-
Previous Article
Random substitution tilings and deviation phenomena
- DCDS Home
- This Issue
-
Next Article
Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five
On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $
1. | Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea |
2. | Department of Mathematics Education, Kongju National University, Kongju 32588, Republic of Korea |
$ Z_{\tau} $ |
$ H^1 $ |
$ \partial_t u = i \Delta u + i\lambda |u|^{p} u,\qquad u (x,0) = \phi (x), $ |
$ p >0 $ |
$ \lambda \in \{-1,1\} $ |
$ (x,t) \in \mathbb{R}^d \times [0,\infty) $ |
$ L^2 $ |
$ \mathcal{O}(\tau^{1/2}) $ |
$ H^1 (\mathbb{R}^d) $ |
$ p $ |
References:
[1] |
R. Altmann and A. Ostermann,
Splitting methods for constrained diffusion-reaction systems, Comput. Math. Appl., 74 (2017), 962-976.
doi: 10.1016/j.camwa.2017.02.044. |
[2] |
C. Besse, B. Bidégaray and S. Descombes,
Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 26-40.
doi: 10.1137/S0036142900381497. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, Amer. Math. Soc/Courant Institute of Mathematical Sciences, Providence, RI/York New, NY, 2003.
doi: 10.1090/cln/010. |
[4] |
M. Christ and A. Kiselev,
Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[5] |
J. Eilinghoff, R. Schnaubelt and K. Schratz,
Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation, J. Math. Anal. Appl., 442 (2016), 740-760.
doi: 10.1016/j.jmaa.2016.05.014. |
[6] |
E. Faou, A. Ostermann and K. Schratz,
Analysis of exponential splitting methods for inhomogeneous parabolic equations, IMA J. Numer. Anal., 35 (2015), 161-178.
doi: 10.1093/imanum/dru002. |
[7] |
L. Gauckler and C. Lubich,
Splitting integrators for nonlinear Schrödinger equations over long times, Found. Comput. Math., 10 (2010), 275-302.
doi: 10.1007/s10208-010-9063-3. |
[8] |
L. Gauckler and C. Lubich,
Nonlinear Schrödinger equations and their spectral semi-discretizations over long times, Found. Comput. Math., 10 (2010), 141-169.
doi: 10.1007/s10208-010-9059-z. |
[9] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education/Prentice Hall, Upper Saddle River, NJ, 2004. |
[10] |
E. Hansen and A. Ostermann,
High-order splitting schemes for semilinear evolution equations (English summary), Bit Numer. Math., 56 (2016), 1303-1316.
doi: 10.1007/s10543-016-0604-2. |
[11] |
L. I. Ignat,
Fully discrete schemes for the Schrödinger equation. Dispersive properties, Math. Models Methods Appl. Sci., 17 (2007), 567-591.
doi: 10.1142/S0218202507002029. |
[12] |
L. I. Ignat,
A splitting method for the nonlinear Schrödinger equation, J. Differential Equations, 250 (2011), 3022-3046.
doi: 10.1016/j.jde.2011.01.028. |
[13] |
L. I. Ignat and E. Zuazua,
A two-grid approximation scheme for nonlinear Schrödinger equations: Dispersive properties and convergence, C. R. Math. Acad. Sci. Paris, 341 (2005), 381-386.
doi: 10.1016/j.crma.2005.07.018. |
[14] |
L. I. Ignat and E. Zuazua,
Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 47 (2009), 1366-1390.
doi: 10.1137/070683787. |
[15] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[16] |
M. Knöller, A. Ostermann and K. Schratz,
A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data (English summary), SIAM J. Numer. Anal., 57 (2019), 1967-1986.
doi: 10.1137/18M1198375. |
[17] |
J. Lu and J. L. Marzuola,
Strang splitting methods for a quasilinear Schrödinger equation: Convergence, instability, and dynamics, Commun. Math. Sci., 13 (2015), 1051-1074.
doi: 10.4310/CMS.2015.v13.n5.a1. |
[18] |
C. Lubich,
On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equation, Math. Comp., 77 (2008), 2141-2153.
doi: 10.1090/S0025-5718-08-02101-7. |
[19] |
A. Ostermann, F. Rousset and K. Schratz, Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity, Found. Comput. Math., (2020), to appear. |
[20] |
A. Ostermann and K. Schratz,
Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math., 18 (2018), 731-755.
doi: 10.1007/s10208-017-9352-1. |
[21] |
K. Schratz, Y. Wang and X. Zhao,
Low-regularity integrators for nonlinear Dirac equations, Math. Comp., 90 (2021), 189-214.
doi: 10.1090/mcom/3557. |
[22] |
R. S. Strichartz,
Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[23] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. |
[24] |
T. Tao, Nonlinear Dispersive Equations. Local And Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/cbms/106. |
[25] |
M. Thalhammer,
Higher-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 46 (2008), 2022-2038.
doi: 10.1137/060674636. |
[26] |
M. Thalhammer, M. Caliari and C. Neuhauser,
High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822-832.
doi: 10.1016/j.jcp.2008.10.008. |
show all references
References:
[1] |
R. Altmann and A. Ostermann,
Splitting methods for constrained diffusion-reaction systems, Comput. Math. Appl., 74 (2017), 962-976.
doi: 10.1016/j.camwa.2017.02.044. |
[2] |
C. Besse, B. Bidégaray and S. Descombes,
Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), 26-40.
