Home Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation
Article
Licensed
Unlicensed Requires Authentication

Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation

  • Hai-Qiang Zhang EMAIL logo , Xiao-Li Liu and Li-Li Wen
Published/Copyright: December 8, 2015
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 71 Issue 2

Abstract

In this paper, a (2+1)-dimensional nonlinear Schrödinger (NLS) equation, which is a generalisation of the NLS equation, is under investigation. The classical and generalised N-fold Darboux transformations are constructed in terms of determinant representations. With the non-vanishing background and iterated formula, a family of the analytical solutions of the (2+1)-dimensional NLS equation are systematically generated, including the bright-line solitons, breathers, and rogue waves. The interaction mechanisms between two bright-line solitons and among three bright-line solitons are both elastic. Several patterns for first-, second, and higher-order rogue wave solutions fixed at space are displayed, namely, the fundamental pattern, triangular pattern, and circular pattern. The two-dimensional space structures of first-, second-, and third-order rogue waves fixed at time are also demonstrated.

Keywords: Breather; Darboux transformation; (2+1)-Dimensional Nonlinear Schrödinger Equation; Soliton; Rogue Wave
Corresponding author: Hai-Qiang Zhang, College of Science, University of Shanghai for Science and Technology, PO Box 253, Shanghai 200093, China, E-mail:

Acknowledgments

This work is supported by the Shanghai Leading Academic Discipline Project under Grant No. XTKX2012, by Hujiang Foundation of China under Grant No. B14005, by the Natural Science Foundation of Shanghai under Grant No. 12ZR1446800, Science and Technology Commission of Shanghai municipality, and by the National Natural Science Foundation of China under Grant Nos. 11201302 and 11171220.

References

[1] D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, Phys. Rev. E 84, 056611 (2011); 88, 013207 (2013).10.1103/PhysRevE.84.056611Search in Google Scholar

[2] C. Kharif and E. Pelinovsky, Eur. J. Mech. B 22, 603 (2003).10.1016/j.euromechflu.2003.09.002Search in Google Scholar

[3] B. L. Guo, L. M. Ling, and Q. P. Liu, Phys. Rev. E 85, 026607 (2012).10.1103/PhysRevE.85.026607Search in Google Scholar

[4] N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, Phys. Lett. A 373, 2137 (2009).10.1016/j.physleta.2009.04.023Search in Google Scholar

[5] Y. Ohta and J. K. Yang, Proc. R. Soc. A 468, 1716 (2012).10.1098/rspa.2011.0640Search in Google Scholar

[6] A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, et al., Phys. Rev. E 86, 056601 (2012).10.1103/PhysRevE.86.056601Search in Google Scholar

[7] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, et al., Nature Phys. 6, 790 (2010).10.1038/nphys1740Search in Google Scholar

[8] B. G. Zhai, W. G. Zhang, X. L. Wang, and H. Q. Zhang, Nonlinear Anal. Real World Appl. 14, 14 (2013).10.1016/j.nonrwa.2012.04.010Search in Google Scholar

[9] R. Guo, Y. F. Liu, H. Q. Hao, and F. H. Qi, Nonlinear Dyn. 80, 1221 (2015).10.1007/s11071-015-1938-zSearch in Google Scholar

[10] R. Guo, H. Q. Hao, and L. L. Zhang, Nonlinear Dyn. 74, 701 (2013).10.1007/s11071-013-0998-1Search in Google Scholar

[11] Y. Ohta and J. K. Yang, J. Phys. A 46, 105202 (2013).10.1088/1751-8113/46/10/105202Search in Google Scholar

[12] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge 1992.10.1017/CBO9780511623998Search in Google Scholar

[13] A. R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform, Academic, New York 2010.10.1016/S0074-6142(10)97003-4Search in Google Scholar

[14] D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, Phys. Rev. E 88, 013207 (2013).10.1103/PhysRevE.88.013207Search in Google Scholar PubMed

