Abstract
In this paper, we investigate two extended higher-order KdV models (i.e., the extended Sawada–Kotera equation and the extended Lax equation), which can successfully describe propagation of dimly nonlinear long waves in fluids and ion-acoustic waves in harmonic sparklers. First, we present a general formula of multisoliton solutions of the two models. We then build the interaction solutions in terms of hyperbolic and sinusoidal functions by using multisoliton solutions with appropriate complex conjugate parameters controlling the phase shifts, propagation direction and energies of the waves. In particular, we present their collision solutions in the identical plane with different parametric constraints, which degenerate to the line rogue waves, x-shaped rogue waves, cnoidal periodic waves, interactions of rogue and bell waves, line breather and double breather waves. The dynamical characteristics of the wave solutions are shown graphically by choosing some special parameter values.
Similar content being viewed by others
References
Y S Kivshar and B A Malomed, Rev. Mod. Phys. 61, 763 (1989)
Y L Dang, H J Li and J Lin, Nonlinear Dyn. 88, 489 (2017)
S Novikov, S V Manakov, L P Pitaevskii and V E Zakharo, Springer, US, XI-276 (1984)
A H Kara and C M Khalique, J. Phys. A 38, 4629 (2005)
V B Matveev and M A Salle, Darboux transformations and solitons (Springer, Berlin, 1991)
W X Ma, T Huang and Y Zhang, Phys. Scr. 82, 065003 (2010)
W Malfliet and W Hereman, Phys. Scr. 54(6), 563 (1996)
H O Roshid and M A Rahman, Results Phys. 4, 150 (2014)
M N Alam, M G Hafez, M A Akbar and H O Roshid, Alexandria Eng. J. 54(3), 635 (2015)
Z Feng, J. Phys. A 35, 343 (2002)
R Hirota (Cambridge University Press, Cambridge, 2004)
R Hirota and J Sutsuma, Phys. Lett. A 85(8–9), 407 (1981)
M B Hossen, H O Roshid and M Z Ali, Phys. Lett. A 382, 1268 (2018)
W Q Peng, S F Tian and T T Zhang, Phys. Fluids 31, 102107 (2019)
H O Roshid and W X Ma, Phys. Lett. A 382, 3262 (2018)
A M Wazwaz, Phys. Scr. 83, 35003 (2011)
T R Marchant and N F Smyth, IMA J. Appl. Math. 56(2), 157 (1996)
T R Marchant and N F Smyth, J. Fluid Mech. 221, 263 (1990)
Y Wang and Y Chen, Nonlinear Anal.: Real World Appl. 31, 533 (2016)
H R Dullin, G A Gottwald, D Holm and C Holm, Fluid Dyn. Res. 33, 73 (2003)
Y H Wang and Y Chen, Pramana – J. Phys. 81(5), 737 (2013)
Y Kodama, Phys. Lett. A 107(6), 245 (1985)
A M Wazwaz, Nonlinear Dyn. 83(1), 591 (2016)
J Satsuma and M J Ablowitz, J. Math. Phys. 20, 1496 (1979)
A M Wazwaz, J. Ocean Eng. Sci. 1, 181 (2016)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rahman, Z., Ali, M.Z., Harun-Or-Roshid et al. Dynamical structures of interaction wave solutions for the two extended higher-order KdV equations. Pramana - J Phys 95, 134 (2021). https://doi.org/10.1007/s12043-021-02155-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12043-021-02155-4
Keywords
- Extended Sawada–Kotera equation
- extended Lax equation
- Hirota bilinear method
- propagation angle
- lump wave
- breather wave
- cnoidal periodic waves