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Numerical simulations of parity–time symmetric nonlinear Schrödinger equations in critical case
Laboratoire de Mathématiques Informatique et Applications, Université des Antilles, BP 250, F-97157 Pointe à Pitre cedex, Guadeloupe FWI |
In this paper, we study the solution behavior of two coupled non–linear Schrödinger equations (CNLS) in the critical case, where one equation includes gain, while the other includes losses. Next, we present two numerical methods for solving the CNLS equations, for which we have made a comparison. These numerical experiments permit to illustrate other theoretical results proven by the authors [
References:
[1] |
G. P. Agrawal, Applications of Nonlinear Fiber Optics, Optics and Photonics Series 2nd edition, Academic Press, Elsevier, (2008). |
[2] |
X. Antoine, C. Besse and S. Descombes,
Artificial boundary conditions for one–dimensional cubic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 43 (2006), 2272-2293.
doi: 10.1137/040606983. |
[3] |
V. A. Baskakov and A. V. Popov,
Implementation of transparent boundaries for numerical solution of the Schrödinger equation, Wave Motion, 14 (1991), 123-128.
doi: 10.1016/0165-2125(91)90053-Q. |
[4] |
C. M. Bender,
Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys., 70 (2007), 947-1018.
doi: 10.1088/0034-4885/70/6/R03. |
[5] |
C. M. Bender, B. Berntson, D. Parker and E. Samuel, Observation of PT phase transition in a simple mechanical system, Am. J. Phys., 81 (2013), 173.
doi: 10.1119/1.4789549. |
[6] |
C. M. Bender and S. Borttcher,
Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}-$symmetry, Phys. Rev. Lett., 80 (1998), 5243-5246.
doi: 10.1103/PhysRevLett.80.5243. |
[7] |
E. Destyl, S. P. Nuiro and P. Poullet,
On the global behavior of solutions of a coupled system of nonlinear Schrödinger equation, Stud. Appl. Math., 138 (2017), 227-244.
doi: 10.1111/sapm.12150. |
[8] |
E. Destyl, S. P. Nuiro, D. E. Pelinovsky and P. Poullet,
Coupled pendula chains under parametric $\mathcal{PT}$-symmetric driving force, Phys. Lett. A, 381 (2017), 3884-3892.
doi: 10.1016/j.physleta.2017.10.021. |
[9] |
E. Destyl, S. P. Nuiro and P. Poullet,
Critical blow up in coupled Parity-Time-symmetric nonlinear Schrödinger equations, AIMS Math., 2 (2017), 195-206.
|
[10] |
E. Destyl, Modélisation et Analyse de Systèmes D'équations de Schrödinger non Linéaire, Thèse de doctorat de l'Universté des Antilles, 28 septembre 2018, Pointe-à-Pitre, Guadeloupe. |
[11] |
J.-P. Dias, M. Figueira, V. V. Konotop and D. A. Zezyulin,
Supercritical blow up in coupled parity-time-symmetric nonlinear Schrödinger equations, Stud. Appl. Math., 133 (2014), 422-440.
doi: 10.1111/sapm.12063. |
[12] |
J. P. Dias, M. M. Figueira and V. V. Konotop,
The Cauchy problem for coupled nonlinear Schrödinger equations with linear damping: local and global existence and blow up of solutions, Chin. Ann. Math., 37 (2016), 665-682.
doi: 10.1007/s11401-016-1006-0. |
[13] |
L. Di Menza,
Transparent and absorbing boundary conditions for the Schrödinger equation in a bounded domain, (English summary), Numer. Funct. Anal. Optim., 18 (1997), 759-775.
doi: 10.1080/01630569708816790. |
[14] |
R. Driben and B. A. Malomed,
Stability of solitons in parity-time-symmetric couplers, Opt. Lett., 36 (2011), 4323-4325.
doi: 10.1364/OL.36.004323. |
[15] |
M. S. Ismail and T. R. Taha,
Numerical simulation of coupled nonlinear Schödinger equation, Math. Comput. Simul., 56 (2001), 547-562.
