TARU PUBLICATIONS
Journal of Interdisciplinary Mathematics cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502

Freq.: MONTHLY - Publishes the methodological and theoretical role of mathematics and mathematical applications underpinning scientific research.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

First integral blow-up solutions to complex-valued KdV equations

* ,

* Corresponding author · click or hover a name for details

pp. 747–759Vol. 26Issue 4June 2023DOI: 10.47974/JIM-1511XML
Received:
01 Aug 2021
Accepted:
01 Feb 2022
Published Online:
19 Aug 2023
Article type:
Research Article
Language:
EN
Article no.:
JIM-1511
Pages:
747–759

Abstract

We apply Lie theory to determine the aone-parameter point transformations which leave complex-valued Korteweg-de Vries equations invariant. The conserved vectors of the systems are constructed. We provide travelling wave reductions that lead to third-order ordinary differential equations. These equations are highly nonlinear to solve directly and, therefore, we establish their first integrals. The latter is of second-order and facilitates the analysis of the complex-valued system, as well as some interesting blow-up solutions.

Keywords

Subject Classifications

35Q7F35A1583C2035L65

References

[1] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scatting (London Mathematical Society Lecture Note Series), Cambridge University Press, Cambridge, 1991. DOI: 10.1017/CBO9780511623998.
[2] H-L. An, Y. Chen, Numerical complexiton solutions for the complex KdV equation by the homotopy perturbation method, Appl. Math. Comput. 203 (2008) 125-133.
[3] J. Bi, Novel solutions of MKdV-equation with the modified Bäcklund transformation, J. Shanghai Univ., 8 (Enlgish Edition) (2004), 286-288.
[4] J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Philos. Trans. Royal Soc. London, Ser. A, 351 (1995), 107-164.
[5] G. Bowtell, A.E.G. Stuart, A particle representation of the Korteweg-de Vries soliton, J. Math. Phys., 24 (1983), 969-981.
[6] S.R. Hejazi, Rindler geodesics: Lie and Hamiltonian symmetries, conservation laws and Hamiltonian equations, J. Interdiscip. Math., 24:8 (2021), 2295-2305, DOI: 10.1080/09720502.2021.1885179.
[7] R. Hirota, The Direct Method in Soliton Theory, (A. Nagai, J. Nimmo, & C. Gilson, Eds.), Cambridge University Press, Cambridge, 2004, DOI: 10.1017/CBO9780511543043.
[8] H.C. Hu, B. Tong, S.Y. Lou, Nonsingular positon and complexiton solutions for the coupled KdV system, Phys. Lett. A, 351 (2006) 403-412.
[9] S. Jamal, Imaging Noise Suppression: Fourth-Order Partial Differential Equations and Travelling Wave Solutions, Mathematics 8 (2020) 2019.
[10] S. Jamal, A. Mathebula, Ghulam Shabbir, Noether generators and the Klein-Gordon potential on spaces with nonzero Weyl tensor, Int. J. Geom. Methods Mod. Phys. 17(7) (2020) 2050110.
[11] M. D. Kruskal, The Korteweg-de Vries equation and related evolution equations, Nonlinear Wave Motion (Lectures in Applied Mathematics 15) (ed. A. C. Newell), (Providence, RI: American Mathematical Society), 1974, 61-83.
[12] Y.-C. Li, Simple explicit formulae for finite time blowup solutions to the complex KdV equation, Chaos, Solitons & Fractals, 39 (2009), 369-372.
[13] S.Y. Lou, Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equation, J. Math. Phys. 35 (1994) 2390-2396.
[14] S.Y. Lou, B. Tong, H.C. Hu, X.Y. Tang, Coupled KdV equations derived from atmospherical dynamics, J. Phys. A: Math. Gen. 39 (2005) 513.
[15] W.X. Ma, Complexiton solutions to the Korteweg-de Vries equation, Phys. Lett. A 301 (2002) 35-44.
[16] Y. Martel, F. Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. of Math., 155 (2002), 235-280.
[17] V.A. Matveev, M.A. Salle, Darboux Transformations and Solitons (Springer Series in Nonlinear Dynamics), Springer-Verlag, Berlin, Heidelberg, 1991. 
[18] M.C. Moroke, B. Muatjetjeja, A.R. Adem, On the symbolic computation of exact solutions and conservation laws of a generalized (2+1)-dimensional Calogaro-Bogoyavlenskii-Schiff equation, J. Interdiscip. Math., 24:6 (2021), 1607-1615, DOI: 10.1080/09720502.2020.1848320.
[19] N. Mnguni, S. Jamal, Invariant solutions of fractional-order spatio-temporal partial differential equations, Int. J. Nonlinear Sci. Numer. Simul. 22 (2021), 1011-1022, DOI: 10.1515/ijnsns-2019-0239.
[20] C. Nwaigwe, S. Mungkasi, Comparison of different numerical schemes for 1D conservation laws, J. Interdiscip. Math., 24:3 (2021), 537-552, DOI: 10.1080/09720502.2020.1792665.
[21] P. Olver, Application of Lie Groups to Differential Equations, vol 107, Springer, New York, 1993. DOI: 10.1007/978-1-4684-0274-2.
[22] H. Steudel, Uber die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssätzen. Z. für Natur. 17 (1962) 129-132.

Views: 68Downloads: 7Citations: 0