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Exact solutions of nonlinear evolution equations of the AKNS class

Published online by Cambridge University Press:  17 February 2009

W. L. Chan  and
Yu-Kun Zheng
W. L. Chan
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Science Centre, Shantin, N. T., Hong Kong.
Yu-Kun Zheng
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Science Centre, Shantin, N. T., Hong Kong.
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Abstract

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The problem of obtaining explicit and exact solutions of soliton equations of the AKNS class is considered. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system; that is, a linear eigenvalue problem in the form of a system of first order partial differential equations. The method of characteristics is used and Bäcklund transformations are employed to generate new solutions from the old. Thus, families of new solutions for the KdV equation, the mKdV equation, the sine-Gordon equation and the nonlinear Schrôdinger equation are obtained, avoiding the solution of some Riccati equations. Our results in the KdV case include those obtained recently by other investigators.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 3 , January 1989 , pp. 313 - 325
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Ablowitz, M. J. and Segur, H., Solitons and the inverse scattering transform, SIAM, Philadelphia, (1981).CrossRefGoogle Scholar
[2]Au, C. and Fung, P. C. W., “Vacuum states of the KdV equation,” Phys. Rev. B 25 (1982), 6460.CrossRefGoogle Scholar
[3]Konno, K. and Wadati, M., “Simple derivation of Bäcklund transformation from Riccati form of inverse method,” Progr. Theoret. Phys. 53 (1975), 1652–1656.CrossRefGoogle Scholar
[4]Oliver, F. W., Asymptotics and special functions, (Academic Press, New York, 1974).Google Scholar
[5]Rogers, C. and Shadwick, W. E., Bäcklund transformation and their applications (Academic Press, New York, 1982).Google Scholar