Abstract
We briefly review the recent progress in obtaining (2+1) dimensional integrable generalizations of soliton equations in (1+1) dimensions. Then, we develop an algorithmic procedure to obtain interesting classes of solutions to these systems. In particular using a Painlevé singularity structure analysis approach, we investigate their integrability properties and obtain their appropriate Hirota bilinearized forms. We identify line solitons and from which we introduce the concept of ghost solitons, which are patently boundary effects characteristic of these (2+1) dimensional integrable systems. Generalizing these solutions, we obtain exponentially localized solutions, namely the dromions which are driven by the boundaries. We also point out the interesting possibility that while the physical field itself may not be localized, either the potential or composite fields may get localized. Finally, the possibility of generating an even wider class of localized solutions is hinted by using curved solitons.
Similar content being viewed by others
References
N J Zabusky and M D Kruskal,Phys. Rev. Lett. 15, 240 (1965)
C S Gardner, J M Greene, M D Kruskal and R M Miura,Phys. Rev. Lett. 19, 1095 (1967)
M J Ablowitz and P A Clarkson,Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991)
M Lakshmanan (ed.),Solitons: Introduction and Applications, (Springer-Verlag, Berlin, 1988)
M Lakshmanan (ed.)Chaos, Solitons and Fractals (Special issue on Solitons in Science and Engineering: Theory and Applications)5, 2213–2656 (1995)
P L Bhatnagar,Nonlinear Waves in One-Dimensional Dispersive Systems (Oxford University Press, Calcutta, 1979)
M Boiti, J J P Leon, L Martina and F Pempinelli,Phys. Lett. A132, 432 (1988)
A S Fokas and P M Santini,Physica D44, 99 (1990)
M J Ablowitz and A S Fokas,Stud. Appl. Math. 69, 135 (1983)
A S Fokas and M J Ablowitz,Stud. Appl. Math. 69, 211 (1983)
A S Fokas and M J Ablowitz,J. Math. Phys. 25, 2494 (1984)
B G Konopelchenko,Solitons in Multidimensions (Springer-Verlag, Berlin, 1993)
V E Zakharov and S V Manakov,Funct. Anal. Appl. 19, 89 (1985)
F Calogero,Lett. Nuovo Cimento 14, 443 (1975)
M Boiti, J J P Leon, M Manna and F Pempinelli,Inv. Prob. 2, 271 (1986)
S P Novikov and A P Veselov,Physica D18, 267 (1986)
A S Fokas,Inv. Prob. 10, L19 (1994)
I A B Strachan,Inv. Prob. 8, L21 (1992);J. Math. Phys. 34, 243 (1993)
B G Konopelchenko and C Rogers,Phys. Lett. A158, 391 (1991);J. Math. Phys. 34, 214 (1993)
J J C Nimmo,Phys. Lett. A168, 113 (1992)
S Chakravarty, S L Kent and E T Newman,J. Math. Phys. 36, 763 (1995)
M Lakshmanan and R Sahadevan,Phys. Rep. 224, 1 (1993)
M J Ablowitz, A Ramani and H Segur,J. Math. Phys. 21, 715 (1980)
M Lakshmanan,Int. J. Bifurcation and Chaos 3, 3 (1993)
J Weiss, M Tabor and G Carnevale,J. Math. Phys. 24, 522 (1984)
M Daniel, M D Kruskal, M Lakshmanan and K Nakamura,J. Math. Phys. 33, 771 (1992)
R Radha and M Lakshmanan,Inv. Problems 10, L29 (1994)
R Radha and M Lakshmanan,J. Math. Phys. 35, 4746 (1994)
R Radha and M Lakshmanan,Phys. Lett. A197, 7 (1995)
R Radha and M Lakshmanan,J. Phys. A29, 1551 (1996)
R Radha and M Lakshmanan,Chaos, Solitons and Fractals 8, 17 (1997)
R Radha and M Lakshmanan,J. Math. Phys. 38, 292 (1997)
R Radha and M Lakshmanan,J. Phys. A30, (1997) to appear
R Radha, Ph.D Thesis (Bharathidasan University, 1996)
Sen-yue Lou,J. Phys. A28, 7227 (1995)
J Hietarinta,Phys. Lett. A149, 113 (1990)
V Dubrovsky and B G Konopelchenko,Inv. Prob. 9, 391 (1993)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lakshmanan, M., Radha, R. Localized coherent structures of (2+1) dimensional generalizations of soliton systems. Pramana - J Phys 48, 163–188 (1997). https://doi.org/10.1007/BF02845629
Issue Date:
DOI: https://doi.org/10.1007/BF02845629