Abstract and Applied Analysis

New Rational Homoclinic and Rogue Waves for Davey-Stewartson Equation

Changfu Liu, Chuanjian Wang, Zhengde Dai, and Jun Liu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A new method, homoclinic breather limit method (HBLM), for seeking rogue wave solution of nonlinear evolution equation is proposed. A new family of homoclinic breather wave solution, and rational homoclinic solution (homoclinic rogue wave) for DSI and DSII equations are obtained using the extended homoclinic test method and homoclinic breather limit method (HBLM), respectively. Moreover, rogue wave solution is exhibited as period of periodic wave in homoclinic breather wave approaches to infinite. This result shows that rogue wave can be generated by extreme behavior of homoclinic breather wave for higher dimensional nonlinear wave fields.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 572863, 8 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412606304

Digital Object Identifier
doi:10.1155/2014/572863

Mathematical Reviews number (MathSciNet)
MR3198214

Zentralblatt MATH identifier
07022637

Citation

Liu, Changfu; Wang, Chuanjian; Dai, Zhengde; Liu, Jun. New Rational Homoclinic and Rogue Waves for Davey-Stewartson Equation. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 572863, 8 pages. doi:10.1155/2014/572863. https://projecteuclid.org/euclid.aaa/1412606304


Export citation

References

  • M. Hopkin, “Sea snapshots will map frequency of freak waves,” Nature, vol. 430, no. 6999, p. 492, 2004.
  • C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue Waves in the Ocean: Observation, Theories and Modeling, Springer, New York, NY, USA, 2009.
  • N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Rogue waves and rational solutions of the nonlinear Schrödinger equation,” Physical Review E, vol. 80, no. 2, Article ID 026601, 9 pages, 2009.
  • Y. C. Ma, “The perturbed plane-wave solutions of the cubic Schrödinger equation,” Studies in Applied Mathematics, vol. 60, no. 1, pp. 43–58, 1979.
  • D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature, vol. 450, no. 7172, pp. 1054–1057, 2007.
  • Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Rogue waves as spatial energy concentrators in arrays of nonlinear waveguides,” Optics Letters, vol. 34, no. 19, pp. 3015–3017, 2009.
  • A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium,” Physical Review Letters, vol. 101, no. 6, Article ID 065303, 4 pages, 2008.
  • Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Physical Review A: Atomic, Molecular, and Optical Physics, vol. 80, no. 3, Article ID 033610, 5 pages, 2009.
  • A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-gaussian statistics and extreme waves in a nonlinear optical cavity,” Physical Review Letters, vol. 103, no. 17, Article ID 173901, 4 pages, 2009.
  • Z. Y. Yan, “Financial rogue waves,” Communications in Theoretical Physics, vol. 54, no. 5, pp. 947–949, 2010.
  • D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” Australian Mathematical Society B: Applied Mathematics, vol. 25, no. 1, pp. 16–43, 1983.
  • Y. Tao and J. He, “Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation,” Physical Review E, vol. 85, no. 2, Article ID 026601, 7 pages, 2012.
  • U. Bandelow and N. Akhmediev, “Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa-Satsuma case,” Physics Letters A: General, Atomic and Solid State Physics, vol. 376, no. 18, pp. 1558–1561, 2012.
  • Y. Ohta and J. Yang, “Dynamics of rogue waves in the Davey-Stewartson II equation,” Journal of Physics A: Mathematical and Theoretical, vol. 46, no. 10, Article ID 105202, 19 pages, 2013.
  • L. C. Zhao and J. Liu, “Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation,” Physical Review E, vol. 87, no. 1, Article ID 013201, 8 pages, 2013.
  • W. P. Zhong, “Rogue wave solutions of the generalized one-dimensional Gross-Pitaevskii equation,” Journal of Nonlinear Optical Physics & Materials, vol. 21, no. 2, Article ID 1250026, 9 pages, 2012.
  • C. Li, J. He, and K. Porseizan, “Rogue waves of the Hirota and the Maxwell-Bloch equations,” Physical Review E, vol. 87, no. 1, Article ID 012913, 13 pages, 2013.
  • G. Mu and Z. Qin, “Rogue waves for the coupled Schrödinger-Boussinesq equation and the coupled Higgs equation,” Journal of the Physical Society of Japan, vol. 81, no. 8, Article ID 084001, 6 pages, 2012.
  • A. Davey and K. Stewartson, “On three-dimensional packets of surface waves,” Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, vol. 338, pp. 101–110, 1974.
  • V. A. Arkadiev, A. K. Pogrebkov, and M. C. Polivanov, “Inverse scattering transform method and soliton solutions for Davey-Stewartson II equation,” Physica D: Nonlinear Phenomena, vol. 36, no. 1-2, pp. 189–197, 1989.
  • M. Boiti, J. J. Leon, L. Martina, and F. Pempinelli, “Scattering of localized solitons in the plane,” Physics Letters A, vol. 132, no. 8-9, pp. 432–439, 1988.
  • M. Boiti, J. P. Leon, and F. Pempinelli, “Multidimensional solitons and their spectral transforms,” Journal of Mathematical Physics, vol. 31, no. 11, pp. 2612–2618, 1990.
  • K. W. Chow, “A class of doubly periodic waves for nonlinear evolution equations,” Wave Motion, vol. 35, no. 1, pp. 71–90, 2002.
  • A. S. Fokas and P. M. Santini, “Coherent structures in multidimensions,” Physical Review Letters, vol. 63, no. 13, pp. 1329–1333, 1989.
  • C. R. Gilson, “Resonant behaviour in the Davey-Stewartson equation,” Physics Letters. A, vol. 161, no. 5, pp. 423–428, 1992.
  • J. Hietarinta and R. Hirota, “Multidromion solutions to the Davey-Stewartson equation,” Physics Letters A, vol. 145, no. 5, pp. 237–244, 1990.
  • J. Satsuma and M. J. Ablowitz, “Two-dimensional lumps in nonlinear dispersive systems,” Journal of Mathematical Physics, vol. 20, no. 7, pp. 1496–1503, 1979.
  • M. Tajiri and T. Arai, “Periodic soliton solutions to the Davey-Stewartson equation,” Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 30, part 1, pp. 210–217, 2000.
  • Y. Watanabe and M. Tajiri, “Periodic soliton resonance: solutions to the Davey-Stewartson I equation,” Journal of the Physical Society of Japan, vol. 67, no. 3, pp. 705–708, 1998.
  • Z. Dai and J. Huang, “Homoclinic tubes for the Davey-Stewartson II equation with periodic boundary conditions,” Chinese Journal of Physics, vol. 43, no. 2, pp. 349–356, 2005.
  • J. Huang and Z. Dai, “Homoclinic solutions for Davey-Stewartson equation,” Chaos, Solitons & Fractals, vol. 35, no. 5, pp. 996–1002, 2008.
  • Z. D. Dai, A. J. Zhu, S. L. Li, and D. L. Li, “Homoclinic tubes for the Davey-Stewartson II equation with periodic boundary conditions,” Chinese Journal of Physics, vol. 45, no. 1, pp. 62–71, 2007.
  • L. B. Santos, M. Rojas-Medar, and G. Ruiz-Garzón, “Existence of weakly efficient solutions in vector optimization,” Acta Mathematica Sinica, vol. 24, no. 4, pp. 599–606, 2008. \endinput