with a nonlinear boundary condition, Differential Integral Equations, 17 (2004), 1293-1307.   Google Scholar

[3]

G. L. Alfimov, V. V. Konotop and P. Pacciani, Stationary localized modes of the quintic nonlinear Schrodinger equation with a periodic potential, Phys. Rev. A, 75 (2007), 023624.   Google Scholar

[4]

R. Balakrishan, Soliton propagation in nonuniform media, Phys. Rev. A, 32 (1985), 1144-1149.   Google Scholar

[5]

A. Batal and T. Özsarı, Nonlinear Schrödinger equation on the half-line with nonlinear boundary condition, Electron. J. Differential Equations, Paper No. 222 (2016), 20 pp.  Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[7]

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

[8]

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

[9]

A. V. Gurevich, Nonlinear Phonomena in the Ionosphere, Berlin: Springer, 1978. Google Scholar

[10]

J. Holmer and C. Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity Ⅰ: Basic theory, preprint, arXiv: 1510.03491. Google Scholar

[11]

V. K. Kalantarov and T. Özsarı, Qualitative properties of solutions for nonlinear Schrödinger equations with nonlinear boundary conditions on the half-line, J. Math. Phys., 18 (2016), 021511.  doi: 10.1063/1.4941459.  Google Scholar

[12]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.   Google Scholar

[13]

E. B. Kolomeisky, T. J. Newman and J. P. Straley, Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation, Phys. Rev. Lett., 85 (2000), 1146.   Google Scholar

[14]

E. H. Lieb, R. Seiringer and J. Yngvason, One-dimensional bosons in three-dimensional traps, Phys. Rev. Lett., 91 (2003), 150401.   Google Scholar

[15]

B. A. Malomed, Nonlinear Schrödinger equations, in Scott Alwyn, Encyclopedia of Nonlinear Science, New York: Routledge, (2005), 639–643.  Google Scholar

[16]

M. I. Molina and C. A. Bustamante, The attractive nonlinear delta-function potential, preprint, arXiv: physics/0102053. Google Scholar

[17]

T. Ogawa and Y. Tsutsumi, Blow-up of $ H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[18]

T. Ogawa and Y. Tsutsumi, Blow-up of $ H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.  doi: 10.2307/2048340.  Google Scholar

[19]

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch and I. Bloch, Tonks-Girardeau gas of ultracold atoms in an optical lattice, Nature, 249 (2004), 277-281.   Google Scholar

[20]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, 116. The Clarendon Press, Oxford University Press, Oxford, 2003.  Google Scholar

[21]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Series in Mathematical Sciences, Volume 139, Springer-Verlag, 1999.  Google Scholar

[22]

P. Yeh, Optical Waves in Layered Media, New York: Wiley, 1988. Google Scholar

[23]

V. E. Zakharov and A. B. Shabat, Exact theory of two dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69.   Google Scholar

[24]

J. Zhang and S. Zhu, Blow-up profile to solutions of NLS with oscillating nonlinearities, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 219-234.  doi: 10.1007/s00030-011-0125-2.  Google Scholar

Figure 1.  The oscillating coefficient $A_\Omega$ over an extended interval
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Mollifiers $l$ and $r$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Weight function $\varphi$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The graph of $m(x)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The graph of $m'(x)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The graph of initial datum $\text{Re}[u_0(x)]$ ($\rho = 0.1$) and $m(x)$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194
[2]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032
[3]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024
[4]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[5]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[6]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060
[7]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030
[8]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013
[9]

Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021100
[10]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[11]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021
[12]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028
[13]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031
[14]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019
[15]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011
[16]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030
[17]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376
[18]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047
[19]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2725-3737. doi: 10.3934/dcds.2020383
[20]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (163)
  • HTML views (180)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences