Split-step quintic B-spline collocation methods for nonlinear Schrödinger equations

Shanshan Wang; Shanshan Wang

doi:10.3934/math.20231009

AIMS Mathematics

2023, Volume 8, Issue 8: 19794-19815. doi: 10.3934/math.20231009

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Split-step quintic B-spline collocation methods for nonlinear Schrödinger equations

Shanshan Wang ^{1,2
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1.
College of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
2.
Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing 211106, China

Received: 31 January 2023 Revised: 24 May 2023 Accepted: 06 June 2023 Published: 13 June 2023
MSC : 65M70, 35Q55, 35Q51

Split-step quintic B-spline collocation (SS5BC) methods are constructed for nonlinear Schrödinger equations in one, two and three dimensions in this paper. For high dimensions, new notations are introduced, which make the schemes more concise and achievable. The solvability, conservation and linear stability are discussed for the proposed methods. Numerical tests are carried out, and the present schemes are numerically verified to be convergent with second-order in time and fourth-order in space. The conserved quantity is also computed which agrees with the exact one. And solitary waves in one, two and three dimensions are simulated numerically which coincide with the exact ones. The SS5BC scheme is compared with the split-step cubic B-spline collocation (SS3BC) method in the numerical tests, and the former scheme is more efficient than the later one. Finally, the SS5BC scheme is also applied to compute Bose-Einstein condensates.
- split-step,
- quintic B-spline collocation,
- nonlinear Schrödinger equation,
- numerical experiment,
- convergence order,
- conservation law,
- soliton,
- Bose-Einstein condensate
Citation: Shanshan Wang. Split-step quintic B-spline collocation methods for nonlinear Schrödinger equations[J]. AIMS Mathematics, 2023, 8(8): 19794-19815. doi: 10.3934/math.20231009

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Abstract

Split-step quintic B-spline collocation (SS5BC) methods are constructed for nonlinear Schrödinger equations in one, two and three dimensions in this paper. For high dimensions, new notations are introduced, which make the schemes more concise and achievable. The solvability, conservation and linear stability are discussed for the proposed methods. Numerical tests are carried out, and the present schemes are numerically verified to be convergent with second-order in time and fourth-order in space. The conserved quantity is also computed which agrees with the exact one. And solitary waves in one, two and three dimensions are simulated numerically which coincide with the exact ones. The SS5BC scheme is compared with the split-step cubic B-spline collocation (SS3BC) method in the numerical tests, and the former scheme is more efficient than the later one. Finally, the SS5BC scheme is also applied to compute Bose-Einstein condensates.

References

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