Bicubic B-spline Galerkin method for solving the two-dimensional nonlinear Schrödinger equation

Emad Omar Altahat; Nur Nadiah Abd Hamid; Adila Aida Azahar; Yazariah Mohd Yatim; Emad Omar Altahat; Nur Nadiah Abd Hamid; Adila Aida Azahar; Yazariah Mohd Yatim

doi:10.3934/math.20261163

AIMS Mathematics

2026, Volume 11, Issue 9: 29256-29284. doi: 10.3934/math.20261163

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Research article

Bicubic B-spline Galerkin method for solving the two-dimensional nonlinear Schrödinger equation

1.
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia
2.
Nur Nadiah Academic Services, Butterworth 13800, Penang, Malaysia

Received: 29 April 2026 Revised: 08 July 2026 Accepted: 15 July 2026 Published: 11 September 2026
MSC : 65N30, 65M60, 35Q55

The nonlinear Schrödinger (NLS) equation appears in many physical contexts, such as nonlinear optics, quantum mechanics, and wave propagation. This paper introduces a Galerkin finite-element method based on bicubic B-spline basis functions for solving the two-dimensional time-dependent NLS equation. The spatial discretization was carried out using bicubic B-splines within a Galerkin framework, while time integration was performed using the Crank-Nicolson scheme, resulting in a fully discrete formulation. Bicubic B-spline basis functions provide higher smoothness across element interfaces and offer better approximation properties compared with standard polynomial finite-element spaces. Linear stability was investigated under periodic boundary conditions using the von Neumann method. A plane wave test problem was used to assess the accuracy, stability, and convergence behavior of the proposed method. To supplement the plane-wave test, a spatially varying amplitude benchmark was included to examine the nonlinear two-dimensional dynamics and the long-time evolution of the discrete mass and Hamiltonian invariants, denoted by $ I_1 $ and $ I_2 $, respectively. The numerical results indicated stable two-dimensional performance, with convergence rates close to second order in both space and time, while the nonconstant-amplitude benchmark illustrated the distinct long-time behavior of the discrete $ I_1 $ and $ I_2 $ invariants.
- two-dimensional NLS equation,
- bicubic B-spline Galerkin method,
- Crank-Nicolson time discretization,
- numerical stability,
- convergence behavior
Citation: Emad Omar Altahat, Nur Nadiah Abd Hamid, Adila Aida Azahar, Yazariah Mohd Yatim. Bicubic B-spline Galerkin method for solving the two-dimensional nonlinear Schrödinger equation[J]. AIMS Mathematics, 2026, 11(9): 29256-29284. doi: 10.3934/math.20261163

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Abstract

The nonlinear Schrödinger (NLS) equation appears in many physical contexts, such as nonlinear optics, quantum mechanics, and wave propagation. This paper introduces a Galerkin finite-element method based on bicubic B-spline basis functions for solving the two-dimensional time-dependent NLS equation. The spatial discretization was carried out using bicubic B-splines within a Galerkin framework, while time integration was performed using the Crank-Nicolson scheme, resulting in a fully discrete formulation. Bicubic B-spline basis functions provide higher smoothness across element interfaces and offer better approximation properties compared with standard polynomial finite-element spaces. Linear stability was investigated under periodic boundary conditions using the von Neumann method. A plane wave test problem was used to assess the accuracy, stability, and convergence behavior of the proposed method. To supplement the plane-wave test, a spatially varying amplitude benchmark was included to examine the nonlinear two-dimensional dynamics and the long-time evolution of the discrete mass and Hamiltonian invariants, denoted by $ I_1 $ and $ I_2 $, respectively. The numerical results indicated stable two-dimensional performance, with convergence rates close to second order in both space and time, while the nonconstant-amplitude benchmark illustrated the distinct long-time behavior of the discrete $ I_1 $ and $ I_2 $ invariants.

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