[15] B. L. Guo, L. M. Ling, and Q. P. Liu, Stud. Appl. Math. 4, 317 (2013).10.1111/j.1467-9590.2012.00568.xSearch in Google Scholar

[16] I. A. B. Strachant, Inverse Problems 8, L21 (1992).10.1088/0266-5611/8/5/001Search in Google Scholar

[17] R. Radha and M. Lakshmanan, Inverse Problems 10, L29 (1994).10.1088/0266-5611/10/4/002Search in Google Scholar

[18] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Press, Berlin 1991.10.1007/978-3-662-00922-2Search in Google Scholar

[19] V. B. Matveev, Theor. Math. Phys. 131, 483 (2002).10.1023/A:1015149618529Search in Google Scholar

[20] H. Q. Zhang, B. Tian, L. L. Li, and Y. S. Xue, Physica A 388, 9 (2009).10.1016/j.physa.2008.09.032Search in Google Scholar

Received: 2015-9-30
Accepted: 2015-11-1
Published Online: 2015-12-8
Published in Print: 2016-2-1

©2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation
  3. The Modified Simple Equation Method, the Exp-Function Method, and the Method of Soliton Ansatz for Solving the Long–Short Wave Resonance Equations
  4. Application of the Reverberation-Ray Matrix to the Non-Fourier Heat Conduction in Functionally Graded Materials
  5. Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations
  6. Structural, Electronic, Magnetic and Optical Properties of Ni,Ti/Al-based Heusler Alloys: A First-Principles Approach
  7. Ab Initio Calculations on the Structural, Mechanical, Electronic, Dynamic, and Optical Properties of Semiconductor Half-Heusler Compound ZrPdSn
  8. Qualitative Behaviour of Generalised Beddington Model
  9. Studying Nuclear Level Densities of 238U in the Nuclear Reactions within the Macroscopic Nuclear Models
  10. Total π-Electron Energy of Conjugated Molecules with Non-bonding Molecular Orbitals
  11. Investigation of Thermal Expansion and Physical Properties of Carbon Nanotube Reinforced Nanocrystalline Aluminum Nanocomposite
  12. Bistable Bright Optical Spatial Solitons due to Charge Drift and Diffusion of Various Orders in Photovoltaic Photorefractive Media Under Closed-Circuit Conditions
  13. Application of a Differential Transform Method to the Transient Natural Convection Problem in a Vertical Tube with Variable Fluid Properties
https://doi.org/10.1515/zna-2015-0408
Keywords for this article
Breather; Darboux transformation; (2+1)-Dimensional Nonlinear Schrödinger Equation; Soliton; Rogue Wave

Articles in the same Issue

  1. Frontmatter
  2. Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation
  3. The Modified Simple Equation Method, the Exp-Function Method, and the Method of Soliton Ansatz for Solving the Long–Short Wave Resonance Equations
  4. Application of the Reverberation-Ray Matrix to the Non-Fourier Heat Conduction in Functionally Graded Materials
  5. Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations
  6. Structural, Electronic, Magnetic and Optical Properties of Ni,Ti/Al-based Heusler Alloys: A First-Principles Approach
  7. Ab Initio Calculations on the Structural, Mechanical, Electronic, Dynamic, and Optical Properties of Semiconductor Half-Heusler Compound ZrPdSn
  8. Qualitative Behaviour of Generalised Beddington Model
  9. Studying Nuclear Level Densities of 238U in the Nuclear Reactions within the Macroscopic Nuclear Models
  10. Total π-Electron Energy of Conjugated Molecules with Non-bonding Molecular Orbitals
  11. Investigation of Thermal Expansion and Physical Properties of Carbon Nanotube Reinforced Nanocrystalline Aluminum Nanocomposite
  12. Bistable Bright Optical Spatial Solitons due to Charge Drift and Diffusion of Various Orders in Photovoltaic Photorefractive Media Under Closed-Circuit Conditions
  13. Application of a Differential Transform Method to the Transient Natural Convection Problem in a Vertical Tube with Variable Fluid Properties
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 29.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/zna-2015-0408/html
Scroll to top button