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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
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Figure 2.  Mollifiers $l$ and $r$
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Figure 3.  Weight function $\varphi$
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Figure 4.  The graph of $m(x)$
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Figure 5.  The graph of $m'(x)$
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Figure 6.  The graph of initial datum $\text{Re}[u_0(x)]$ ($\rho = 0.1$) and $m(x)$
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