Bargmann Symmetry Constraint and Binary Nonlinearization of Super NLS-MKdV Hierarchy

Affiliation(s)

School of Mathematics and Information Science, Shangqiu Normal University, Shangqiu, China.

School of Physics and Electronic Information, Shangqiu Normal University, Shangqiu, China.

School of Mathematics and Information Science, Shangqiu Normal University, Shangqiu, China.

School of Physics and Electronic Information, Shangqiu Normal University, Shangqiu, China.

ABSTRACT

An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super NLS-MKdV hierarchy. Under the obtained symmetry constraint, the n-th flow of the super NLS-MKdV hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold R^{4N｜2N }with the corresponding dynamical variables *x *and* t _{n}*. The integrals of motion required for Liouville integrability are explicitly given.

Cite this paper

S. Tao and H. Shi, "Bargmann Symmetry Constraint and Binary Nonlinearization of Super NLS-MKdV Hierarchy,"*Journal of Modern Physics*, Vol. 4 No. 5, 2013, pp. 5-11. doi: 10.4236/jmp.2013.45B002.

S. Tao and H. Shi, "Bargmann Symmetry Constraint and Binary Nonlinearization of Super NLS-MKdV Hierarchy,"

References

[1] Y. Cheng and Y. S. Li, “The Constraint of The Kadomt-sev-Petviashvili Equation and Its Special Solutions,” Physics Letters A , Vol. 157, No. 1, 1991, pp. 22-26. doi: 10.1016/0375-9601(91)90403-U

[2] Y. Cheng, “Con-straints of the Kadomtsev-Petviashvili Hierarchy,” Journal of Mathematical Physics, Vol. 33, No. 11, 1992, pp. 3774-3782. doi: 10.1063/1.529875

[3] W. X. Ma and W Strampp, “An Explicit Symmetry Constraint for the Lax Pairs of AKNS Sys-tems,” Physics Letters A, Vol. 185, No. 3, 1994, pp. 277-286. doi: 10.1016/0375-9601(94)90616-5

[4] W. X. Ma, “New Finite-Dimensional Integrable Systems by Symmetry Constraint of the KdV Equations,” Journal of the Physical Society of Japan, Vol. 64, No. 4, 1995, pp. 1085-1091. doi: 10.1143/JPSJ.64.1085

[5] Y. B. Zeng and Y. S. Li, “The Constraints of Potentials and the Finite-Dimensional Integrable Systems,” Journal of Mathematical Physics, Vol. 30, No. 8, 1989, pp.1679-1689. doi:10.1063/1.528253

[6] C. W. Cao and X. G. Geng, “A Monconfocal Generator of Involutive Sys-tems and Three Associated Soliton Hierarchies,” Journal of Mathematical Physics, Vol. 2, No. 9, 1991, pp. 2323-2328. doi:10.1063/1.529156

[7] W. X. Ma, “Symmetry Constraint of MKdV Equations by Binary Nonlinearization,” Physica A, Vol. 219, No. 3-4, 1995, pp. 467-481. doi:10.1016/0378-4371(95)00161-Y

[8] W. X. Ma and R. G. Zhou, “Adjoint Symmetry Constraints Leading to Binaary Nonlinearization,” Journal of Nonlinear Mathematical Physics, Vol. 9, No. Suppl. 1, 2002, pp. 106-126.

[9] W. X. Ma, “Bi-nary Bonlinearization for the Dirac Systems,” Chinese Annals of Mathematics, Series B, Vol. 18, No. 1, 1997, pp. 79-88.

[10] J. S. He, J. Yu, Y. Cheng and R. G. Zhou, “Binary Bon linearization of the Super AKNS System,” Modern Physics Letters B, Vol. 22, No. 4, 2008, pp. 275-288. doi:10.1142/S0217984908014778

[11] J. Yu, J. W. Han and J. S. He, “Binary Nonlinearization of the Super AKNS System under an Implicit Symmetry constraint,” Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 46, 2009, 465201. doi:10.1088/1751-8113/42/46/465201

[12] J. Yu, J. S. He, W. X. Ma and Y. Cheng, “The Bargmann Symmetry Constraint and Binary Nonlinearization of the Super Dirac System,” Chinese Annals of Mathematics, Series B, Vol. 31, No. 3, 2010, pp. 361-372. doi:10.1007/s11401-009-0032-6

[13] H. H. Dong and X. Z. Wang, “Lie Algebra and Lie Super Algebra for the Integrable Couplings of NLS-MKdV Hierarchy,” Communications in Nonlinear Science and Numeical Simulation, Vol. 14, No. 12, 2009, pp. 4071-4077. doi:10.1016/j.cnsns.2009.03.010

[14] W. X. Ma, J. S. He and Z. Y. Qin, “A Supertrace Identity and Its Applications to Super Integrable Systems,” Journal of Mathe-matical Physics, Vol. 49, No. 3, 2008, 033511. doi:10.1063/1.2897036

[15] X. B. Hu, “An Approach to Gen-erate Superextensions of Integrable Systems,” Journal of Physics A: Mathematical and General, Vol. 30, No. 2, 1997, pp. 619-632. doi:10.1088/0305-4470/30/2/023

