## Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation.(English)Zbl 1291.35329

Summary: In this paper, we propose new efficient and accurate numerical methods for computing dark solitons and review some existing numerical methods for bright and/or dark solitons in the nonlinear Schrödinger equation (NLSE), and compare them numerically in terms of accuracy and efficiency. We begin with a review of dark and bright solitons of NLSE with defocusing and focusing cubic nonlinearities, respectively. For computing dark solitons, to overcome the nonzero and/or non-rest (or highly oscillatory) phase background at far field, we design efficient and accurate numerical methods based on accurate and simple artificial boundary conditions or a proper transformation to rest the highly oscillatory phase background. Stability and conservation laws of these numerical methods are analyzed. For computing interactions between dark and bright solitons, we compare the efficiency and accuracy of the above numerical methods and different existing numerical methods for computing bright solitons of NLSE, and identify the most efficient and accurate numerical methods for computing dark and bright solitons as well as their interactions in NLSE. These numerical methods are applied to study numerically the stability and interactions of dark and bright solitons in NLSE. Finally, they are extended to solve NLSE with general nonlinearity and/or external potential and coupled NLSEs with vector solitons.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35C08 Soliton solutions 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs

### Software:

imagetime1d; GP-SCL; AEDU; Gross-Pitaevskii
Full Text:

