-
Previous Article
A kinetic theory description of liquid menisci at the microscale
- KRM Home
- This Issue
-
Next Article
Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds
Numerical methods for a class of generalized nonlinear Schrödinger equations
1. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany, Germany |
2. | Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius, Lithuania |
References:
[1] |
M. J. Ablowitz and T. P. Horikis, Solitons and spectral renormalization methods in nonlinear optics,, Eur. Phys. J. Special Topics, 173 (2009), 147.
doi: 10.1140/epjst/e2009-01072-0. |
[2] |
G. P. Agrawal, Nonlinear Fiber Optics,, $4^{th}$ edition, (2007). Google Scholar |
[3] |
N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams,, Chapman and Hall, (1997). Google Scholar |
[4] |
N. Akhmediev and M. Karlsson, Cherenkov radiation emitted by solitons in optical fibers,, Phys. Rev. A, 51 (1995), 2602.
doi: 10.1103/PhysRevA.51.2602. |
[5] |
G. D. Akrivis and V. D. Dougalis, On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation,, RAIRO Model. Math. Anal. Numer., 25 (1991), 643.
|
[6] |
G. D. Akrivis, V. D. Dougalis and V. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation,, Numer. Math., 59 (1991), 31.
doi: 10.1007/BF01385769. |
[7] |
Sh. Amiranashvili, U. Bandelow and A. Mielke, Padé approximant for refractive index and nonlocal envelope equations,, Opt. Comm., 283 (2010), 480.
doi: 10.1016/j.optcom.2009.10.034. |
[8] |
Sh. Amiranashvili and A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses,, Phys. Rev. A, 82 (2010).
doi: 10.1103/PhysRevA.82.013812. |
[9] |
Sh. Amiranashvili and A. Demircan, Ultrashort optical pulse propagation in terms of analytic signal,, Advances in Optical Technologies, 2011 (2011).
doi: 10.1155/2011/989515. |
[10] |
X. Antoine, A. Arnold, Ch. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,, Commun. Comput. Physics, 4 (2008), 729.
|
[11] |
X. Antoine, W. Bao and C. Besse, Computational methods for the dynamics of nonlinear schrödinger/gross-pitaevskii equations,, Comput. Phys. Commun., 184 (2013), 2621.
doi: 10.1016/j.cpc.2013.07.012. |
[12] |
W. Bao, Sh. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes,, SIAM J. Sci. Comput., 25 (2003), 27.
doi: 10.1137/S1064827501393253. |
[13] |
W. Bao, Q. Tang and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear schrödinger equation,, J. Comput. Phys., 235 (2013), 423.
doi: 10.1016/j.jcp.2012.10.054. |
[14] |
M. Bass, E. W. Van Stryland, D. R. Williams and W. L. Wolfe, editors, Handbook of Optics, Volume 1,, $2^{nd}$ edition, (1995). Google Scholar |
[15] |
A. Borzi and E. Decker, Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation,, J. Comput. Appl. Math., 193 (2006), 65.
doi: 10.1016/j.cam.2005.04.066. |
[16] |
R. W. Boyd, Nonlinear Optics,, $3^rd$ edition, (2008).
|
[17] |
Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear schrödinger equation,, J. Comput. Phys., 148 (1999), 397.
doi: 10.1006/jcph.1998.6120. |
[18] |
R. Čiegis, I. Laukaitytė and M. Radziunas, Numerical algorithms for Schrödinger equations with artificial boundary conditions,, Numer. Funct. Anal. Optim., 30 (2009), 903.
doi: 10.1080/01630560903393097. |
[19] |
R. Čiegis and M. Radziunas, Effective numerical integration of traveling wave model for edge-emitting broad-area semiconductor lasers and amplifiers,, Math. Model. Anal., 15 (2010), 409.
doi: 10.3846/1392-6292.2010.15.409-430. |
[20] |
A. Demircan, Sh. Amiranashvili, C. Brée and G. Steinmeyer, Compressible octave spanning supercontinuum generation by two-pulse collisions,, Phys. Rev. Lett., 110 (2013).
