Numerical results for the nonlinear Schrödinger equation via a cubic Hermite spline method
DOI: https://doi.org/10.3846/mma.2026.25083Abstract
In this paper, we investigate the numerical solution of the nonlinear Schrödinger (NLS) equation using a cubic Hermite spline collocation method. The proposed scheme is applied to various test problems, including single soliton propagation, soliton collisions, and both standing and traveling soliton solutions, in order to evaluate its accuracy and efficiency. A Von Neumann stability analysis is performed for the linearized form of the scheme, demonstrating that the method is unconditionally stable in the linear sense. Numerical experiments demonstrate that the method provides stable and accurate approximations, with good agreement between the computed and exact solutions. The results confirm that the cubic Hermite spline approach is a reliable tool for solving the NLS equation, particularly in capturing the qualitative behavior of the wave profiles. Comparisons with existing numerical methods further highlight the effectiveness of the proposed technique.
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cubic Hermite spline, collocation method, nonlinear Schrödinger equation, Legendre and Chebyschev rootsHow to Cite
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Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
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S. Arora, R. Jain and V.K. Kukreja. Solution of Benjamin-BonaMahony-Burgers equation using collocation method with quintic Hermite splines. Applied Numerical Mathematics, 154:1–16, 2020. https://doi.org/10.1016/j.apnum.2020.03.015
İ. Daǧ. A quadratic B-spline finite element method for solving nonlinear Schrödinger equation. Computer Methods in Applied Mechanics and Engineering, 174:247–258, 1999. https://doi.org/10.1016/S0045-7825(98)00257-6
L. Debnath. Nonlinear Partial Differential Equations for Scientist and Engineers. Birkhauser, Boston, 1997. https://doi.org/10.1007/978-1-4899-2846-7
M. Delfour, M. Fortin and G. Payr. Finite-difference solutions of a non-linear Schro¨dinger equation. Journal of Computational Physics, 44(2):277–288, 1981. https://doi.org/10.1016/0021-9991(81)90052-8
Y. Dereli. The meshless kernel-based method of lines for the numerical solution of the nonlinear Schrödinger equation. Engineering Analysis with Boundary Elements, 36(9):1416–1423, 2012. https://doi.org/10.1016/j.enganabound.2012.02.018
Y. Dereli, D. Irk and İ. Daǧ. Soliton solutions for NLS equation using radial basis functions. Chaos, Solitons & Fractals, 42(2):1227–1233, 2009. https://doi.org/10.1016/j.chaos.2009.03.030
I.A. Ganaie, S. Arora and V.K. Kukreja. Cubic Hermite collocation method for solving boundary value problems with Dirichlet, Neumann, and Robin conditions. International Journal of Engineering Mathematics, 2014:365209, 2014. https://doi.org/10.1155/2014/365209
I.A. Ganaie, S. Arora and V.K. Kukreja. Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation. International Journal of Computer Mathematics, 93(1):223–235, 2016. https://doi.org/10.1080/00207160.2014.999052
I.A. Ganaie, B. Gupta, N. Parumasur, P. Singh and V.K. Kukreja. Asymptotic convergence of cubic Hermite collocation method for parabolic partial differential equation. Applied Mathematics and Computation, 220:560–567, 2013. https://doi.org/10.1016/j.amc.2013.05.073
I.A. Ganaie and V.K. Kukreja. Numerical solution of Burgers’ equation by cubic Hermite collocation method. Applied Mathematics and Computation, 237:571– 581, 2014. https://doi.org/10.1016/j.amc.2014.03.102
L.R.T. Gardner, G.A. Gardner, S.I. Zaki and Z. El Sahrawi. B-spline finite element studies of the non-linear Schrödinger equation. Computer Methods in Applied Mechanics and Engineering, 108(3–4):303–318, 1993. https://doi.org/10.1016/0045-7825(93)90007-K
L.R.T. Gardner, G.A. Gardner, S.I. Zaki and Z. El Sahrawi. A leapfrog algorithm and stability studies for the nonlinear Schrödinger equation. Arabian Journal for Science and Engineering, 18:23–32, 1993.
