Abstract
Numerical simulations of the nonlinear Schrödinger equations are studied using Delta-shaped basis functions, which recently proposed by Reutskiy. Propagation of a soliton, interaction of two solitons, birth of standing and mobile solitons and bound state solutions are simulated. Some conserved quantities are computed numerically for all cases. Then we extend application of the method to solve some coupled nonlinear Schrödinger equations. Obtained systems of ordinary differential equations are solved via the fourth- order Runge–Kutta method. Numerical solutions coincide with the exact solutions in desired machine precision and invariant quantities are conserved sensibly. Some comparisons with the methods applied in the literature are carried out.
Similar content being viewed by others
References
Chegini, N.G., Salaripanah, A., Mokhtari, R., Isvand, D.: Numerical solution of the regularized long wave equation using nonpolynomial splines. Nonlinear Dyn. 69(1–2), 459–471 (2012)
Dag, I.: A quartic B-spline finite element method for solving nonlinear Schrödinger equation. Comput. Methods Appl. Mech. Eng. 174, 247–258 (1999)
Delfour, M., Fortin, M., Payne, G.: Finite-difference solutions of non-linear Schrödinger equation. J. Comput. Phys. 44, 277–288 (1981)
Gardner, L.R.T., Gardner, G.A., Zaki, S.I., Sharawi, Z.El.: B-spline finite element studies of the non-linear Schrödinger equation. Comput. Methods Appl. Mech. Eng. 108, 303–318 (1993)
Hon, Y.C., Yang, Z.: Meshless collocation method by Delta-shape basis functions for default barrier model. Eng. Anal. Bound. Elem. 33, 951–958 (2009)
Ismail, M.S.: A fourth-order explicit schemes for the coupled nonlinear Schrödinger equation. Appl. Math. Comput. 196(1), 273–284 (2008)
Ismail, M.S.: Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method. Math. Comput. Simul. 78(4), 532–547 (2008)
Ismail, M.S., Taha, T.R.: Numerical simulation of coupled nonlinear Schrödinger equation. Math. Comput. Simul. 56(6), 547–562 (2001)
Kong, L., Duan, Y., Wang, L., Yin, X., Ma, Y.: Spectral-like resolution compact ADI finite difference method for the multi-dimensional Schrödinger equations. Math. Comput. Model. 55, 1798–1812 (2012)
Korkmaz, A., Dag, I.: A differential quadrature algorithm for simulations of nonlinear Schrödinger equation. Comput. Math. Appl. 56, 2222–2234 (2008)
Korkmaz, A., Dag, I.: A differential quadrature algorithm for nonlinear Schrödinger equation. Nonlinear Dyn. 56, 69–83 (2009)
Miles, J.M.: An envelope soliton problems. SIAM J. Appl. Math. 41, 227 (1981)
Mohebbi, A.: Solitary wave solutions of the nonlinear generalized Pochhammer–Chree and regularized long wave equations. Nonlinear Dyn. 70(4), 2463–2474 (2012)
Mokhtari, R., Mohseni, M.: A meshless method for solving mKdV equation. Comput. Phys. Commun. 183, 1259–1268 (2012)
Mokhtari, R., Samadi Toodar, A., Chegini, N.G.: Numerical simulation of coupled nonlinear Schrödinger equations using the generalized differential quadrature method. Chin. Phys. Lett. 28(2), 020202 (2011)
Mokhtari, R., Ziaratgahi, S.T.: Numerical solution of RLW equation using integrated radial basis functions. Appl. Comput. Math. 10(3), 428–448 (2011)
Rashid, A., Ismail, A.I.: A Chebishev spectral collocation method for the coupled nonlinear Schrödinger equation. Appl. Comput. Math. 9(9), 104–115 (2010)
Reutskiy, S.Y.: A boundary method of Trefftz type for PDEs with scattered data. Eng. Anal. Bound. Elem. 29, 713–724 (2005)
Reutskiy, S.Y.: A meshless method for one-dimensional Stefan problems. Appl. Math. Comput. 217, 9689–9701 (2011)
Tian, H.Y., Reutskiy, S., Chen, C.S.: New basis functions and their applications to PDEs. Int. Conf. Civ. Eng. Sci. 3(4), 169–175 (2007)
Tian, H.Y., Reutskiy, S., Chen, C.S.: A basis function for approximation and the solutions of partial differential equations. Numer. Methods Partial Differ. Equ. 24, 1018–1036 (2008)
Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equations. J. Comput. Phys. 55, 203–230 (1984)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley-Interscience, New York (1974)
Zakharov, V.E., Shabat, A.B.: Exact theory of two dimensional self focusing and one dimensional self waves in non-linear media. Sov. Phys. JETP 34, 62 (1972)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mokhtari, R., Isvand, D., Chegini, N.G. et al. Numerical solution of the Schrödinger equations by using Delta-shaped basis functions. Nonlinear Dyn 74, 77–93 (2013). https://doi.org/10.1007/s11071-013-0950-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-013-0950-4