Solution of The KdV Equation with Asymptotic Degeneracy
Recently T. C. Au-Yeung, C.Au, and P. C. W. Fung [2] have given the solution of the KdV equation [1] to the boundary condition , where b is a constant. We have further extended the method of [2] to find the solution of the KdV equation with asymptotic degeneracy. Via simulations we find both bright and dark Solitons (i.e. Solitons with opposite phases).
[1] C.S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Muira, Phys.
Rev. Lett. 19 (1968) pp. 467.
[2] C.S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Muira, Commun.
Pure Appl. Math., 27 (1974) pp. 97.
[3] T. C. Au-Yeung, C. Au and P. C. W. Fung, Physical Review A, 29 (1984)
pp. 2370-2374.
[4] W. P. Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. B, 22 (1980)
pp.2099
[5] I. M. Gel-fand and B. M. Levitan, Izv. Akad. Nauk SSSR, Ser. Mat.
15 (1951) pp.309 (translated in Am. Math. Soc. Transl., Ser.2 1,(1955)
pp253).
[6] P. D. Lax, Commun. Pure Appl. Math., 21,(1968) pp.467.
[7] A. J. Bray, B. Derrida and C. Godreche, J. Phys. A 27, L357 (1994);
[8] For a review, see S. N. Majumdar, Curr. Sci. India 77 370 (1999).
[9] S.N. Majumdar, A. J. Bray, S. J. Corwell and C. Sire, Phys. Rev. Lett.
77 3704 (1996).
[10] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 435 (1977).
[11] B. Derrida, V. Hakim and V. Pasquier, Phys. Rev. Lett. 75 751 (1995).
[12] A. J. Bray, B. Derrida and C. Godreche, Eur. Phys. Lett. 27 175 (1994);
D. Stauffer, J. Phys. A 27 5029 (1994); S. N. Majumdar and C. Sire,
Phys. Rev.Lett. 77 1420 (1996).
[13] Abraham, I., Hunter, R. J., Sampson, K. E., Smith, S., Gottesman, M.
M. & Mayo, J. K. (1987) Mol. Cell. Biol. 7 , 3098-3106.
[14] Cho-Chung, Y. S., Nesterova, M., Becker, K. G., Srivastava, R., Park,
Y. G., Lee, Y. N., Cho, Y. S., Kim, M. K., Neary, C. & Cheadle, C.
(2002) Ann. N.Y. Acad. Sci. 968 , 22-36.
[15] Weissinger, E. M., Oettrich, K., Evans, C., Genieser, H. G., Schwede,
F., Dangers, M., Dammann, E., Kolb, H. J., Mischak, H., Ganser, A. &
Kolch, W. (2004) Br. J. Cancer 91, 186-192.
[16] M. Mostovoy, M. T. Figge, and J. Knoester, Europhys.Lett. 38 (1997)
687.
[17] M. Mostovoy, M. T. Figge, and J. Knoester, Phys. Rev.B 57 (1998)
2861.
[18] M. T. Figge, M. Mostovoy, and J. Knoester, Phys. Rev.B 58 (5) (1998);
cond-mat/9803385.
[19] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, 1973 Phys.
Rev. Lett. 30, 1262.
[20] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, "The Inverse
Scattering transform -Fourier Analysis for nonlinear problems", Stud.
Appl. Math, 53(1974) 249-315.
[1] C.S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Muira, Phys.
Rev. Lett. 19 (1968) pp. 467.
[2] C.S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Muira, Commun.
Pure Appl. Math., 27 (1974) pp. 97.
[3] T. C. Au-Yeung, C. Au and P. C. W. Fung, Physical Review A, 29 (1984)
pp. 2370-2374.
[4] W. P. Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. B, 22 (1980)
pp.2099
[5] I. M. Gel-fand and B. M. Levitan, Izv. Akad. Nauk SSSR, Ser. Mat.
15 (1951) pp.309 (translated in Am. Math. Soc. Transl., Ser.2 1,(1955)
pp253).
[6] P. D. Lax, Commun. Pure Appl. Math., 21,(1968) pp.467.
[7] A. J. Bray, B. Derrida and C. Godreche, J. Phys. A 27, L357 (1994);
[8] For a review, see S. N. Majumdar, Curr. Sci. India 77 370 (1999).
[9] S.N. Majumdar, A. J. Bray, S. J. Corwell and C. Sire, Phys. Rev. Lett.
77 3704 (1996).
[10] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 435 (1977).
[11] B. Derrida, V. Hakim and V. Pasquier, Phys. Rev. Lett. 75 751 (1995).
[12] A. J. Bray, B. Derrida and C. Godreche, Eur. Phys. Lett. 27 175 (1994);
D. Stauffer, J. Phys. A 27 5029 (1994); S. N. Majumdar and C. Sire,
Phys. Rev.Lett. 77 1420 (1996).
[13] Abraham, I., Hunter, R. J., Sampson, K. E., Smith, S., Gottesman, M.
M. & Mayo, J. K. (1987) Mol. Cell. Biol. 7 , 3098-3106.
[14] Cho-Chung, Y. S., Nesterova, M., Becker, K. G., Srivastava, R., Park,
Y. G., Lee, Y. N., Cho, Y. S., Kim, M. K., Neary, C. & Cheadle, C.
(2002) Ann. N.Y. Acad. Sci. 968 , 22-36.
[15] Weissinger, E. M., Oettrich, K., Evans, C., Genieser, H. G., Schwede,
F., Dangers, M., Dammann, E., Kolb, H. J., Mischak, H., Ganser, A. &
Kolch, W. (2004) Br. J. Cancer 91, 186-192.
[16] M. Mostovoy, M. T. Figge, and J. Knoester, Europhys.Lett. 38 (1997)
687.
[17] M. Mostovoy, M. T. Figge, and J. Knoester, Phys. Rev.B 57 (1998)
2861.
[18] M. T. Figge, M. Mostovoy, and J. Knoester, Phys. Rev.B 58 (5) (1998);
cond-mat/9803385.
[19] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, 1973 Phys.
Rev. Lett. 30, 1262.
[20] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, "The Inverse
Scattering transform -Fourier Analysis for nonlinear problems", Stud.
Appl. Math, 53(1974) 249-315.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51007", author = "Tapas Kumar Sinha and Joseph Mathew", title = "Solution of The KdV Equation with Asymptotic Degeneracy", abstract = "Recently T. C. Au-Yeung, C.Au, and P. C. W. Fung [2] have given the solution of the KdV equation [1] to the boundary condition , where b is a constant. We have further extended the method of [2] to find the solution of the KdV equation with asymptotic degeneracy. Via simulations we find both bright and dark Solitons (i.e. Solitons with opposite phases).
", keywords = "KdV equation, Asymptotic Degeneracy, Solitons, Inverse Scattering", volume = "6", number = "3", pages = "215-4", }
