Figure 1.
The oscillating coefficient $A_\Omega$ over an extended interval
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Semiclassical states for fractional Choquard equations with critical growth
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 |
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 [
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 and Applied Analysis,
2019, 18
(1)
: 539-558.
doi: 10.3934/cpaa.2019027
![]()
References:
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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.
|
| [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.
|
| [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.
|
| [4] |
R. Balakrishan,
Soliton propagation in nonuniform media, Phys. Rev. A, 32 (1985), 1144-1149.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
A. V. Gurevich,
Nonlinear Phonomena in the Ionosphere, Berlin: Springer, 1978. |
| [10] |
J. Holmer and C. Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity Ⅰ: Basic theory, preprint, arXiv: 1510.03491. |
| [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. |
| [12] |
T. Kato,
On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.
|
| [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.
|
| [14] |
E. H. Lieb, R. Seiringer and J. Yngvason,
One-dimensional bosons in three-dimensional traps, Phys. Rev. Lett., 91 (2003), 150401.
|
| [15] |
B. A. Malomed, Nonlinear Schrödinger equations, in Scott Alwyn, Encyclopedia of Nonlinear Science, New York: Routledge, (2005), 639–643. |
| [16] |
M. I. Molina and C. A. Bustamante, The attractive nonlinear delta-function potential, preprint, arXiv: physics/0102053. |
| [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. |
| [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. |
| [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.
|
| [20] |
L. Pitaevskii and S. Stringari,
Bose-Einstein Condensation, International Series of Monographs on Physics, 116. The Clarendon Press, Oxford University Press, Oxford, 2003. |
| [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. |
| [22] |
P. Yeh,
Optical Waves in Layered Media, New York: Wiley, 1988. |
| [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.
|
| [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. |
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.
|
| [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.
|
| [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.
|
| [4] |
R. Balakrishan,
Soliton propagation in nonuniform media, Phys. Rev. A, 32 (1985), 1144-1149.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
A. V. Gurevich,
Nonlinear Phonomena in the Ionosphere, Berlin: Springer, 1978. |
| [10] |
J. Holmer and C. Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity Ⅰ: Basic theory, preprint, arXiv: 1510.03491. |
| [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. |
| [12] |
T. Kato,
On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.
|
| [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.
|
| [14] |
E. H. Lieb, R. Seiringer and J. Yngvason,
One-dimensional bosons in three-dimensional traps, Phys. Rev. Lett., 91 (2003), 150401.
|
| [15] |
B. A. Malomed, Nonlinear Schrödinger equations, in Scott Alwyn, Encyclopedia of Nonlinear Science, New York: Routledge, (2005), 639–643. |
| [16] |
M. I. Molina and C. A. Bustamante, The attractive nonlinear delta-function potential, preprint, arXiv: physics/0102053. |
| [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. |
| [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. |
| [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.
|
| [20] |
L. Pitaevskii and S. Stringari,
Bose-Einstein Condensation, International Series of Monographs on Physics, 116. The Clarendon Press, Oxford University Press, Oxford, 2003. |
| [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. |
| [22] |
P. Yeh,
Optical Waves in Layered Media, New York: Wiley, 1988. |
| [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.
|
| [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. |
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