Abstract
We propose a new method to find the spectral problem for the inverse scattering equations for integrable nonlinear evolution equations. This approach is able to construct the Miura transformation directly and utilize a factorization scheme to construct a linear recursion operator of gradients of constants of motion which can be taken as the spectral operator of a pair of Lax operators for the integrable equation. We have applied this method to the KdV, MKdV, and sine-Gordon equations. We have also obtained the Lax pair operators for their modified equations.
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References
Gardner, C., Greene, J., Kruskal, M. and Miura, R., Commun. Pure Appl. Math. 27, 97–133 (1974).
Lax, P., Commun. Pure Appl. Math. 21, 467–490 (1968).
Scott, A. C., Chu, F. Y. F. and McLaughlin, D. W., Proc. IEEE, 61, 1443–1483 (1973).
Ablowitz, M. J., Kaup, D. J., Newell, A. C., and Segur, H., Studies in Appl. Math. 53, 249–315 (1974).
Miura, R. M., SIAM Review 18, 412–459 (1976).
Ablowitz, M. J. and Segur, H., Solitons and the Inverse Scattering Transform (SIAM, Philadelphia 1981).
Newell, A. C., Solitons in Mathematics and Physics (SIAM, Philadelphia 1985).
Novikov, S., Manakov, S., Pitaevskii, L. and Zakharov, V. E., Theory of Solitons (Consultants Bureau, New York 1984).
Miura, R., J. Math. Phys. 9, 1202–1204 (1968).
Chen, H. H., Lee, Y. C. and Liu, C. S., Physica Scripta 20, 490–492 (1979).
Gardner, C. S., J. Math. Phys. 12, 1548–1551 (1971).
Arnold, V. I., Mathematical Methods of Classical Mechanics (Springer, New York 1978).
Lax, P., SIAM Review 18, 351–375 (1976).
Chen, H. H., Lee, Y. C. and Lin, J. E., Physica 26D, 165–170 (1987).
Fuchssteiner, B. and Fokas, A. S., Physica 4D, 47–66 (1981).
McKean, H. P., in Topics in Functional Analysis (Academic Press, New York 1978) 217–226.
Zhu, G. C. and Chen, H. H., J. Math. Phys. 27, 100–103 (1986).
Chen, H. H., in Backlund Transformations (Lecture Notes in Mathematics, Vol. 515. Springer-Verlag, New York 1976) 241–252.
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Chen, H.H., Lin, J.E. (1989). On the integrability of nonlinear evolution equations. In: Gill, T.L., Zachary, W.W. (eds) Nonlinear Semigroups, Partial Differential Equations and Attractors. Lecture Notes in Mathematics, vol 1394. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086750
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DOI: https://doi.org/10.1007/BFb0086750
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