The article proposes an iterative method to find soliton solutions of the three-dimensional Gross–Pitaevskii equation that describes the interaction of a Bose–Einstein condensate with an external potential (a magnetic trap, an obstacle, etc.). The method finds both primary and reflected soliton solutions. It can also be applied to find soliton solutions of other nonlinear differential evolution equations. The method can be efficiently implemented on parallel computer systems, producing high-accuracy soliton solutions.
Similar content being viewed by others
References
V. S. Laponin, N. P. Savenkova, and V. P. Il’yutko, “Numerical method to find soliton solutions,” Prikl. Matem. Informat., No. 38, 69–80 (2011).
A. Newell, Solitons in Mathematics and Physics [Russian translation], Mir, Moskva (1989).
V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Soliton Theory: The Inverse Problem Method [in Russian], Nauka, Moscow (1980).
N. N. Rozanov, Yu. V. Rozhdestvenskii, V. A. Smirnov, and S. V. Fedorov, “Atomic “needles” and “bullets” in Bose-Einstein condensate and the formation of nanosized structures,” Pis’ma v ZhETF, 77, No. 2, 89–92 (2003).
A. V. Borisov, A. Yu. Trifonov, and A. V. Shapovalov, “Quasi-classical solutions of the Gross–Pitaevskii equation localized in the neighborhood of a circle,” Komp. Issled. Model., 1, No. 4, 359–365 (2009).
A. M. Kamchatnov and S. V. Korneev, “Dynamics of ring dark solitons in Bose–Einstein condensates and nonlinear optics,” Phys. Lett. A, 374, 4625–4628 (2010).
A. M. Kamchatnov and M. Salerno, “Dark soliton oscillations in Bose–Einstein condensates with multi-body interactions,” J. Phys. B, 42, 185303 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Matematika i Informatika, No. 43, 2013, pp. 15–24.
Rights and permissions
About this article
Cite this article
Laponin, V.S. Search for Soliton Solutions in the Three-Dimensional Gross–Pitaevskii Equation. Comput Math Model 25, 306–314 (2014). https://doi.org/10.1007/s10598-014-9227-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10598-014-9227-0