[16] W. X. Ma, B. Fuch-ssteiner and W. Oevel, “A 3×3 Matrix Spectral Problem for AKNS Hierarchy and Its Binary Nonlinearization,” Physica A, Vol. 233, No. 1-2, 1996, pp. 331-354. doi: 10.1016/S0378-4371(96)00225-7

[17] W. X. Ma and Z. X. Zhou, “Binary Symmetry Constraints of N-wave Intersection Equations in 1+1 and 2+1 Dimensions,” Journal of Mathemat-ical Physics, Vol. 42, No. 9, 2001, pp. 4345-4382. doi:10.1063/1.1388898

[1] Y. Cheng and Y. S. Li, “The Constraint of The Kadomt-sev-Petviashvili Equation and Its Special Solutions,” Physics Letters A , Vol. 157, No. 1, 1991, pp. 22-26. doi: 10.1016/0375-9601(91)90403-U

[2] Y. Cheng, “Con-straints of the Kadomtsev-Petviashvili Hierarchy,” Journal of Mathematical Physics, Vol. 33, No. 11, 1992, pp. 3774-3782. doi: 10.1063/1.529875

[3] W. X. Ma and W Strampp, “An Explicit Symmetry Constraint for the Lax Pairs of AKNS Sys-tems,” Physics Letters A, Vol. 185, No. 3, 1994, pp. 277-286. doi: 10.1016/0375-9601(94)90616-5

[4] W. X. Ma, “New Finite-Dimensional Integrable Systems by Symmetry Constraint of the KdV Equations,” Journal of the Physical Society of Japan, Vol. 64, No. 4, 1995, pp. 1085-1091. doi: 10.1143/JPSJ.64.1085

[5] Y. B. Zeng and Y. S. Li, “The Constraints of Potentials and the Finite-Dimensional Integrable Systems,” Journal of Mathematical Physics, Vol. 30, No. 8, 1989, pp.1679-1689. doi:10.1063/1.528253

[6] C. W. Cao and X. G. Geng, “A Monconfocal Generator of Involutive Sys-tems and Three Associated Soliton Hierarchies,” Journal of Mathematical Physics, Vol. 2, No. 9, 1991, pp. 2323-2328. doi:10.1063/1.529156

[7] W. X. Ma, “Symmetry Constraint of MKdV Equations by Binary Nonlinearization,” Physica A, Vol. 219, No. 3-4, 1995, pp. 467-481. doi:10.1016/0378-4371(95)00161-Y

[8] W. X. Ma and R. G. Zhou, “Adjoint Symmetry Constraints Leading to Binaary Nonlinearization,” Journal of Nonlinear Mathematical Physics, Vol. 9, No. Suppl. 1, 2002, pp. 106-126.

[9] W. X. Ma, “Bi-nary Bonlinearization for the Dirac Systems,” Chinese Annals of Mathematics, Series B, Vol. 18, No. 1, 1997, pp. 79-88.

[10] J. S. He, J. Yu, Y. Cheng and R. G. Zhou, “Binary Bon linearization of the Super AKNS System,” Modern Physics Letters B, Vol. 22, No. 4, 2008, pp. 275-288. doi:10.1142/S0217984908014778

[11] J. Yu, J. W. Han and J. S. He, “Binary Nonlinearization of the Super AKNS System under an Implicit Symmetry constraint,” Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 46, 2009, 465201. doi:10.1088/1751-8113/42/46/465201

[12] J. Yu, J. S. He, W. X. Ma and Y. Cheng, “The Bargmann Symmetry Constraint and Binary Nonlinearization of the Super Dirac System,” Chinese Annals of Mathematics, Series B, Vol. 31, No. 3, 2010, pp. 361-372. doi:10.1007/s11401-009-0032-6

[13] H. H. Dong and X. Z. Wang, “Lie Algebra and Lie Super Algebra for the Integrable Couplings of NLS-MKdV Hierarchy,” Communications in Nonlinear Science and Numeical Simulation, Vol. 14, No. 12, 2009, pp. 4071-4077. doi:10.1016/j.cnsns.2009.03.010

[14] W. X. Ma, J. S. He and Z. Y. Qin, “A Supertrace Identity and Its Applications to Super Integrable Systems,” Journal of Mathe-matical Physics, Vol. 49, No. 3, 2008, 033511. doi:10.1063/1.2897036

[15] X. B. Hu, “An Approach to Gen-erate Superextensions of Integrable Systems,” Journal of Physics A: Mathematical and General, Vol. 30, No. 2, 1997, pp. 619-632. doi:10.1088/0305-4470/30/2/023

[16] W. X. Ma, B. Fuch-ssteiner and W. Oevel, “A 3×3 Matrix Spectral Problem for AKNS Hierarchy and Its Binary Nonlinearization,” Physica A, Vol. 233, No. 1-2, 1996, pp. 331-354. doi: 10.1016/S0378-4371(96)00225-7

[17] W. X. Ma and Z. X. Zhou, “Binary Symmetry Constraints of N-wave Intersection Equations in 1+1 and 2+1 Dimensions,” Journal of Mathemat-ical Physics, Vol. 42, No. 9, 2001, pp. 4345-4382. doi:10.1063/1.1388898