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Published: 2022-12-22

The Generalized Petviashvili Iteration Method for Determined Stationary Waves Solutions of Nonlinear Schrödinger Equations with Potential Function V(x)

Program Studi Matematika, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Mataram
Prodi Tadris Fisika, Fakultas Tarbiyah dan Keguruan, Universitas Islam Negeri Mataram
a generalized Petviashvili iteration method; stationary wave; Nonlinear Schrödinger equation; Bose-Einstein condensation; potential function


This research discusses a numerical method for determining the stationary waves as a solution of Nonlinear Schrödinger (NLS) equations. In general, solutions for the partial differential equations can be solved analytically. However, most the solutions of the nonlinear wave equations are difficult to determine analytically. Therefore, a numerical approach is needed to determine the solution of the NLS equation. One of the numerical methods can be used to find the solution of the NLS equation is the Petviashvili iteration method. For case study, the NLS equation has been generated by the theory of Bose-Einstein condensation which contain potential function . To solve this problem, we generalized Petviashvili iteration method to determine the stationary waves solution easily. The most interesting result for this study is by modification of Petviashvili iteration method, we can make it easier to find a stationary solution for the nonlinear Schrodinger equation which containing the Bose-Einstein condensation potential function .


  1. Achdou, Y., Buera, F. J., Lions, P., & Moll, B. (2014). Partial differential equation models in macroeconomics. Journal Philosophical Transaction of The Royal Society, A 372:0397, 1–19. Trans. R. Soc
  2. Álvarez, J., & Durán, A. (2014). Journal of Computational and Applied Petviashvili type methods for traveling wave computations : I . Analysis of convergence. Journal of Computational and Applied Mathematics, 266, 39–51.
  3. Cho, Y. (2015). A modified Petviashvili method using simple stabilizing. Journal of Engineering Mathematics, 91(1), 37–57.
  4. Koleva, M. N., & Vulkov, L. G. (2013). A second-order positivity preserving numerical method for Gamma equation. Applied Mathematics and Computation, 220, 722–734.
  5. Lakoba, T. I., & Yang, J. (2007). A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity, 226, 1668–1692.
  6. Michel Remoissenet. (2010). Analytical and Numerical Study of Soliton Collisions. Waves Called Solitons: Concepts and Experiments, Springer- Verlag (1999).
  7. Nisar, K. S., Ali, K. K., Inc, M., Mehanna, M. S., Rezazadeh, H., & Akinyemi, L. (2022). New solutions for the generalized resonant nonlinear Schrödinger equation. Results in Physics, 33(November 2021).
  8. Pelinovsky & Stepanyants, Y. A. (2004). Convergences of Petviashvili’s Iteration Method for Numerical Approximation of Stationary Solutions of Nonlinear Wave Equations. SIAM J, 42(2004), 1110–1127.
  9. Rezazadeh, H., Ullah, N., Akinyemi, L., Shah, A., Mirhosseini-Alizamin, S. M., Chu, Y. M., & Ahmad, H. (2021). Optical soliton solutions of the generalized non-autonomous nonlinear Schrödinger equations by the new Kudryashov’s method. Results in Physics, 24.
  10. Serkin, V. N., & Hasegawa, A. (2000). Novel soliton solutions of the nonlinear Schrodinger equation model. Physical Review Letters, 85(21), 4502–4505.
  11. Weiss, J. (2012). Modified equations , rational solutions , and the Painlevé property for the Kadomtsev – Petviashvili and Hirota – Satsuma equations Kadomtsev-Petviashvili and Hirota-Satsuma equations, 2174(1985).
  12. Yang, J., & Lakoba, T. I. (2007). Accelerated Imaginary-time Evolution Methods for the Computation of Solitary Waves. J. Comput. Phys, 226, 265–292.
  13. Zada, L., & Aziz, I. (2022). Numerical solution of fractional partial differential equations via Haar wavelet. Numerical Methods for Partial Differential Equations, 38(2), 222–242.

How to Cite

Robbaniyyah, N. A., & Irwan. (2022). The Generalized Petviashvili Iteration Method for Determined Stationary Waves Solutions of Nonlinear Schrödinger Equations with Potential Function V(x). EIGEN MATHEMATICS JOURNAL, 5(2), 47–53.