Abstract
We define a Darboux transformation in terms of a quasideterminant Darboux matrix on the solutions of a semidiscrete short-pulse equation. We also give a quasideterminant formula for N-loop soliton solutions and obtain a general expression for the multiloop solution expressed in terms of quasideterminants. Using quasideterminants properties, we find explicit solutions and as an example compute one- and two-loop soliton solutions in explicit form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Grammaticos, T. Tamizhmani, and Y. Kosmann-Schwarzbach, eds., Discrete Integrable Systems (Lect. Notes Phys., Vol. 644), Springer, Berlin (2004).
Y. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach (Progr. Math., Vol. 219), Birkh äuser, Basel (2003).
M. J. Ablowitz and J. F. Ladik, “Nonlinear differential–difference equations and Fourier analysis,” J. Math. Phys., 17, 1011–1018 (1976).
M. J. Ablowitz and J. F. Ladik, “A nonlinear difference scheme and inverse scattering,” Stud. Appl. Math., 55, 213–229 (1976).
R. Hirota, “Nonlinear partial difference equations: I. A difference analogue of the Korteweg–de Vries equation,” J. Phys. Soc. Japan, 43, 1424–1433 (1977).
R. Hirota, “Nonlinear partial difference equations: II. Discrete-time Toda equation,” J. Phys. Soc. Japan, 43, 2074–2078 (1977).
B. F. Feng, K. Maruno, and Y. Ohta, “Integrable discretizations of the short pulse equation,” J. Phys. A: Math. Theor., 43, 085203 (2010).
B.-F. Feng, K. Maruno, and Y. Ohta, “Self-adaptive moving mesh schemes for short pulse type equations and their Lax pairs,” Pac. J. Math. Ind., 6, 8 (2014).
T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Phys. D, 196, 90–105 (2004).
Y. Chung, C. K. T. Jones, T. Schäfer, and C. E. Wayne, “Ultra-short pulses in linear and nonlinear media,” Nonlinearity, 18, 1351–1374 (2005).
A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Japan, 74, 239–241 (2005).
A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A: Math. Gen., 39, L361–L367 (2006).
B.-F. Feng, J. Inoguchi, K. Kajiwara, K. Maruno, and Y. Ohta, “Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves,” J. Phys. A: Math. Theor., 44, 395201 (2011).
V. K. Kuetche, T. B. Bouetou, and T. C. Kofane, “On two-loop soliton solution of the Schäfer–Wayne short-pulse equation using Hirota’s method and Hodnett–Moloney approach,” J. Phys. Soc. Japan, 76, 024004 (2007).
Y. Matsuno, “Multiloop soliton and multibreather solutions of the short pulse model equation,” J. Phys. Soc. Japan, 76, 084003 (2007).
U. Saleem and M. Hassan, “Darboux transformation and multisoliton solutions of the short pulse equation,” J. Phys. Soc. Japan, 81, 094008 (2012).
L. Ling, B.-F. Feng, and Z. Zhu, “Multi-soliton, multi-breather, and higher order rogue wave solutions to the complex short pulse equation,” Phys. D, 327, 13–29 (2016).
H. Wajahat A. Riaz and M. ul Hassan, “Darboux transformation of a semi-discrete coupled dispersionless integrable system,” Commun. Nonlinear Sci. Numer. Simul., 48, 387–397 (2017).
V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin (1991).
M. Hassan, “Darboux transformation of the generalized coupled dispersionless integrable system,” J. Phys. A: Math. Theor., 42, 065203 (2009).
I. M. Gel’fand and V. S. Retakh, “Determinants of matrices over noncommutative rings,” Funct. Anal. Appl., 25, No. 2, 91–102 (1991).
I. M. Gelfand, S. Gelfand, V. M. Retakh, and R. L. Wilson, “Quasideterminants,” Adv. Math., 193, 56–141 (2005).
C. X. Li and J. J. C. Nimmo, “Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation,” Proc. Roy. Soc. London Ser. A, 464, 951–966 (2008); arXiv:0711.2594v2 [nlin.SI] (2007).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wajahat, H., Riaz, A. & Hassan, M. Darboux Transformation for a Semidiscrete Short-Pulse Equation. Theor Math Phys 194, 360–376 (2018). https://doi.org/10.1134/S0040577918030042
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577918030042