Abstract
A new integrable discretization of the nonlinear Schr¨odinger (NLS) equation is presented. Different from the one given by Ablowitz and Ladik, we discretize the time variable in this paper. The new discrete system converges to the NLS equation when we take a standard limit and has the same scattering operator as the original NLS equation. This indicates that both the new system and the NLS equation possess the same set of infinite conservation quantities. By applying the Fourier pseudo-spectral method to the space variable, we calculate the first five conservation quantities at different times. The numerical results indeed verify the conservation properties.
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Zabusky, N.J., Kruskal, M.D.: Interaction of “Solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)
Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and continuous nonlinear Schrdinger systems. Cambridge University Press (2004)
Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical. J. Comput. Phys. 55(2), 192–02 (1984)
Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation. J. Comput. Phys. 55(2), 203–30 (1984)
Ablowitz, M.J., Herbst, B.M.: On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation. SIAM J. Appl. Math. 50(2), 339–351 (1990)
Herbst, B.M., Ablowitz, M.J.: Numerically induced chaos in the nonlinear Schrödinger equation. Phys. Rev. Lett. 62(18), 2065 (1989)
Hirota, R., Nagai, J.N., Gilson, C.: Direct Method in Soliton Theory. Cambridge University Press), Cambridge (2004)
Ablowitz, M.J., Segur, H.: Solitons and the inverse scattering transform. SIAM, Philadelphia (1981)
Yang, J.: Nonlinear waves in integrable and non-integrable systems. SIAM (2010)
Fornberg, B.: A practical guide to pseudospectral methods. Cambridge University Press, Cambridge (1998)
Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–9 (1972)
Zakharov, V.E., Shabat, A.B.: Interaction between solitons in a stable medium. Sov. Phys. JETP 37, 823–28 (1973)
Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995)
Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers: I. Anomalous dispersion. Appl. Phys. Lett. 23, 142–44 (1973)
HASEGAWA, A., TAPPERT, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers: II. Normal dispersion. Appl. Phys. Lett. 23, 171–72 (1973)
Kivshar, Y.S., Luther-Davies, B.: Dark optical solitons: physics and applications. Phys. Rep. 298, 81–97 (1998)
Carr, L.D., Kutz, J.N., Reinhardt, W.P.: Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose-Einstein condensate. Phys. Rev. E 63, 066604 (2001)
Carretero-González, R., Frantzeskakis, D.J., Kevrekidis, P.G.: Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques. Nonlinearity 21, 139–202 (2008)
Frantzeskakis, D.J.: Dark solitons in atomic Bose-Einstein condensates: from theory to experiments. J. Phys. A: Math. Theor. 43(21), 213001 (2010)
Akrivis, G.D.: Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal. 13, 115–24 (1993)
Chang, Q.S., Jia, E., Sun, W.: Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys. 148, 397–215 (1999)
Delfour, M., Fortin, M., Payre, G.: Finite-difference solutions of a nonlinear Schrödinger equation. J. Comput. Phys. 44, 277–88 (1981)
Bao, W.: Numerical methods for the nonlinear Schrödinger equation with nonzero far-field conditions. Methods Appl. Anal. 367–87, 11 (2004)
Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science, Beijing (2006)
Zouraris, G.: On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation. ESAIM Math. Model. Numer. Anal. 35(03), 389–405 (2001)
Bao, W., Jin, S., Markowich, P.A.: On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 487–524, 175 (2002)
Besse, C., Bidegaray, B., Descombes, S.: Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 26–30, 40 (2002)
Weideman, J.A.C., Herbst, B.M.: Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 485–507, 23 (1986)