doi: 10.1103/PhysRevLett.110.233901. |
[21] |
A. Demircan, Sh. Amiranashvili and G. Steinmeyer, Controlling light by light with an optical event horizon,, Phys. Rev. Lett., 106 (2011).
doi: 10.1103/PhysRevLett.106.163901. |
[22] |
J. M. Dudley, G. Genty and S. Coen, Supercontinuum generation in photonic crystal fiber,, Rev. Mod. Phys., 78 (2006), 1135.
doi: 10.1103/RevModPhys.78.1135. |
[23] |
M. D. Feit, J. A. Fleck and A. Steiger, Solution of the Schrödinger equation by a spectral method,, J. Comput. Phys., 47 (1982), 412.
doi: 10.1016/0021-9991(82)90091-2. |
[24] |
A. Hasegawa, Optical Solitons in Fibers,, Springer, (1980). Google Scholar |
[25] |
W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Difusion-Reaction Equations,, Springer Series in Computational Mathematics, (2003).
doi: 10.1007/978-3-662-09017-6. |
[26] |
A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics,, Applied Mathematical Sciences, (1992).
doi: 10.1007/978-1-4757-2184-3. |
[27] |
C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations,, Math. Comput., 77 (2008), 2141.
doi: 10.1090/S0025-5718-08-02101-7. |
[28] |
G. Muslu and H. Erbay, Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation,, Mathematics and Computers in Simulation, 67 (2005), 581.
doi: 10.1016/j.matcom.2004.08.002. |
[29] |
Ch. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential,, BIT Numer. Math., 49 (2009), 199.
doi: 10.1007/s10543-009-0215-2. |
[30] |
D. Pathria and J. L. Morris, Exact solutions for a generalized nonlinear Schrödinger equation,, Phys. Scripta, 39 (1989), 673.
doi: 10.1088/0031-8949/39/6/001. |
[31] |
D. Pathria and J. L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations,, J. Comput. Phys., 87 (1990), 108.
doi: 10.1016/0021-9991(90)90228-S. |
[32] |
M. Radziunas, R. Čiegis and A. Mirinavičius, On compact high order finite difference schemes for linear Schrödinger problem on non-uniform meshes,, Int. Journal of Num. Anal. Model., 11 (2014), 303.
|
[33] |
J. M. Sanz-Serna and J. G. Verwer, Conservative and non-conservative schemes for the solution of the nonlinear Schrödinger equation,, IMA J. Numer. Anal., 6 (1986), 25.
doi: 10.1093/imanum/6.1.25. |
[34] |
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems,, Applied Mathematics and Mathematical Computation, (1994).
|
[35] |
J. M. Stone and J. C. Knight, Visibly white light generation in uniform photonic crystal fiber using a microchip laser,, Opt. Express, 16 (2008), 2670.
doi: 10.1364/OE.16.002670. |
[36] |
G. Strang, On the construction and comparison of difference schemes,, SIAM J. Numer. Anal., 5 (1968), 506.
doi: 10.1137/0705041. |
[37] |
T. R. Taha and M. J. Ablovitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical Schrödinger equation,, J. Comp. Phys., 55 (1984), 203.
doi: 10.1016/0021-9991(84)90003-2. |
[38] |
M. Thalhammer, M. Caliari and Ch. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods,, J. Comput. Phys., 228 (2009), 822.
doi: 10.1016/j.jcp.2008.10.008. |
[39] |
D. E. Vakman and L. A. Vainshtein, Amplitude, phase, frequency - fundamental concepts of oscillation theory,, Usp. Fiz. Nauk, 20 (1977), 1002.
doi: 10.1070/PU1977v020n12ABEH005479. |
[40] |
P. K. A. Wai, H. H. Chen and Y. C. Lee, Radiations by "solitons" at the zero group-dispersion wavelength of single-mode optical fibers,, Phys. Rev. A, 41 (1990), 426.
doi: 10.1103/PhysRevA.41.426. |
[41] |
V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,, Sov. Phys. JETP, 34 (1972), 62.
|
show all references
References:
[1] |
M. J. Ablowitz and T. P. Horikis, Solitons and spectral renormalization methods in nonlinear optics,, Eur. Phys. J. Special Topics, 173 (2009), 147.
