An efficient ADI compact scheme with Strang splitting for the two-dimensional cubic-quintic-septic nonlinear Schrödinger equation

Jiaqi Chen; Weizhong Dai; Anjan Biswas; Jiaqi Chen; Weizhong Dai; Anjan Biswas

doi:10.3934/era.2026349

Electronic Research Archive

2026, Volume 34, Issue 11: 8203-8231. doi: 10.3934/era.2026349

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An efficient ADI compact scheme with Strang splitting for the two-dimensional cubic-quintic-septic nonlinear Schrödinger equation

1.
School of Electrical Engineering and Automation, Xiamen University of Technology, Xiamen 361024, China
2.
Mathematics & Statistics, College of Engineering & Science, Louisiana Tech University, Ruston, LA 71272, USA
3.
Department of Mathematics and Physics, Grambling State University, Grambling, LA 71245-2715, USA
4.
Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon 61080, Türkiye
5.
Applied Science Research Center, Applied Science Private University, Amman 11937, Jordan
6.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa, Pretoria 0204, South Africa
7.
Department of Physics and Electronics, Khazar University, Baku, AZ 1096, Azerbaijan

Received: 08 July 2026 Revised: 29 August 2026 Accepted: 16 September 2026 Published: 22 September 2026

We studied the two-dimensional cubic-quintic-septic nonlinear Schrödinger equation. We first established its continuous mass, energy, and momentum conservation laws, and then proposed an alternating direction implicit (ADI) compact finite difference scheme combined with Strang splitting. The splitting technique isolates the nonlinear term for an exact analytical solution via phase rotation, while a 3-point Padë approximation resolves the linear term to fourth-order spatial accuracy without requiring boundary ghost points. We rigorously proved the scheme is unconditionally stable with a convergence rate of $ O(\tau^2 + h_x^4 + h_y^4) $. Numerical experiments validated the theoretical analysis and demonstrated the method's high-order accuracy and efficiency.
- nonlinear Schrödinger equation,
- finite difference method,
- ADI method,
- Strang splitting
Citation: Jiaqi Chen, Weizhong Dai, Anjan Biswas. An efficient ADI compact scheme with Strang splitting for the two-dimensional cubic-quintic-septic nonlinear Schrödinger equation[J]. Electronic Research Archive, 2026, 34(11): 8203-8231. doi: 10.3934/era.2026349

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Abstract

We studied the two-dimensional cubic-quintic-septic nonlinear Schrödinger equation. We first established its continuous mass, energy, and momentum conservation laws, and then proposed an alternating direction implicit (ADI) compact finite difference scheme combined with Strang splitting. The splitting technique isolates the nonlinear term for an exact analytical solution via phase rotation, while a 3-point Padë approximation resolves the linear term to fourth-order spatial accuracy without requiring boundary ghost points. We rigorously proved the scheme is unconditionally stable with a convergence rate of $ O(\tau^2 + h_x^4 + h_y^4) $. Numerical experiments validated the theoretical analysis and demonstrated the method's high-order accuracy and efficiency.

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