doi: 10.1016/S0378-4754(01)00324-X. |
[16] |
A. Jüngel and R.-M. Weishäupl,
Blow up in two-component nonlinear Schrödinger systems with an external driven field, Math. Models Methods Appl. Sci., 23 (2013), 1699-1727.
doi: 10.1142/S0218202513500206. |
[17] |
V. V. Konotop, J. Yang and D. A. Zezyulin, Nonlinear waves in PT–symmetric system, Rev. Mod., 88 (2016), 035002(59). |
[18] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Springer, LLC, 2009. |
[19] |
D. E. Pelinovsky, D. A. Zezyulin and V. V. Konotop,
Global existence of solutions to coupled PT–symmetric nonlinear Schrodiger equations, Int. J. Theor. Phys., 54 (2015), 3920-3931.
doi: 10.1007/s10773-014-2422-0. |
[20] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer, New-York, 1999. |
[21] |
T. R. Taha and M. J. Ablowitz,
Analytical and numerical aspects of certain nonlinear evolution equations II. Numerical nonlinear Schrödinger equation, J. Comp. Phys., 55 (2006), 203-230.
doi: 10.1016/0021-9991(84)90003-2. |
[22] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576.
|
[23] |
C. Zheng,
A perfectly matched layer approach to the nonlinear Schrödinger wave equations, J. Comput. Phys., 227 (2007), 537-556.
doi: 10.1016/j.jcp.2007.08.004. |
show all references
References:
[1] |
G. P. Agrawal, Applications of Nonlinear Fiber Optics, Optics and Photonics Series 2nd edition, Academic Press, Elsevier, (2008). |
[2] |
X. Antoine, C. Besse and S. Descombes,
Artificial boundary conditions for one–dimensional cubic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 43 (2006), 2272-2293.
doi: 10.1137/040606983. |
[3] |
V. A. Baskakov and A. V. Popov,
Implementation of transparent boundaries for numerical solution of the Schrödinger equation, Wave Motion, 14 (1991), 123-128.
doi: 10.1016/0165-2125(91)90053-Q. |
[4] |
C. M. Bender,
Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys., 70 (2007), 947-1018.
doi: 10.1088/0034-4885/70/6/R03. |
[5] |
C. M. Bender, B. Berntson, D. Parker and E. Samuel, Observation of PT phase transition in a simple mechanical system, Am. J. Phys., 81 (2013), 173.
doi: 10.1119/1.4789549. |
[6] |
C. M. Bender and S. Borttcher,
Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}-$symmetry, Phys. Rev. Lett., 80 (1998), 5243-5246.
doi: 10.1103/PhysRevLett.80.5243. |
[7] |
E. Destyl, S. P. Nuiro and P. Poullet,
On the global behavior of solutions of a coupled system of nonlinear Schrödinger equation, Stud. Appl. Math., 138 (2017), 227-244.
doi: 10.1111/sapm.12150. |
[8] |
E. Destyl, S. P. Nuiro, D. E. Pelinovsky and P. Poullet,
Coupled pendula chains under parametric $\mathcal{PT}$-symmetric driving force, Phys. Lett. A, 381 (2017), 3884-3892.
doi: 10.1016/j.physleta.2017.10.021. |
[9] |
E. Destyl, S. P. Nuiro and P. Poullet,
Critical blow up in coupled Parity-Time-symmetric nonlinear Schrödinger equations, AIMS Math., 2 (2017), 195-206.
|
[10] |
E. Destyl, Modélisation et Analyse de Systèmes D'équations de Schrödinger non Linéaire, Thèse de doctorat de l'Universté des Antilles, 28 septembre 2018, Pointe-à-Pitre, Guadeloupe. |
[11] |
J.-P. Dias, M. Figueira, V. V. Konotop and D. A. Zezyulin,
Supercritical blow up in coupled parity-time-symmetric nonlinear Schrödinger equations, Stud. Appl. Math., 133 (2014), 422-440.
