Abstract.
In this paper, we study a semi-discrete sine-Gordon (sd-SG) equation and compute various types of solutions analytically. We apply Darboux transformation to the associated spectral problem and construct N-soliton solutions of sd-SG equation in terms of ratio of ordinary determinants. In addition, we also construct explicit expressions of discrete one-kink, two-kink, kink-antikink, breather and degenerate soliton solutions of sd-SG equation in zero background.
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A. Barone, F. Esposito, C.J. Magee, A.C. Scott, Riv. Nuovo Cimento 1, 227 (1971)
A.C. Scott, F.Y.F. Chu, D.W. McLaughlin, Proc. IEEE 61, 1443 (1973)
G.M. Lamb, Elements of soliton theory (Wiley, New York, 1980)
R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equation (Academic Press, London, 1982)
M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Phys. Lett. 30, 1262 (1973)
M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Phys. Rev. Lett. 31, 125 (1973)
L.D. Faddeev, L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, in Springer Series in Soviet Mathematics (Springer, Berlin, 1987)
C. Rogers, W.F. Shadwick, Bäcklund Transformations and their Applications, in Mathematics in Science and Engineering, Vol. 161 (Academic, New York, 1982)
J. Rubinstein, J. Math. Phys. 11, 258 (1970)
R. Hirota, J. Phys. Soc. Jpn. 33, 1459 (1972)
M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981)
E. Bour, J. Ecole Imperiale Polytechn. 19, 1 (1862)
J.C. Maraver, P.G. Kevrekidis, F. Williams, The sine-Gordon Model and its Applications, From Pendulas and Josphosen Junction to Gravity and High Energy Physics (Springer Cham, Heidelberg, New York, Dordrecht, London, 2014)
I.O. Kulik, Sov. Phys. JETP 24, 1307 (1967)
J. Frenkel, T. Kontorova, J. Phys. (USSR) 1, 137 (1939)
C.P. Bean, R.W. deBlois, Bull. Am. Phys. Soc. 4, 53 (1959)
S. Yomosa, Am. Phys. Soc. A 27, 2120 (1983)
M.J. Ablowitz, J.F. Ladik, J. Math. Phys. 17, 1011 (1976)
M.J. Ablowitz, J.F. Ladik, Stud. Appl. Math. 55, 213 (1977)
R. Hirota, J. Phys. Soc. Jpn. 43, 4116 (1977)
R. Hirota, J. Phys. Soc. Jpn. 43, 2074 (1977)
R.E. Mickens, Difference Equations: Theory and Applications (Chapman & Hall, USA, 1990)
G. Fulford, P. Forrester, A. Jones, Modelling with Differential and Difference Equations (Cambridge University Press, UK, 1997)
M.J. Ablowitz, Y. Ohta, A.D. Trubatch, Chaos, Soliton Fractals 11, 159 (2000)
J. Hietarinta, F.W. Nijhoff, J. Satsuma, J. Phys. A 34, 10337 (2001)
M.J. Ablowitz, B. Prinariet, A.D. Trubatch, Discrete and continuous nonlinear Schrödinger systems (Cambridge University Press, 2004)
Y.B. Suris, The problem of integrable discretization: Hamiltonian approach (Birkhäuser, 2012)
J. Hietarinta, N. Joshi, F.W. Nijhoff, Discrete Systems and Integrability (Cambridge University Press, 2016)
T. Tsuchida, J. Phys. A 35, 7827 (2002)
J.J.C. Nimmo, J. Phys. A 30, 8693 (1997)
S.L. Zhao, J. Nonlinear Math. Phys. 23, 544 (2016)
S.L. Zhao, Y.Y. Sun, Z. Naturforsch. A 71, 1151 (2016)
R. Hirota, J. Phys. Soc. Jpn. 43, 2079 (1977)
S.J. Orfanidis, Phys. Rev. D 18, 3822 (1978)
S.J. Orfanidis, Phys. Rev. D 18, 3828 (1978)
D. Levi, O. Ragnisco, M. Bruschi, Nuovo Cimento A 58, 56 (1980)
L. Pilloni, D. Levi, Phys. Lett. A 92, 5 (1982)
M. Boiti, F. Pempinelli, B. Prinari, A. Spire, Inverse Prob. 18, 1309 (2002)
X. Kou, D.J. Zhang, Y. Si, S.L. Zhao, Commun. Theor. Phys. 55, 545 (2011)
J. Zhou, D.J. Zhang, S.L. Zhao, Phys. Lett. A 373, 3248 (2009)
V.B. Matveev, M.A. Salle, Darboux Transformations and Soliton (Springer-Verlag, Germany, 1991)
J. Cen, F. Correa, A. Fring, J. Phys. A 50, 435201 (2017)
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Hanif, Y., Saleem, U. Exact solutions of semi-discrete sine-Gordon equation. Eur. Phys. J. Plus 134, 200 (2019). https://doi.org/10.1140/epjp/i2019-12544-y
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DOI: https://doi.org/10.1140/epjp/i2019-12544-y