Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 096, 8 pages      nlin.SI/0701003      https://doi.org/10.3842/SIGMA.2006.096
Contribution to the Vadim Kuznetsov Memorial Issue

Restricted Flows and the Soliton Equation with Self-Consistent Sources

Runliang Lin a, Haishen Yao b and Yunbo Zeng a
a) Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
b) Dept. of Math and Computer Science, QCC, The City University of New York, USA

Received October 28, 2006, in final form December 22, 2006; Published online December 30, 2006

Abstract
The KdV equation is used as an example to illustrate the relation between the restricted flows and the soliton equation with self-consistent sources. Inspired by the results on the Bäcklund transformation for the restricted flows (by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations for the KdV equation with self-consistent sources (KdVES). These Darboux transformations are used to get some explicit solutions of the KdVES, which include soliton, rational, positon, and negaton solutions.

Key words: the KdV equation with self-consistent sources; restricted flows; Lax pair; Darboux transformation; soliton solution.

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references

  1. Mel'nikov V.K., Integration method of the Korteweg-de Vries equation with a self-consistent source, Phys. Lett. A, 1988, V.133, 493-496.
  2. Shchesnovich V.S., Doktorov E.V., Modified Manakov system with self-consistent source, Phys. Lett. A, 1996, V.213, 23-31.
  3. Urazboev G.U., Khasanov A.B., Integrating the Korteweg-de Vries equation with a self-consistent source and "steplike" initial data, Theoret. and Math. Phys., 2001, V.129, 1341-1356.
  4. Zeng Y.B., New factorization of the Kaup-Newell hierarchy, Phys. D, 1994, V.73, 171-188.
  5. Lin R.L., Zeng Y.B., Ma W.X., Solving the KdV hierarchy with self-consistent sources by inverse scattering method, Phys. A, 2001, V.291, 287-298.
  6. Zeng Y.B., Ma W.-X., Lin R.L., Integration of the soliton hierarchy with self-consistent sources, J. Math. Phys., 2000, V.41, 5453-5489.
  7. Zeng Y.B., Shao Y.J., Xue W.M., Negaton and positon solutions of the soliton equation with self-consistent sources, J. Phys. A: Math. Gen., 2003, V.36, 5035-5043.
  8. Ma W.X., Complexiton solutions of the Korteweg-de Vries equation with self-consistent sources, Chaos Solitons Fractals, 2005, V.26, 1453-1458.
  9. Hu X.B., The higher-order KdV equation with a source and nonlinear superposition formula, Chaos Solitons Fractals, 1996, V.7, 211-215.
  10. Zhang D.J., Chen D.Y., The N-soliton solutions of the sine-Gordon equation with self-consistent sources, Phys. A, 2003, V.321, 467-481.
  11. Gegenhasi, Hu X.B., On a integrable differential-difference equation with a source, J. Nonlinear Math. Phys., 2006, V.13, 183-192.
  12. Zeng Y.B., Bi-Hamiltonian structure of JM hierarchy with self-consistent sources, Phys. A, 1999, V.262, 405-419.
  13. Zeng Y.B., Lin R.L., Wang X.H., Bi-Hamiltonian structure of the KdV hierarchy with self-consistent sources, Adv. Math. (China), 1998, V.27, 451-463.
  14. Zeng Y.B., Li Y.S., The deduction of the Lax representation for constrained flows from the adjoint representation, J. Phys. A: Math. Gen., 1993, V.26, L273-L278.
  15. Matveev V.B., Salle M.A., Darboux transformations and solitons, Berlin, Springer, 1991.
  16. Kuznetsov V.B., Sklyanin E.K., On Bäcklund transformations for many-body systems, J. Phys. A: Math. Gen., 1998, V.31, 2241-2251, solv-int/9711010.
  17. Hone A.N.W., Kuznetsov V.B., Ragnisco O., Bäcklund transformations for many-body systems related to KdV, J. Phys. A: Math. Gen., 1999, V.32, L299-L306, solv-int/9904003.
  18. Hone A.N.W., Kuznetsov V.B., Ragnisco O., Bäcklund transformations for the Hénon-Heiles and Garnier systems, in Proceedings of the SIDE III (1998, Sabaudia), Editors D. Levi and O. Ragnisco, CRM Proceedings and Lecture Notes Series, Vol. 25, American Mathematical Society, 2000, 231-235.
  19. Newell A.C., Solitons in mathematics and physics, Philadelphia, SIAM, 1985.


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