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Abstract: In this paper, an implicit symmetry constraint is calculated and its associated binary nonlinearization of the Lax pairs and the adjoint Lax pairs is carried out for the modified Korteweg-de Vries (mKdV) equation. After introducing two new independent variables, we find that under the implicit symmetry constraint, the spatial part and the temporal part of the mKdV equation are decomposed into two finite-dimensional systems. Furthermore we prove that the obtained finite-dimensional systems are Hamiltonian systems and completely integrable in the Liouville sense.

[1] Cao, C.W., 1988. A cubic system which generates Bargmann potential and N-gap potential. Chin. Quart. J. Math., 3(1):90-96.

[2] Cao, C.W., Geng, X.G., 1990. C Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy. J. Phys. A, 23(18):4117-4125.

[3] Cao, C.W., Geng, X.G., 1992. A Bargmann system and an involutive representation of solutions to the coupled Harry Dym equation. Acta Math. Sinica, 35(3):314-322.

[4] Geng, X.G., Cao, C.W., 1999. Quasi-periodic solutions of the 2+1 dimensional modified Korteweg-de Vries equation. Phys. Lett. A, 261(5-6):289-296.

[5] Gesztesy, F., Schweiger, W., Simon, B., 1991. Commutation methods applied to the mKdV-equation. Trans. Amer. Math. Soc., 324(2):465-525.

[6] He, J.S., Chen, S.R., 2005. Hamiltonian formalism of mKdV equation with non-vanishing boundary values. Commun. Theor. Phys., 44(2):321-325.

[7] Li, Y.S., Ma, W.X., 2000. Binary nonlinearization of AKNS spectral problem under higher-order symmetry constraints. Chaos Solitons & Fractals, 11(5):697-710.

[8] Li, Y.S., Ma, W.X., 2002. A nonconfocal involutive system and constrained flows associated with the MKdV equation. J. Math. Phys., 43(10):4950-4962.

[9] Ma, W.X., 1995a. New finite-dimensional integrable systems by symmetry constraint of the KdV equations. J. Phys. Soc. Jpn., 64(4):1085-1091.

[10] Ma, W.X., 1995b. Symmetry constraint of MKdV equations by binary nonlinearization. Phys. A, 219(3-4):467-481.

[11] Ma, W.X., 1997. Binary nonlinearization for the Dirac systems. Chin. Ann. Math. Ser. B, 18(1):79-88.

[12] Ma, W.X., Strampp, W., 1994. An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems. Phys. Lett. A, 185(3):277-286.

[13] Ma, W.X., Zhou, R.G., 2001. Nonlinearization of spectral problems for the perturbation KdV systems. Phys. A, 296(1-2):60-74.

[14] Ma, W.X., Fuchssteiner, B.A., Oevel, W., 1996. A 3×3 matrix spectral problem for AKNS hierarchy and its binary nonlinearization. Phys. A, 233(1-2):331-354.

[15] Tu, G.Z., 1989. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J. Math. Phys., 30(2):330-338.

[16] Wu, H.X., Zeng, Y.B., Fan, T.Y., 2007. Positon and negaton solutions of the mKdV equation with self-consistent sources. J. Phys. A, 40(34):10505-10517.

[17] Yan, Z.Y., 2002. New Jacobian elliptic function solutions to modified KdV equation. I. Commun. Theor. Phys., 38(2):143-146.

[18] Yu, J., Zhou, R.G., 2006. Two kinds of new integrable decompositions of the mKdV equation. Phys. Lett. A, 349(6):452-461.

[19] Zeng, Y.B., Li, Y.S., 1989. The constraints of potentials and the finite-dimensional integrable systems. J. Math. Phys., 30(8):1679-1689.

[20] Zeng, Y.B., Li, Y.S., 1990. An approach to the integrability of Hamiltonian systems obtained by reduction. J. Phys. A, 23(3):L89-L94.

[21] Zeng, Y.B., Shao, Y.J., Ma, W.X., 2002. Integral-type Darboux transformations for the mKdV hierarchy with selfconsistent sources. Commun. Theor. Phys., 38(6):641-648.

[22] Zhang, D.J., 2002. The N-soliton solutions for the modified KdV equation with self-consistent sources. J. Phys. Soc. Jpn., 71(11):2649-2656.

[23] Zhou, R.G., 1998. Lax representation, r-matrix method, and separation of variables for the Neumann-type restricted flow. J. Math. Phys., 39(5):2848-2858.

[24] Zhou, R.G., Ma, W.X., 1998. New classical and quantum integrable systems related to the MKdV integrable hierarchy. J. Phys. Soc. Jpn., 67(12):4045-4050.