doi: 10.1137/S0036142900381497. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, Amer. Math. Soc/Courant Institute of Mathematical Sciences, Providence, RI/York New, NY, 2003.
doi: 10.1090/cln/010. |
[4] |
M. Christ and A. Kiselev,
Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[5] |
J. Eilinghoff, R. Schnaubelt and K. Schratz,
Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation, J. Math. Anal. Appl., 442 (2016), 740-760.
doi: 10.1016/j.jmaa.2016.05.014. |
[6] |
E. Faou, A. Ostermann and K. Schratz,
Analysis of exponential splitting methods for inhomogeneous parabolic equations, IMA J. Numer. Anal., 35 (2015), 161-178.
doi: 10.1093/imanum/dru002. |
[7] |
L. Gauckler and C. Lubich,
Splitting integrators for nonlinear Schrödinger equations over long times, Found. Comput. Math., 10 (2010), 275-302.
doi: 10.1007/s10208-010-9063-3. |
[8] |
L. Gauckler and C. Lubich,
Nonlinear Schrödinger equations and their spectral semi-discretizations over long times, Found. Comput. Math., 10 (2010), 141-169.
doi: 10.1007/s10208-010-9059-z. |
[9] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education/Prentice Hall, Upper Saddle River, NJ, 2004. |
[10] |
E. Hansen and A. Ostermann,
High-order splitting schemes for semilinear evolution equations (English summary), Bit Numer. Math., 56 (2016), 1303-1316.
doi: 10.1007/s10543-016-0604-2. |
[11] |
L. I. Ignat,
Fully discrete schemes for the Schrödinger equation. Dispersive properties, Math. Models Methods Appl. Sci., 17 (2007), 567-591.
doi: 10.1142/S0218202507002029. |
[12] |
L. I. Ignat,
A splitting method for the nonlinear Schrödinger equation, J. Differential Equations, 250 (2011), 3022-3046.
doi: 10.1016/j.jde.2011.01.028. |
[13] |
L. I. Ignat and E. Zuazua,
A two-grid approximation scheme for nonlinear Schrödinger equations: Dispersive properties and convergence, C. R. Math. Acad. Sci. Paris, 341 (2005), 381-386.
doi: 10.1016/j.crma.2005.07.018. |
[14] |
L. I. Ignat and E. Zuazua,
Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 47 (2009), 1366-1390.
doi: 10.1137/070683787. |
[15] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[16] |
M. Knöller, A. Ostermann and K. Schratz,
A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data (English summary), SIAM J. Numer. Anal., 57 (2019), 1967-1986.
doi: 10.1137/18M1198375. |
[17] |
J. Lu and J. L. Marzuola,
Strang splitting methods for a quasilinear Schrödinger equation: Convergence, instability, and dynamics, Commun. Math. Sci., 13 (2015), 1051-1074.
doi: 10.4310/CMS.2015.v13.n5.a1. |
[18] |
C. Lubich,
On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equation, Math. Comp., 77 (2008), 2141-2153.
doi: 10.1090/S0025-5718-08-02101-7. |
[19] |
A. Ostermann, F. Rousset and K. Schratz, Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity, Found. Comput. Math., (2020), to appear. |
[20] |
A. Ostermann and K. Schratz,
Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math., 18 (2018), 731-755.
doi: 10.1007/s10208-017-9352-1. |
[21] |
K. Schratz, Y. Wang and X. Zhao,
Low-regularity integrators for nonlinear Dirac equations, Math. Comp., 90 (2021), 189-214.
doi: 10.1090/mcom/3557. |
[22] |
R. S. Strichartz,
Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[23] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. |
[24] |
T. Tao, Nonlinear Dispersive Equations. Local And Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/cbms/106. |
[25] |
M. Thalhammer,
Higher-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 46 (2008), 2022-2038.
doi: 10.1137/060674636. |
[26] |
M. Thalhammer, M. Caliari and C. Neuhauser,
High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822-832.
doi: 10.1016/j.jcp.2008.10.008. |
[1] |
Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337 |
[2] |
Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024 |
[3] |
Alexander Pankov. Nonlinear Schrödinger Equations on Periodic Metric Graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030 |
[4] |
Nobu Kishimoto. A remark on norm inflation for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1375-1402. doi: 10.3934/cpaa.2019067 |
[5] |
Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359 |
[6] |
Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565 |
[7] |
Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703 |
[8] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 |
[9] |
Thierry Colin, Pierre Fabrie. Semidiscretization in time for nonlinear Schrödinger-waves equations. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 671-690. doi: 10.3934/dcds.1998.4.671 |
[10] |
Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253 |
[11] |
Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205 |
[12] |
Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136 |
[13] |
Jaeseop Ahn, Jimyeong Kim, Ihyeok Seo. On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 423-439. doi: 10.3934/dcds.2020016 |
[14] |
Juan Belmonte-Beitia, Vladyslav Prytula. Existence of solitary waves in nonlinear equations of Schrödinger type. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1007-1017. doi: 10.3934/dcdss.2011.4.1007 |
[15] |
Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071 |
[16] |
Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003 |
[17] |
Masahito Ohta. Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1671-1680. doi: 10.3934/cpaa.2018080 |
[18] |
Chuangye Liu, Rushun Tian. Normalized solutions for 3-coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5115-5130. doi: 10.3934/cpaa.2020229 |
[19] |
Van Duong Dinh. A unified approach for energy scattering for focusing nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6441-6471. doi: 10.3934/dcds.2020286 |
[20] |
Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]