doi: 10.1140/epjst/e2009-01072-0. |
[2] |
G. P. Agrawal, Nonlinear Fiber Optics,, $4^{th}$ edition, (2007). Google Scholar |
[3] |
N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams,, Chapman and Hall, (1997). Google Scholar |
[4] |
N. Akhmediev and M. Karlsson, Cherenkov radiation emitted by solitons in optical fibers,, Phys. Rev. A, 51 (1995), 2602.
doi: 10.1103/PhysRevA.51.2602. |
[5] |
G. D. Akrivis and V. D. Dougalis, On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation,, RAIRO Model. Math. Anal. Numer., 25 (1991), 643.
|
[6] |
G. D. Akrivis, V. D. Dougalis and V. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation,, Numer. Math., 59 (1991), 31.
doi: 10.1007/BF01385769. |
[7] |
Sh. Amiranashvili, U. Bandelow and A. Mielke, Padé approximant for refractive index and nonlocal envelope equations,, Opt. Comm., 283 (2010), 480.
doi: 10.1016/j.optcom.2009.10.034. |
[8] |
Sh. Amiranashvili and A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses,, Phys. Rev. A, 82 (2010).
doi: 10.1103/PhysRevA.82.013812. |
[9] |
Sh. Amiranashvili and A. Demircan, Ultrashort optical pulse propagation in terms of analytic signal,, Advances in Optical Technologies, 2011 (2011).
doi: 10.1155/2011/989515. |
[10] |
X. Antoine, A. Arnold, Ch. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,, Commun. Comput. Physics, 4 (2008), 729.
|
[11] |
X. Antoine, W. Bao and C. Besse, Computational methods for the dynamics of nonlinear schrödinger/gross-pitaevskii equations,, Comput. Phys. Commun., 184 (2013), 2621.
doi: 10.1016/j.cpc.2013.07.012. |
[12] |
W. Bao, Sh. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes,, SIAM J. Sci. Comput., 25 (2003), 27.
doi: 10.1137/S1064827501393253. |
[13] |
W. Bao, Q. Tang and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear schrödinger equation,, J. Comput. Phys., 235 (2013), 423.
doi: 10.1016/j.jcp.2012.10.054. |
[14] |
M. Bass, E. W. Van Stryland, D. R. Williams and W. L. Wolfe, editors, Handbook of Optics, Volume 1,, $2^{nd}$ edition, (1995). Google Scholar |
[15] |
A. Borzi and E. Decker, Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation,, J. Comput. Appl. Math., 193 (2006), 65.
doi: 10.1016/j.cam.2005.04.066. |
[16] |
R. W. Boyd, Nonlinear Optics,, $3^rd$ edition, (2008).
|
[17] |
Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear schrödinger equation,, J. Comput. Phys., 148 (1999), 397.
doi: 10.1006/jcph.1998.6120. |
[18] |
R. Čiegis, I. Laukaitytė and M. Radziunas, Numerical algorithms for Schrödinger equations with artificial boundary conditions,, Numer. Funct. Anal. Optim., 30 (2009), 903.
doi: 10.1080/01630560903393097. |
[19] |
R. Čiegis and M. Radziunas, Effective numerical integration of traveling wave model for edge-emitting broad-area semiconductor lasers and amplifiers,, Math. Model. Anal., 15 (2010), 409.
doi: 10.3846/1392-6292.2010.15.409-430. |
[20] |
A. Demircan, Sh. Amiranashvili, C. Brée and G. Steinmeyer, Compressible octave spanning supercontinuum generation by two-pulse collisions,, Phys. Rev. Lett., 110 (2013).
doi: 10.1103/PhysRevLett.110.233901. |
[21] |
A. Demircan, Sh. Amiranashvili and G. Steinmeyer, Controlling light by light with an optical event horizon,, Phys. Rev. Lett., 106 (2011).
doi: 10.1103/PhysRevLett.106.163901. |
[22] |
J. M. Dudley, G. Genty and S. Coen, Supercontinuum generation in photonic crystal fiber,, Rev. Mod. Phys., 78 (2006), 1135.
