We modified the rational Jacobi elliptic functions method to construct some new exact solutions for nonlinear differential difference equations in mathematical physics via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the discrete nonlinear Klein-Gordon equation, and the quintic discrete nonlinear Schrodinger equation. Some new types of the Jacobi elliptic solutions are obtained for some nonlinear differential difference equations in mathematical physics. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

## References

E. Fermi, J. Pasta, and S. Ulam,

*Collected Papers of Enrico Fermi II*, University of Chicago Press, Chicago, Ill, USA, 1965. MR205808 1083.01508 E. Fermi, J. Pasta, and S. Ulam,*Collected Papers of Enrico Fermi II*, University of Chicago Press, Chicago, Ill, USA, 1965. MR205808 1083.01508 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. 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. 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. MR1121839 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. MR1121839 M. J. Ablowitz and J. Ladik, “Nonlinear differential-difference equations,”

*Journal of Mathematical Physics*, vol. 16, pp. 598–603, 1975. MR377223 10.1063/1.522558 M. J. Ablowitz and J. Ladik, “Nonlinear differential-difference equations,”*Journal of Mathematical Physics*, vol. 16, pp. 598–603, 1975. MR377223 10.1063/1.522558 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. MR1889299 0985.35072 10.1016/S0375-9601(01)00850-7 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. MR1889299 0985.35072 10.1016/S0375-9601(01)00850-7 D. Baldwin, U. Goktas, 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. MR2112229 10.1016/j.cpc.2004.07.002 1196.68324 D. Baldwin, U. Goktas, 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. MR2112229 10.1016/j.cpc.2004.07.002 1196.68324 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. MR2489647 10.1088/0253-6102/49/5/14 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. MR2489647 10.1088/0253-6102/49/5/14 C. Qiong and L. Bin, “Applications of Jacobi elliptic function expansion method for nonlinear differential-difference equations,”

*Communications in Theoretical Physics*, vol. 43, no. 3, pp. 385–388, 2005. MR2185672 10.1088/0253-6102/43/3/001 C. Qiong and L. Bin, “Applications of Jacobi elliptic function expansion method for nonlinear differential-difference equations,”*Communications in Theoretical Physics*, vol. 43, no. 3, pp. 385–388, 2005. MR2185672 10.1088/0253-6102/43/3/001 F. Xie, M. Jia, and H. Zhao, “Some solutions of discrete sine-Gordon equation,”

*Chaos, Solitons & Fractals*, vol. 33, no. 5, pp. 1791–1795, 2007. MR2321512 1129.35456 10.1016/j.chaos.2006.03.018 F. Xie, M. Jia, and H. Zhao, “Some solutions of discrete sine-Gordon equation,”*Chaos, Solitons & Fractals*, vol. 33, no. 5, pp. 1791–1795, 2007. MR2321512 1129.35456 10.1016/j.chaos.2006.03.018 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. 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. 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. MR2592609 1222.65114 10.1016/j.cnsns.2009.08.008 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. MR2592609 1222.65114 10.1016/j.cnsns.2009.08.008 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. MR2509911 10.1016/j.amc.2009.01.010 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. MR2509911 10.1016/j.amc.2009.01.010 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. MR2557377 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. MR2557377 S. Zhang, L. Dong, J. Ba, and Y. Sun, “The $(\dfrac{{G}^{'}}{G})$-expansion method for nonlinear differential-difference equations,”

*Physics Letters A*, vol. 373, no. 10, pp. 905–910, 2009. MR2499365 1228.34096 10.1016/j.physleta.2009.01.018 S. Zhang, L. Dong, J. Ba, and Y. Sun, “The $(\dfrac{{G}^{'}}{G})$-expansion method for nonlinear differential-difference equations,”*Physics Letters A*, vol. 373, no. 10, pp. 905–910, 2009. MR2499365 1228.34096 10.1016/j.physleta.2009.01.018 I. Aslan, “The Ablowitz-Ladik lattice system by means of the extended $({G}^{'}/G)$-expansion method,”

*Applied Mathematics and Computation*, vol. 216, no. 9, pp. 2778–2782, 2010. MR2653093 1193.35179 10.1016/j.amc.2010.03.124 I. Aslan, “The Ablowitz-Ladik lattice system by means of the extended $({G}^{'}/G)$-expansion method,”*Applied Mathematics and Computation*, vol. 216, no. 9, pp. 2778–2782, 2010. MR2653093 1193.35179 10.1016/j.amc.2010.03.124 A. Khaled Gepreel, “Rational Jacobi elliptic solutions for nonlinear difference differential equations,”