### References:

 [1] Abdullaev, F.; Darmanyan, S.; Khabibullaev, P., Optical solitons, (1993), Springer Verlag New York [2] Ablowitz, M. J.; Segur, H., Solitons and the inverse scattering transform, (1981), SIAM Philadelphia · Zbl 0472.35002 [3] Akrivis, G. D., Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal., 13, 115-124, (1993) · Zbl 0762.65070 [4] Akrivis, G. D.; Dougalis, V. A.; Karakashian, O., Solving the systems of equations arising in the discretization of some nonlinear PDEs by implicit runge – kutta methods, RAIRO Model. Math. Anal. Numer., 31, 251-287, (1997) · Zbl 0869.65060 [5] 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) [6] Antoine, X.; Arnold, A.; Besse, C.; Ehrhardt, M.; Schädle, A., A review of transparent and artificial boundary condition techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4, 729-796, (2008) · Zbl 1364.65178 [7] Antoine, X.; Besse, C.; Klein, P., Absorbing boundary conditions for general nonlinear Schrödinger equations, SIAM J. Sci. Comput., 33, 1008-1033, (2011) · Zbl 1231.35223 [8] Bao, W., Numerical methods for the nonlinear Schrödinger equation with nonzero far-field conditions, Methods Appl. Anal., 11, 367-387, (2004) · Zbl 1090.65116 [9] Bao, W., Ground states and dynamics of multi-component bose – einstein condensates, Multiscale Model. Simul., 2, 210-236, (2004) · Zbl 1062.82034 [10] Bao, W.; Jaksch, D.; Markowich, P. A., Numerical solution of the gross – pitaevskii equation for bose – einstein condensation, J. Comput. Phys., 187, 318-342, (2003) · Zbl 1028.82501 [11] Bao, W.; Jin, S.; Markowich, P. A., On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175, 487-524, (2002) · Zbl 1006.65112 [12] Bao, W.; Lim, F. Y., Computing ground states of spin-1 bose – einstein condensates by the normalized gradient flow, SIAM J. Sci. Comput., 30, 1925-1948, (2008) · Zbl 1173.81028 [13] Bao, W.; Zhang, Y., Dynamics of the ground state and central vortex states in bose – einstein condensation, Math. Models Meth. Appl. Sci., 15, 1863-1896, (2005) · Zbl 1154.82313 [14] Bao, W.; Zheng, C., A time-splitting spectral method for three-wave interactions in media with competing quadratic and cubic nonlinearities, Commun. Comput. Phys., 2, 123-140, (2007) · Zbl 1164.78320 [15] Barashenkov, I. V.; Harin, A. O., Nonrelativistic cherns – simons theory for the repulsive Bose gas, Phys. Rev. Lett., 72, 1575-1579, (1994) · Zbl 0973.82501 [16] Barashenkov, I. V.; Panova, E. Y., Stability and evolution of the quiescent and travelling solitonic bubbles, Physica D, 69, 114-134, (1993) · Zbl 0791.35126 [17] Besse, C.; Bidegaray, B.; Descombes, S., Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40, 26-40, (2002) · Zbl 1026.65073 [18] 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) [19] 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) · Zbl 1216.82023 [20] Chang, Q. S.; Jia, E.; Sun, W., Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys, 148, 397-415, (1999) · Zbl 0923.65059 [21] Dauxois, T.; Peyrard, M., Physics of solitons, (2006), Cambridge University Press · Zbl 1192.35001 [22] Delfour, M.; Fortin, M.; Payre, G., Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys., 44, 277-288, (1981) · Zbl 0477.65086 [23] Frantzeskakis, D. J., Dark solitons in atomic bose – einstein condensates: from theory to experiments, J. Phys. A Math. Theor., 43, 213001, (2010) · Zbl 1192.82033 [24] Griffiths, D. F.; Mitchell, A. R.; Morris, J. L., A numerical study of the nonlinear Schrödinger equation, Comput. Methods Appl. Mech. Eng., 45, 177-215, (1984) · Zbl 0555.65060 [25] Hasegawa, A.; Tappert, F., Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers: I. anomalous dispersion, Appl. Phys. Lett., 23, 142-144, (1973) [26] Hasegawa, A.; Tappert, F., Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers: II. normal dispersion, Appl. Phys. Lett., 23, 171-172, (1973) [27] Han, H.; Huang, Z., Exact artificial boundary conditions for Schrödinger equation in $$\mathbb{R}^2$$, Commun. Math. Sci., 2, 79-94, (2004) · Zbl 1089.35524 [28] Karakashian, O.; Akrivis, G. D.; Dougalis, V. A., On optimal order error estimates for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 30, 377-400, (1993) · Zbl 0774.65091 [29] Kivshar, Y. S.; Luther-Davies, B., Dark optical solitons: physics and applications, Phys. Rep., 298, 81-197, (1998) [30] Kivshar, Y. S.; Yang, X., Perturbation-induced dynamics of dark solitons, Phys. Rev. E, 49, 1657-1670, (1994) [31] Muruganandam, P.; Adhikari, S. K., Fortran programs for the time-dependent gross – pitaevskii equation in a fully anisotropic trap, Comput. Phys. Commun., 180, 1888-1912, (2009) · Zbl 1353.35002 [32] Newell, A. C., Solitons in mathematics and physics, (1985), SIAM Philadelphia · Zbl 0565.35003 [33] Novikov, S.; Manakov, S. V.; Pitaevskij, L. P.; Zakharov, V. E., Theory of solitons: the inverse scattering methods, (1984), Plenum New York · Zbl 0598.35002 [34] J.S. Russell, Report on waves, in: Fourteenth Meeting of the British Association for the Advancement of Science, 1844. [35] Shamardan, A. B., The numerical treatment of the nonlinear Schrödinger equation, Comput. Math. Appl., 19, 67-73, (1990) · Zbl 0702.65096 [36] Shen, J.; Tang, T., Spectral and high-order methods with applications, (2006), Science Beijing · Zbl 1234.65005 [37] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 505-517, (1968) · Zbl 0184.38503 [38] Taha, T. R.; Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations: II. numerical, nonlinear schrodinger equation, J. Comput. Phys., 55, 203-230, (1984) · Zbl 0541.65082 [39] Q. Tang, Dynamics and interaction of the multi-component solitons in coupled nonlinear Schrödinger equations, in preparation. [40] Uzunov, I. M.; Gerdjikov, V. S., Self-frequency shift of dark solitons in optical fibers, Phys. Rev. A, 47, 1582-1585, (1993) [41] Vudragovic, D.; Vidanovic, I.; Balaz, A.; Muruganandam, P.; Adhikari, S. K., C programs for solving the time-dependent gross – pitaevskii equation in a fully anisotropic trap, Comput. Phys. Commun., 183, 2021-2025, (2012) · Zbl 1353.35003 [42] Weideman, J. A.C.; Herbst, B. M., Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 23, 485-507, (1986) · Zbl 0597.76012 [43] 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, 62-69, (1972) [44] Zakharov, V. E.; Shabat, A. B., Interaction between solitons in a stable medium, Sov. Phys. JETP, 37, 823-828, (1973) [45] Zheng, C., Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations, J. Comput. Phys., 215, 552-562, (2006) · Zbl 1094.65086 [46] Zouraris, G., On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation, M2AN Math. Model. Numer. Anal., 35, 389-405, (2001) · Zbl 0991.65088
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.