Karakashian, O., Akrivis, G.D., Dougalis, V.A.: On optimal order error estimates for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 377–400, 30 (1993)
Islas, A.L., Karpeev, D.A., Schober, C.M.: Geometric integrators for the nonlinear Schrödinger equation. J. Comput. Phys. 173(1), 116–148 (2001)
Islas, A.L., Schober, C.M.: On the preservation of phase space structure under multisymplectic discretization. J. Comput. Phys. 197(2), 585–609 (2004)
Chen, J.B., Qin, M.Z., Tang, Y.F.: Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation. Comput. Math. Appl. 43(8), 1095–1106 (2002)
Matsuo, T., Furihata, D.: Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Math. Phys. 171(2), 425–447 (2001)
Chiu, S.C., Ladik, J.F.: Generating exactly soluble nonlinear discrete evolution equations by a generalized Wronskian technique. J Math Phys 18, 690 (1977)
Levi, D.: Nonlinear differential difference equations as Bäcklund transformations. J. Phys. A Math. Theor. 14, 1083–1098 (1981)
Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys. 16, 598–603 (1975)
Hirota, R.: Nonlinear partial difference equations I. A difference analogue of the Korteweg-de Vries equation. J. Phys. Soc. Jpn. 43, 1424–1433 (1977)
Hirota, R.: Nonlinear partial difference equations II. Discrete-time Toda equation. J. Phys. Soc. Jpn. 43, 2074–2078 (1977)
Hirota, R.: Nonlinear partial difference equations III. Discrete Sine-Gordon equation. J. Phys. Soc. Jpn. 43, 2079–2086 (1977)
Date, E., Jimbo, M., Miwa, T.: J. Phys. Soc. Jpn. 51, 4116–4124 (1982)
Date, E., Jimbo, M., Miwa, T.: Methods for generating discrete soliton equations II. J. Phys. Soc. Jpn. 51, 4125–4131 (1982)
Date, E., Jimbo, M., Miwa, T.: Methods for generating discrete soliton equations III. J. Phys. Soc. Jpn. 52, 388–393 (1983)
Date, E., Jimbo, M., Miwa, T.: Methods for generating discrete soliton equations IV. J. Phys. Soc. Jpn. 52, 761–765 (1983)
Date, E., Jimbo, M., Miwa, T.: Methods for generating discrete soliton equations V. J. Phys. Soc. Jpn. 52, 766–771 (1983)
Levi, D., Pilloni, L., Santini, P.M.: Integrable three-dimensional lattices. J. Phys. A. Math. Gen. 14, 1567–1575 (1981)
Levi, D., Benguria, R.: Bäcklund transformations nonlinear differential difference equations. Proc. Natl. Acad. Sci. (USA) 77, 5025–5027 (1980)
Nijhoff, F.W., Capel, H.W., Wiersma, G.L., Quispel, G.R.W.: Linearizing integral transform and partial difference equations. Phys. Lett. A 103, 293 (1984)
Quispel, G.R.W., Nijhoff, F.W., Capel, H.W., Van-der-Linden, J.: Linear integral equations and nonlinear differrence-difference equations. Physica A 125, 344–380 (1984)
Faddeev, L., Volkov, A.Y.: Hirota equation as an example of integrable symplectic map. Lett. Math. Phys. 32, 125–135 (1994)
Suris, Y.B.: The problem of integrable discretization: Hamiltonian approach. Birkhäuser (2003)
Schiff, J.: Loop groups and discrete KdV equations. Nonlinearity 16, 257–275 (2003)
Rogers, C., Shadwick, W.F.: Bäcklund transformations and their applications (Volume 161 Mathematics in Science and Engineering)
Nimmo, J.J.C.: A bilinear Bäcklund transformation for the nonlinear Schrödinger equation. Phys. Lett. A 99(6), 279–280 (1983)
Zhang, Y., Tam, H.W., Hu, X.B.: Integrable discretization of ’time’ and its application on the Fourier pseudospectral method to the Korteweg-de Vries equation. J. Phys. A. Math. Theor. 47, 045202 (2014)
Miles, J.W.: An envelope soliton problem. SIAM J. Appl. Math. 41(2), 227–230 (1981)
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Zhang, Y., Hu, X.B. & Tam, H.W. Integrable discretization of nonlinear Schrödinger equation and its application with Fourier pseudo-spectral method. Numer Algor 69, 839–862 (2015). https://doi.org/10.1007/s11075-014-9928-7
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DOI: https://doi.org/10.1007/s11075-014-9928-7