doi: 10.1111/sapm.12063. |
[12] |
J. P. Dias, M. M. Figueira and V. V. Konotop,
The Cauchy problem for coupled nonlinear Schrödinger equations with linear damping: local and global existence and blow up of solutions, Chin. Ann. Math., 37 (2016), 665-682.
doi: 10.1007/s11401-016-1006-0. |
[13] |
L. Di Menza,
Transparent and absorbing boundary conditions for the Schrödinger equation in a bounded domain, (English summary), Numer. Funct. Anal. Optim., 18 (1997), 759-775.
doi: 10.1080/01630569708816790. |
[14] |
R. Driben and B. A. Malomed,
Stability of solitons in parity-time-symmetric couplers, Opt. Lett., 36 (2011), 4323-4325.
doi: 10.1364/OL.36.004323. |
[15] |
M. S. Ismail and T. R. Taha,
Numerical simulation of coupled nonlinear Schödinger equation, Math. Comput. Simul., 56 (2001), 547-562.
doi: 10.1016/S0378-4754(01)00324-X. |
[16] |
A. Jüngel and R.-M. Weishäupl,
Blow up in two-component nonlinear Schrödinger systems with an external driven field, Math. Models Methods Appl. Sci., 23 (2013), 1699-1727.
doi: 10.1142/S0218202513500206. |
[17] |
V. V. Konotop, J. Yang and D. A. Zezyulin, Nonlinear waves in PT–symmetric system, Rev. Mod., 88 (2016), 035002(59). |
[18] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Springer, LLC, 2009. |
[19] |
D. E. Pelinovsky, D. A. Zezyulin and V. V. Konotop,
Global existence of solutions to coupled PT–symmetric nonlinear Schrodiger equations, Int. J. Theor. Phys., 54 (2015), 3920-3931.
doi: 10.1007/s10773-014-2422-0. |
[20] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer, New-York, 1999. |
[21] |
T. R. Taha and M. J. Ablowitz,
Analytical and numerical aspects of certain nonlinear evolution equations II. Numerical nonlinear Schrödinger equation, J. Comp. Phys., 55 (2006), 203-230.
doi: 10.1016/0021-9991(84)90003-2. |
[22] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576.
|
[23] |
C. Zheng,
A perfectly matched layer approach to the nonlinear Schrödinger wave equations, J. Comput. Phys., 227 (2007), 537-556.
doi: 10.1016/j.jcp.2007.08.004. |













Spatial step | Error | Order | Spatial step | Error | Order | ||
2.32 | 2.32 | ||||||
2.07 | 2.07 | ||||||
2.01 | 2.01 | ||||||
(a) Results for the 1st equation | (b) Results for the 2nd equation |
Spatial step | Error | Order | Spatial step | Error | Order | ||
2.32 | 2.32 | ||||||
2.07 | 2.07 | ||||||
2.01 | 2.01 | ||||||
(a) Results for the 1st equation | (b) Results for the 2nd equation |
Time step | Error | Order | Time step | Error | Order | ||
|
|||||||
1.62 |
|
1.62 | |||||
1.30 |
|
1.30 | |||||
4.22 |
|
1.66 | |||||
(a) Results for the 1st equation | (b) Results for the 2nd equation |
Time step | Error | Order | Time step | Error | Order | ||
|
|||||||
1.62 |
|
1.62 | |||||
1.30 |
|
1.30 | |||||
4.22 |
|
1.66 | |||||
(a) Results for the 1st equation | (b) Results for the 2nd equation |
|
Time step | Error | Order |
|
Time step | Error | Order |
|
|
||||||
|
2.06 |
|
1.97 | ||||
|
2.06 |
|
1.99 | ||||
|
2.17 |
|
2.140 | ||||
(a) Results for the 1st equation | (b) Results for the 2nd equation |
|
Time step | Error | Order |
|
Time step | Error | Order |
|
|
||||||
|
2.06 |
|
1.97 | ||||
|
2.06 |
|
1.99 | ||||
|
2.17 |
|
2.140 | ||||
(a) Results for the 1st equation | (b) Results for the 2nd equation |
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