doi: 10.1103/RevModPhys.78.1135. |
[23] |
M. D. Feit, J. A. Fleck and A. Steiger, Solution of the Schrödinger equation by a spectral method,, J. Comput. Phys., 47 (1982), 412.
doi: 10.1016/0021-9991(82)90091-2. |
[24] |
A. Hasegawa, Optical Solitons in Fibers,, Springer, (1980). Google Scholar |
[25] |
W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Difusion-Reaction Equations,, Springer Series in Computational Mathematics, (2003).
doi: 10.1007/978-3-662-09017-6. |
[26] |
A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics,, Applied Mathematical Sciences, (1992).
doi: 10.1007/978-1-4757-2184-3. |
[27] |
C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations,, Math. Comput., 77 (2008), 2141.
doi: 10.1090/S0025-5718-08-02101-7. |
[28] |
G. Muslu and H. Erbay, Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation,, Mathematics and Computers in Simulation, 67 (2005), 581.
doi: 10.1016/j.matcom.2004.08.002. |
[29] |
Ch. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential,, BIT Numer. Math., 49 (2009), 199.
doi: 10.1007/s10543-009-0215-2. |
[30] |
D. Pathria and J. L. Morris, Exact solutions for a generalized nonlinear Schrödinger equation,, Phys. Scripta, 39 (1989), 673.
doi: 10.1088/0031-8949/39/6/001. |
[31] |
D. Pathria and J. L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations,, J. Comput. Phys., 87 (1990), 108.
doi: 10.1016/0021-9991(90)90228-S. |
[32] |
M. Radziunas, R. Čiegis and A. Mirinavičius, On compact high order finite difference schemes for linear Schrödinger problem on non-uniform meshes,, Int. Journal of Num. Anal. Model., 11 (2014), 303.
|
[33] |
J. M. Sanz-Serna and J. G. Verwer, Conservative and non-conservative schemes for the solution of the nonlinear Schrödinger equation,, IMA J. Numer. Anal., 6 (1986), 25.
doi: 10.1093/imanum/6.1.25. |
[34] |
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems,, Applied Mathematics and Mathematical Computation, (1994).
|
[35] |
J. M. Stone and J. C. Knight, Visibly white light generation in uniform photonic crystal fiber using a microchip laser,, Opt. Express, 16 (2008), 2670.
doi: 10.1364/OE.16.002670. |
[36] |
G. Strang, On the construction and comparison of difference schemes,, SIAM J. Numer. Anal., 5 (1968), 506.
doi: 10.1137/0705041. |
[37] |
T. R. Taha and M. J. Ablovitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical Schrödinger equation,, J. Comp. Phys., 55 (1984), 203.
doi: 10.1016/0021-9991(84)90003-2. |
[38] |
M. Thalhammer, M. Caliari and Ch. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods,, J. Comput. Phys., 228 (2009), 822.
doi: 10.1016/j.jcp.2008.10.008. |
[39] |
D. E. Vakman and L. A. Vainshtein, Amplitude, phase, frequency - fundamental concepts of oscillation theory,, Usp. Fiz. Nauk, 20 (1977), 1002.
doi: 10.1070/PU1977v020n12ABEH005479. |
[40] |
P. K. A. Wai, H. H. Chen and Y. C. Lee, Radiations by "solitons" at the zero group-dispersion wavelength of single-mode optical fibers,, Phys. Rev. A, 41 (1990), 426.
doi: 10.1103/PhysRevA.41.426. |
[41] |
V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,, Sov. Phys. JETP, 34 (1972), 62.
|
[1] |
Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 |
[2] |
Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 |
[3] |
Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101 |
[4] |
Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 |
[5] |
Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020462 |
[6] |
Ke Su, Yumeng Lin, Chun Xu. A new adaptive method to nonlinear semi-infinite programming. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021012 |
[7] |
Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 |
[8] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020450 |
[9] |
Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 |
[10] |
Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020319 |
[11] |
Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104 |
[12] |
Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 |
[13] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[14] |
Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127 |
[15] |
Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351 |
[16] |
Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 |
[17] |
Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020180 |
[18] |
Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020456 |
[19] |
Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260 |
[20] |
Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328 |
2019 Impact Factor: 1.311
Tools
Metrics
Other articles
by authors
[Back to Top]