×

Jacobi elliptic function solutions of the Ablowitz-Ladik discrete nonlinear Schrödinger system. (English) Zbl 1197.81121

Summary: A new general Jacobi elliptic function expansion algorithm is developed to obtain exact solutions for discrete nonlinear systems. Applying this method, many exact Jabobi elliptic function travelling wave solutions for Ablowitz-Ladik discrete nonlinear Schrödinger system are derived. These doubly periodic solutions may degenerate to hyperbolic function solutions including discrete soliton solutions as the modulus \(m \rightarrow 1\) and trigonometric function solutions as \(m \rightarrow 0\), respectively.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35J99 Elliptic equations and elliptic systems
35Q55 NLS equations (nonlinear Schrödinger equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fermi, E.; Pasta, J.; Ulam, S., Collected papers of enrico fermi (1965), Chicago Press: Chicago Press Chicago, IL, p. 978
[2] Levi, D.; Yamilov, R. I., J Math Phys, 38, 6648 (1997)
[3] Sokolov, V. V.; Shabat, A. B., Sov Sci Rev C: Math Phys Rev, 4, 221 (1984)
[4] Yamilov, R. I., J Phys A: Math Gen, 27, 6839 (1994)
[5] Adler, V. E.; Shabat, A. B.; Yamilov, R. I., Theor Math Phys, 125, 1603 (2000)
[6] Cherdantsev, IYu; Yamilov, R. I., Phys D, 87, 140 (1995)
[7] Shabat, A. B.; Yamilov, R. I., Phys Lett A, 227, 15 (1997) · Zbl 0962.37509
[8] Yang, H. X.; Xu, X. X.; Ding, H. Y., Chaos, Solitons & Fractals, 26, 1091 (2005)
[9] Zhang, D. J., Chaos, Solitons & Fractals, 23, 1333 (2005)
[10] Yu, F. J.; Zhang, H. Q., Chaos, Solitons & Fractals, 29, 1173 (2006)
[11] Narita, K., Chaos, Solitons & Fractals, 13, 1121 (2002)
[12] Christodoulides, D. N.; Efremidis, N. K., Opt Lett, 27, 568 (2002)
[13] Kevrekidis, P. G.; Rasmussen, K. O.; Bishop, A. R., Int J Mod Phys B, 15, 2833 (2001)
[14] Schieer, A. J.; Takeno, S., Phys Rev Lett, 61, 970 (1988)
[15] Su, W. P.; Schrieffer, J. R.; Heeger, A. J., Phys Rev Lett, 42, 1698 (1979)
[16] Marquii, P.; Bilbaut, J. M.; Remoissenet, M., Phys Rev E, 51, 6127 (1995)
[17] Eisenberg, H.; Silberberg, Y.; Moraudotti, R.; Boyd, A.; Aitchison, J., Phys Rev Lett, 81, 3383 (1998)
[18] Trombettoni, A.; Smerzi, A., Phys Rev Lett, 86, 2353 (2001)
[19] Wadati, M.; Toda, M., J Phy Soc Jpn, 39, 1196 (1975)
[20] Deng, S. F.; Chen, D. Y., Chaos, Solitons & Fractals, 23, 1169 (2005)
[21] Hirota, R.; Hu, X. B.; Tang, X. Y., J Math Anal Appl, 288, 326 (2003)
[22] Hu, X. B.; Tam, Hon-Wah, Phys Lett A, 276, 65 (2000)
[23] Nimmo, J. J., Chaos, Solitons & Fractals, 11, 115 (2000)
[24] Qian, X. M.; Lou, S. L.; Hu, X. B., J Phys A Math Gen, 37, 2401 (2003)
[25] Wang, Z.; Zhang, H. Q., Chaos, Solitons & Fractals, 31, 197 (2007)
[26] Dai, C. Q.; Yang, Q.; Zhang, J. F., Z Naturforsch, 59a, 1 (2004)
[27] Xie, F. D.; Lu, Z. S.; Wang, D. K., Chaos, Solitons & Fractals, 27, 217 (2006)
[28] Yan, Z. Y., Nonlinear Anal, 64, 1798 (2006)
[29] Yang, Z. H.; Hon, Y. C., Chaos, Solitons & Fractals, 33, 1694 (2007)
[30] Yu, Y. X.; Wang, Q.; Gao, C. X., Chaos, Solitons & Fractals, 33, 1642 (2007)
[31] Xie, F. D.; Wang, J. Q., Chaos, Solitons & Fractals, 27, 1067 (2006)
[32] Lai, X. J.; Zhang, J. F., Z Naturforsch, 60a, 573 (2005)
[33] Dai, C. Q.; Zhang, J. F., Int J Mod Phys B, 19, 2129 (2005)
[34] Zhu, J. M.; Ma, Z. Y., Chin Phys, 14, 17 (2005)
[35] Dai, C. Q.; Zhang, J. F., Chaos, Solitons & Fractals, 27, 1042 (2006)
[36] Lü, D. Z., Chaos, Solitons & Fractals, 24, 1373 (2005)
[37] Ablowitz, M. J.; Ladik, J. F., Stud Appl Math, 55, 213 (1977)
[38] Quispel, G. R.W.; Nijhoff, F. W.; Capel, H. W.; vander Linden, J., Physica A, 125, 344 (1984)
[39] Dayydov, A., J Theor Biol, 38, 559 (1973)
[40] Takeno, S.; Hori, G., J Phys Soc Jpn, 60, 947 (1991)
[41] Aceves, A.; Angelis, C.; Luther, G.; Rubenchik, G.; Turitsyn, S., Phys Rev Lett, 75, 73 (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.