An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems. (English) Zbl 0941.37530

Summary: An explicit symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of AKNS systems. The corresponding Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative, finite-dimensional integrable Hamiltonian systems in the Liouville sense and thus an involutive representation of solutions of AKNS systems is obtained. The purpose of this Letter is to elucidate that the nonlinearization method (i.e. a kind of symmetry constraint method) of integrable systems can be applied to the Lax pairs and the adjoint Lax pairs associated with integrable systems.


37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q58 Other completely integrable PDE (MSC2000)
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[1] Cao, C.W., Chin. Q. J. math., Sci. China A, 33, 528, (1990)
[2] Konopelchenko, B.; Sidorenko, J.; Strampp, W.; Cheng, Y.; Li, Y.S., Phys. lett. A, Phys. lett. A, 157, 22, (1991)
[3] Cao, C.W.; Geng, X.G.; Zeng, Y.B.; Li, Y.S.; Zeng, Y.B., J. phys. A, J. math. phys., J. math. phys., Phys. lett. A, 160, 541, (1991)
[4] Antonowicz, M.; Wojciechowski, S.; Ragnisco, O.; Wojciechowski, S., Phys. lett. A, J. math. phys., Inverse probl., Inverse probl., 8, 245, (1992)
[5] Ma, W.X., (), Acta math. appl. sin., 9, 92, (1993)
[6] Gu, Z.Q.; Xu, T.X.; Mu, W.H.; Geng, X.G.; Qiao, Z.J., J. math. phys., Phys. lett. A, Physica A, Phys. lett. A, 172, 224, (1993)
[7] Konopelchenko, B.; Strampp, W.; Sidorenko, J.; Strampp, W., Inverse probl., J. math. phys., Inverse probl., 7, L37, (1991)
[8] W. Oevel and W. Strampp, Constrained KP hierarchy and bi-Hamiltonian structures, to appear in Commun. Math. Phys.
[9] Lax, P.D., Commun. pure appl. math., SIAM rev., 18, 351, (1976)
[10] Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H., Stud. appl. math., 53, 249, (1974)
[11] Ma, W.X., J. math. phys., J. phys. A, 25, 5329, (1992)
[12] Tu, G.Z., J. math. phys., J. phys. A, 22, 2375, (1989)
[13] Newell, A.C., Solitons in mathematics and physics, (1985), SIAM Philadelphia · Zbl 0565.35003
[14] Arnold, V.I.; Abraham, R.; Marsden, J., Foundations of mechanics, (1978), Springer Reading, MA
[15] Cao, C.W., Acta math. sin.., 1, 216, (1991), New Series
[16] Ma, W.X., New finite dimensional integrable systems by the symmetry constraint of the KdV equations, Binary nonlinearization for the Dirac systems, (1993), preprint
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