A.G. Kaplan and Y. Dereli. A meshless method and stability analysis for the nonlinear Schrödinger equation. Waves in Random and Complex Media, 27(4):602– 614, 2017. https://doi.org/10.1080/17455030.2017.1290301
S.A. Khuri. A complex tanh-function method applied to nonlinear equations of Schrödinger type. Chaos, Solitons & Fractals, 20(5):1037–1040, 2004. https://doi.org/10.1016/j.chaos.2003.09.042
A. Korkmaz and İ. Daǧ. A differential quadrature algorithm for simulations of˙ nonlinear Schrödinger equation. Computers & Mathematics with Applications, 56(9):2222–2234, 2008. https://doi.org/10.1016/j.camwa.2008.03.047
A. Korkmaz and İ. Daǧ. A differential quadrature algorithm for nonlinear Schrödinger equation. Nonlinear Dynamics, 56:69–83, 2009. https://doi.org/10.1007/s11071-008-9380-0
A. Kumari and V.K. Kukreja. Septic Hermite collocation method for the numerical solution of Benjamin–Bona–Mahony–Burgers equation. Journal of Difference Equations and Applications, 27(8):1193–1217, 2021. https://doi.org/10.1080/10236198.2021.1972985
A. Kumari and V.K. Kukreja. Robust septic Hermite collocation technique for singularly perturbed generalized Hodgkin–Huxley equation. International Journal of Computer Mathematics, 99(5):909–923, 2022. https://doi.org/10.1080/00207160.2021.1939317
S. Kutluay, M. Yaǧmurlu and A.S. Karakaș. A robust quintic Hermite collocation method for one-dimensional heat conduction equation. Journal of Mathematical Sciences and Modelling, 7(2):82–89, 2024. https://doi.org/10.33187/jmsm.1475294
S. Kutluay, N.M. Yaǧmurlu and A.S. Karakaș. A powerful robust cubic Hermite collocation method for the numerical calculations and simulations of the equal width wave equation. arXiv preprint, 2023. https://doi.org/10.48550/arXiv.2309.02439
S. Kutluay, N.M. Yaǧmurlu and A.S. Karakaș. A novel perspective for simulations of the modified equal-width wave equation by cubic Hermite B-spline collocation method. Wave Motion, 129:103342, 2024. https://doi.org/10.1016/j.wavemoti.2024.103342
A.K. Mittal, I.A. Ganaie, V.K. Kukreja, N. Parumasur and P. Singh. Solution of diffusion–dispersion models using a computationally efficient technique of orthogonal collocation on finite elements with cubic Hermite as basis. Computers and Chemical Engineering, 58:203–210, 2013. https://doi.org/10.1016/j.compchemeng.2013.07.007
T. Öziș and A. Yıldırım. Reliable analysis for obtaining exact soliton solutions of nonlinear Schrödinger (NLS) equation. Chaos, Solitons & Fractals, 38(1):209– 212, 2008. https://doi.org/10.1016/j.chaos.2006.11.006
Priyanka, S. Arora, F. Mebrek-Oudina and S. Sahani. Super convergence analysis of fully discrete Hermite splines to simulate wave behaviour of Kuramoto–Sivashinsky equation. Wave Motion, 121:103187, 2023. https://doi.org/10.1016/j.wavemoti.2023.103187
M.P. Robinson. The solution of nonlinear Schrödinger equations using orthogonal spline collocation. Computers & Mathematics with Applications, 33(7):39–57, 1997. https://doi.org/10.1016/S0898-1221(97)00042-4
M.P. Robinson and G. Fairweather. Orthogonal spline collocation methods for Schrödinger-type equation in one space variable. Numerische Mathematik, 68:355–376, 1994. https://doi.org/10.1007/s002110050067
B. Saka. A quintic B-spline finite-element method for solving the nonlinear Schrödinger equation. Solitons and Chaos, Physics of Wave Phenomena, 20:107– 117, 2012. https://doi.org/10.3103/S1541308X12020033
E.H. Twizell, A. Bratros and J.C. Newby. A finite-difference method for solving the cubic Schrödinger equation. Mathematics and Computers in Simulation, 43(1):67–75, 1997. https://doi.org/10.1016/S0378-4754(96)00056-0
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Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
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