Jacobi elliptic solutions for nonlinear differential difference equations in mathematical physics. (English) Zbl 1244.39004

Gepreel, Khaled A.; Shehata, A. R.

J. Appl. Math. 2012, Article ID 710375, 15 p. (2012).

Summary: We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

MSC:

39A12

Discrete version of topics in analysis

Software:

DDESpecialSolutions

References:

[1]	E. Fermi, J. Pasta, and S. Ulam, Collected Papers of Enrico Fermi, vol. 2, The University of Chicago Press, Chicago, Ill, USA, 1965. · Zbl 1083.01508
[2]	W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Physical Review Letters, vol. 42, no. 25, pp. 1698-1701, 1979.
[3]	A. S. Davydov, “The theory of contraction of proteins under their excitation,” Journal of Theoretical Biology, vol. 38, no. 3, pp. 559-569, 1973.
[4]	P. Marquié, J. M. Bilbault, and M. Remoissenet, “Observation of nonlinear localized modes in an electrical lattice,” Physical Review E, vol. 51, no. 6, pp. 6127-6133, 1995.
[5]	M. Toda, Theory of Nonlinear Lattices, vol. 20 of Springer Series in Solid-State Sciences, Springer, Berlin, Germany, 2nd edition, 1989. · Zbl 0694.70001
[6]	M. Wadati, “Transformation theories for nonlinear discrete systems,” Progress of Theoretical Physics Supplement, vol. 59, pp. 36-63, 1976.
[7]	Y. Ohta and R. Hirota, “A discrete KdV equation and its Casorati determinant solution,” Journal of the Physical Society of Japan, vol. 60, no. 6, p. 2095, 1991.
[8]	M. J. Ablowitz and J. F. Ladik, “Nonlinear differential-difference equations,” Journal of Mathematical Physics, vol. 16, pp. 598-603, 1975. · Zbl 0296.34062
[9]	X.-B. Hu and W.-X. Ma, “Application of Hirota’s bilinear formalism to the Toeplitz lattice-some special soliton-like solutions,” Physics Letters A, vol. 293, no. 3-4, pp. 161-165, 2002. · Zbl 0985.35072
[10]	D. Baldwin, Ü. Gökta\cs, and W. Hereman, “Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations,” Computer Physics Communications, vol. 162, no. 3, pp. 203-217, 2004. · Zbl 1196.68324
[11]	S.-K. Liu, Z.-T. Fu, Z.-G. Wang, and S.-D. Liu, “Periodic solutions for a class of nonlinear differential-difference equations,” Communications in Theoretical Physics, vol. 49, no. 5, pp. 1155-1158, 2008. · Zbl 1392.34085
[12]	G.-Q. Xu and Z.-B. Li, “Applications of Jacobi elliptic function expansion method for nonlinear differential-difference equations,” Communications in Theoretical Physics, vol. 43, no. 3, pp. 385-388, 2005.
[13]	F. Xie, M. Ji, and H. Zhao, “Some solutions of discrete sine-Gordon equation,” Chaos, Solitons and Fractals, vol. 33, no. 5, pp. 1791-1795, 2007. · Zbl 1129.35456
[14]	S.-D. Zhu, “Exp-function method for the Hybrid-Lattice system,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 461-464, 2007. · Zbl 06942293
[15]	I. Aslan, “A discrete generalization of the extended simplest equation method,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 1967-1973, 2010. · Zbl 1222.65114
[16]	P. Yang, Y. Chen, and Z.-B. Li, “ADM-Padé technique for the nonlinear lattice equations,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 362-375, 2009. · Zbl 1162.65399
[17]	S.-D. Zhu, Y.-M. Chu, and S.-l. Qiu, “The homotopy perturbation method for discontinued problems arising in nanotechnology,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2398-2401, 2009. · Zbl 1189.65186
[18]	S. Zhang, L. Dong, J.-M. Ba, and Y.-N. Sun, “The (G\(^{\prime}\)/G)-expansion method for nonlinear differential-difference equations,” Physics Letters A, vol. 373, no. 10, pp. 905-910, 2009. · Zbl 1228.34096
[19]	I. Aslan, “The Ablowitz-Ladik lattice system by means of the extended (G\(^{\prime}\)/G)-expansion method,” Applied Mathematics and Computation, vol. 216, no. 9, pp. 2778-2782, 2010. · Zbl 1193.35179
[20]	S. Zhang and H.-Q. Zhang, “Discrete Jacobi elliptic function expansion method for nonlinear differential-difference equations,” Physica Scripta, vol. 80, no. 4, Article ID 045002, 2009. · Zbl 1179.35337
[21]	K. A. Gepreel, “Rational jacobi elliptic solutions for nonlinear difference differential equations,” Nonlinear Science Letters, vol. 2, pp. 151-158, 2011.
[22]	G.-C. Wu and T.-C. Xia, “A new method for constructing soliton solutions to differential-difference equation with symbolic computation,” Chaos, Solitons and Fractals, vol. 39, no. 5, pp. 2245-2248, 2009. · Zbl 1197.35250
[23]	F. Xie and J. Wang, “A new method for solving nonlinear differential-difference equation,” Chaos, Solitons and Fractals, vol. 27, no. 4, pp. 1067-1071, 2006. · Zbl 1094.34058
[24]	C.-S. Liu, “Exponential function rational expansion method for nonlinear differential-difference equations,” Chaos, Solitons and Fractals, vol. 40, no. 2, pp. 708-716, 2009. · Zbl 1197.35243
[25]	Q. Wang and Y. Yu, “New rational formal solutions for (1+1)-dimensional Toda equation and another Toda equation,” Chaos, Solitons and Fractals, vol. 29, no. 4, pp. 904-915, 2006. · Zbl 1142.37370
[26]	A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein condensates,” Physical Review Letters, vol. 86, no. 11, pp. 2353-2356, 2001.
[27]	L. Had\vzievski, A. Maluckov, M. Stepić, and D. Kip, “Power controlled soliton stability and steering in lattices with saturable nonlinearity,” Physical Review Letters, vol. 93, no. 3, Article ID 033901, 4 pages, 2004.
[28]	S. Gate and J. Herrmann, “Soliton Propagation in materials with saturable nonlinearity,” Journal of the Optical Society of America B, vol. 9, pp. 2296-2302, 1991.
[29]	S. Gate and J. Herrmann, “Soliton Propagation and soliton collision in double doped fibers with a non- Kerr- like nonlinear refractive- index change,” Optics Letters, vol. 17, pp. 484-486, 1992.