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

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

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.

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. AlfimovV. 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. KolomeiskyT. 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. LiebR. 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. ParedesA. WideraV. MurgO. MandelS. FöllingI. CiracG. V. ShlyapnikovT. 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. AlfimovV. 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. KolomeiskyT. 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. LiebR. 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. ParedesA. WideraV. MurgO. MandelS. FöllingI. CiracG. V. ShlyapnikovT. 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 2.  Mollifiers $l$ and $r$
Figure 3.  Weight function $\varphi$
Figure 4.  The graph of $m(x)$
Figure 5.  The graph of $m'(x)$
Figure 6.  The graph of initial datum $\text{Re}[u_0(x)]$ ($\rho = 0.1$) and $m(x)$
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