Abstract
New several classes of exact solutions are obtained in terms of the Weierstrass elliptic function for some nonlinear partial differential equations modeling ion-acoustic waves as well as dusty plasmas in laboratory and space sciences. The Weierstrass elliptic function solutions of the Schamel equation, a fifth order dispersive wave equation and the Kawahara equation are constructed. Moreover, Jacobi elliptic function solutions and solitary wave solutions of the Schamel equation are also given. The stability of some periodic wave solutions is computationally studied.
Similar content being viewed by others
References
M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991)
M. Wadati, H. Sanuki, K. Koono, Progr. Theor. Phys. 53, 419 (1975); K. Koono, M. Wadati, Progr. Theor. Phys. 53, 1652 (1975); G.L. Lamb, Rev. Mod. Phys. 43, 99 (1971); A.H. Khater, D.K. Callebaut, W. Malfliet, E.S. Kamel, J. Comput. Appl. Math. 140, 435 (2002)
C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Phys. Rev. Lett. 19, 1095 (1967)
R. Hirota, Phys. Rev. Lett. 27, 1192 (1971)
W. Malfliet, Am. J. Phys. 60, 650 (1992); A.H. Khater, W. Malfliet, D.K. Callebaut, E.S. Kamel, J. Comput. Appl. Math. 140, 469 (2002)
W. Hereman, A. Korpel, P.P. Banerjee, Wave Motion 7, 283 (1985)
E.V. Krishnan, Int. J. Diff. Eqs. Appl. 8, 13 (2003)
S.K. Liu, Z. Fu, S.D. Liu, Q. Zhao, Phys. Lett. A 289, 69 (2001)
Z. Fu, S.K Liu, S.D. Liu, Q. Zhao, Phys. Lett. A 290, 72 (2001)
Z.Y. Yan, Academia Sinica 22, 363 (2003); A.H. Khater, M.M. Hassan, Z. Naturforsch. A 59, 389 (2004)
P.P. Banerjee, F.D. Daud, W. Hereman, J. Phys. A: Math. Gen. 23, 521 (1990); E.G. Fan, J. Phys. A: Math. Gen. 35, 6853 (2002)
Z.Y. Yan, Chaos Solit. Fract. 16, 291 (2003)
Y. Peng, J. Phys. Soc. Jpn 72, 1356 (2003): Y. Peng, J. Phys. Soc. Jpn 72, 1889 (2003)
Y. Peng, Phys. Lett. A. 314, 401 (2003); Y. Peng, Z. Naturforsch. A 60, 321 (2005)
Y. Peng, J. Phys. Soc. Jpn 73, 1156 (2004)
Y.A.V. Porubov, Phys. Lett. A 221, 391 (1996)
Y. Chen, Z.Y. Yan, Chaos Solit. Fract. 26, 393 (2005)
Y. Chen, Z.Y. Yan, Chaos Solit. Fract. 29, 948 (2006)
E.V. Krishnan, Y. Peng, Int. J. Pure Appl. Math. Sci. 2, 14 (2005)
M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, 1991)
H.T. Davis, Introduction to Nonlinear Differrential and Integral Equations (Dover Publications, New York, 1960)
W. Hereman, M. Takaoka, J. Phys. A: Math. Gen. 23,4805 (1990); M. Wadati, J. Phys. Soc. Jpn 32, 1681(1972); A.H. Khater, M.M. Hassan, R.S. Temsah, J.Phys. Soc. Jpn 74, 1431 (2005)
H. Schamel, Plasma Phys. 14, 905 (1972); H. Schamel,J. Plasma Phys. 9, 377 (1973); A.H. Khater, A.A. Abdallah, O.H. El-Kalaawy, D.K. Callebaut, Phys. Plasmas 6, 4542 (1999)
T. Kawahara, J. Phys. Soc. Jpn 33, 260 (1972)
W. Hereman, P.P. Banerjee, A. Korpel, G. Assanto, A. Van Immerzeele, A. Meerpoel, J. Phys. A: Math. Gen. 19, 607 (1986)
D. Chakraboty, K.P. Das, J. Plasma Phys. 60, 151 (1998)
A. Bandyopadhyoy, K.P. Das, J. Plasma Phys. 62, 255 (1999)
E. Infeld, J. Plasma Phys. 33, 171 (1985); K.P. Das, F. Verhest, J. Plasma Phys. 41, 139 (1989); A.H. Khater, W. Malfliet, E.S. Kamel, Math. Comput. Simul. 64, 247 (2004)
E. Infeld, G. Rowlands, J. Plasma Phys. 10, 293 (1973)
Taogetusang, Sirendaoerji, Chin. Phys. 15, 2809 (2006)
A. Luque, H. Schamel, Phys. Rep. 415, 261 (2005); H. Schamel, A. Luque, New J. Phys. 7, 69 (2005)
B. Eliasson, P.K. Shukla, Phys. Rep. 422, 225 (2006)
H. Schamel, Phys. Plasmas 7, 4831 (2000)
J. Korn, H. Schamel, J. Plasma Phys. 56, 307 (1996)
E.J. Parkes, B.R. Duffy, P.C. Abbott, Phys. Lett. A 295, 280 (2002)
A.H. Khater, M.M. Hassan, R.S. Temsah, Math. Comput. Simul. 70, 221 (2005)
S. Torven, Phys. Scr. 33, 362 (1986); M.A. Raadu, Phys. Rep. 78, 25 (1989); G.C. Das, K.M. Sen, Contrib. Plasma Phys. 31, 647 (1991)
D.J. Wu, D.Y. Huang, C.G. Falthammar, Phys. Plasmas 3, 2879 (1996); A.Y. Wang, B.H. Quon, Phys. Rev. Lett. 24, 1499 (1975)
H. Alfven, Cosmic Plasmas (Riedel, Dordrecht, 1981), Chap. VI
A. Luque, H. Schamel, J.-M. Griessmeier, Phys. Plasmas 9, 4841 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khater, A., Hassan, M., Krishnan, E. et al. Applications of elliptic functions to ion-acoustic plasma waves. Eur. Phys. J. D 50, 177–184 (2008). https://doi.org/10.1140/epjd/e2008-00205-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjd/e2008-00205-1