*Nonlinear Science Letters A*, vol. 2, pp. 151–158, 2011. A. Khaled Gepreel, “Rational Jacobi elliptic solutions for nonlinear difference differential equations,”*Nonlinear Science Letters A*, vol. 2, pp. 151–158, 2011. K. A. Gepreel and A. R. Shehata, “Rational Jacobi elliptic solutions for nonlinear differential difference lattice equations,”

*Applied Mathematics Letters*, vol. 25, no. 9, pp. 1173–1178, 2012. MR2930741 10.1016/j.aml.2012.02.028 1252.34013 K. A. Gepreel and A. R. Shehata, “Rational Jacobi elliptic solutions for nonlinear differential difference lattice equations,”*Applied Mathematics Letters*, vol. 25, no. 9, pp. 1173–1178, 2012. MR2930741 10.1016/j.aml.2012.02.028 1252.34013 G. Wu and T. Xia, “A new method for constructing soliton solutions to differential-difference equation with symbolic computation,”

*Chaos, Solitons & Fractals*, vol. 39, no. 5, pp. 2245–2248, 2009. MR2519429 1197.35250 10.1016/j.chaos.2007.06.107 G. Wu and T. Xia, “A new method for constructing soliton solutions to differential-difference equation with symbolic computation,”*Chaos, Solitons & Fractals*, vol. 39, no. 5, pp. 2245–2248, 2009. MR2519429 1197.35250 10.1016/j.chaos.2007.06.107 F. Xie and J. Wang, “A new method for solving nonlinear differential-difference equation,”

*Chaos, Solitons & Fractals*, vol. 27, no. 4, pp. 1067–1071, 2006. MR2166688 10.1016/j.chaos.2005.04.078 F. Xie and J. Wang, “A new method for solving nonlinear differential-difference equation,”*Chaos, Solitons & Fractals*, vol. 27, no. 4, pp. 1067–1071, 2006. MR2166688 10.1016/j.chaos.2005.04.078 C. Liu, “Exponential function rational expansion method for nonlinear differential-difference equations,”

*Chaos, Solitons & Fractals*, vol. 40, no. 2, pp. 708–716, 2009. MR2527821 1197.35243 10.1016/j.chaos.2007.08.018 C. Liu, “Exponential function rational expansion method for nonlinear differential-difference equations,”*Chaos, Solitons & Fractals*, vol. 40, no. 2, pp. 708–716, 2009. MR2527821 1197.35243 10.1016/j.chaos.2007.08.018 Q. Wang and Y. Yu, “New rational formal solutions for $(1+1)$-dimensional Toda equation and another Toda equation,”

*Chaos, Solitons & Fractals*, vol. 29, no. 4, pp. 904–915, 2006. MR2222140 1142.37370 10.1016/j.chaos.2005.08.053 Q. Wang and Y. Yu, “New rational formal solutions for $(1+1)$-dimensional Toda equation and another Toda equation,”*Chaos, Solitons & Fractals*, vol. 29, no. 4, pp. 904–915, 2006. MR2222140 1142.37370 10.1016/j.chaos.2005.08.053 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. 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. L. Hadžievski, 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. L. Hadžievski, 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. 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. 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. 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. 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. P. Kevrekidis, V. Konotop, and A. Bishop, “Discrete compactons: some exact results,”

*Journal of Physics A*, vol. 35, no. 45, pp. L641–L652, 2002. MR1946931 1040.37067 10.1088/0305-4470/35/2/307 P. Kevrekidis, V. Konotop, and A. Bishop, “Discrete compactons: some exact results,”*Journal of Physics A*, vol. 35, no. 45, pp. L641–L652, 2002. MR1946931 1040.37067 10.1088/0305-4470/35/2/307 K. Maruno, Y. Ohta, and N. Joshi, “Exact localized solutions of quintic discrete nonlinear Schrödinger equation,”

*Physics Letters A*, vol. 311, no. 2-3, pp. 214–220, 2003. MR1982345 10.1016/S0375-9601(03)00499-7 K. Maruno, Y. Ohta, and N. Joshi, “Exact localized solutions of quintic discrete nonlinear Schrödinger equation,”*Physics Letters A*, vol. 311, no. 2-3, pp. 214–220, 2003. MR1982345 10.1016/S0375-9601(03)00499-7 Z. Wang and W.-X. Ma, “Discrete Jacobi sub-equation method for nonlinear differential-difference equations,”

*Mathematical Methods in the Applied Sciences*, vol. 33, no. 12, pp. 1463–1472, 2010. MR2680686 1200.34104 10.1002/mma.1275 Z. Wang and W.-X. Ma, “Discrete Jacobi sub-equation method for nonlinear differential-difference equations,”*Mathematical Methods in the Applied Sciences*, vol. 33, no. 12, pp. 1463–1472, 2010. MR2680686 1200.34104 10.1002/mma.1275