Symmetry Constraint of the Differential-difference KP Hierarchy and a Second Discretization of the ZS-AKNS System

References

• M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Nonlinear-evolution equations of physical significance, Phys. Rev. Lett. 31 (1973) 125–127. doi: 10.1103/PhysRevLett.31.125
   Web of Science ®Google Scholar
• M.J. Ablowitz and J.F. Ladik, Nonlinear differential-difference equations, J. Math. Phys. 16 (1975) 598–603. doi: 10.1063/1.522558
   Web of Science ®Google Scholar
• M.J. Ablowitz and J.F. Ladik, Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys. 17 (1976) 1011–1018. doi: 10.1063/1.523009
   Web of Science ®Google Scholar
• V.E. Adler and R.I. Yamilov, Explicit auto-transformations of integrable chains, J. Phys. A: Math. Gen. 27 (1994) 477–492. doi: 10.1088/0305-4470/27/2/030
   Google Scholar
• C.W. Cao and G.Y. Zhang, Lax pairs for discrete integrable equations via Darboux transformations, Chin. Phys. Lett. 29 (2012) No.050202 (2pp).
   Web of Science ®Google Scholar
• C.W. Cao and G.Y. Zhang, Integrable symplectic maps associated with the ZS-AKNS spectral problem, J. Phys. A: Math. Theor. 45 (2012) No.265201 (15pp).
   Web of Science ®Google Scholar
• Y. Cheng and Y.S. Li, The constraint of the Kadomtsev-Petviashvili equation and its special solutions, Phys. Lett. A 157 (1991) 22–26. doi: 10.1016/0375-9601(91)90403-U
   Web of Science ®Google Scholar
• Y. Cheng and Y.S. Li, Constraints of the 2+1 dimensional integrable soliton systems, J. Phys. A: Math. Gen. 25 (1992) 419–431. doi: 10.1088/0305-4470/25/2/022
   Google Scholar
• E. Date, M. Jimbo and T. Miwa, Method of generating discrete soliton equations II, J. Phys. Soc. Japan 51 (1982) 4125–4131. doi: 10.1143/JPSJ.51.4125
   Google Scholar
• E.G. Fan and Z.H. Yang, A lattice hierarchy with a free function and its reductions to the Ablowitz-Ladik and Volterra hierarchies, Int. J. Theor. Phys. 48 (2009) 1–9. doi: 10.1007/s10773-008-9773-3
   Web of Science ®Google Scholar
• W. Fu, L. Huang, K.M. Tamizhmani and D.J. Zhang, Integrable properties of the differential-difference Kadomtsev-Petviashvili hierarchy and continuum limits, Nonlinearity 26 (2013) 3197–3229. doi: 10.1088/0951-7715/26/12/3197
   Web of Science ®Google Scholar
• W. Fu, Z.J. Qiao, J.W. Sun and D.J. Zhang, Continuous correspondence of conservation laws of the semi-discrete AKNS system, J. Nonlin. Math. Phys. 22 (2015) 321–341. doi: 10.1080/14029251.2015.1056612
   Web of Science ®Google Scholar
• B. Fuchssteiner and A.S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D 4 (1981) 47–66. doi: 10.1016/0167-2789(81)90004-X
   Web of Science ®Google Scholar
• B. Fuchssteiner, Hamiltonian structure and integrability, Math. Sci. Eng. 185 (1991) 211–256. doi: 10.1016/S0076-5392(08)62801-5
   Google Scholar
• L. Haine and P. Iliev, Commutative rings of difference operators and an adelic flag manifold, Int. Math. Res. Notices 2000 (2000) 281–323. doi: 10.1155/S1073792800000179
   Google Scholar
• F. Khanizadeh, A.V. Mikhailov and J.P. Wang, Darboux transformations and recursion operators for differential-difference equations, Theore. Math. Phys. 177 (2013) 387–440.
   Web of Science ®Google Scholar
• B. Konopelchenko, J. Sidorenko and W. Strampp, (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems, Phys. Lett. A 157 (1991) 17–21. doi: 10.1016/0375-9601(91)90402-T
   Web of Science ®Google Scholar
• B. Konopelchenko and W. Strampp, The AKNS hierarchy as symmetry constraint of the KP hierarchy, Inverse Problem 7 (1991) L17–24. doi: 10.1088/0266-5611/7/2/002
   Web of Science ®Google Scholar
• D. Levi and R. Benguria, Bäcklund transformations and nonlinear differential difference equations, Proc. Natl. Acad. Sci. U.S.A. 77 (1980) 5025–5027. doi: 10.1073/pnas.77.9.5025
   PubMed Web of Science ®Google Scholar
• D. Levi, Nonlinear differential difference equations as Bäcklund transformations, J. Phys. A: Math. Gen. 14 (1981) 1083–1098. doi: 10.1088/0305-4470/14/5/028
   Google Scholar
• C.Z. Li, J.P. Cheng, K.L. Tian, M.H. Li and J.S. He, Ghost symmetry of the discrete KP hierarchy, Monatsh. Math. 180 (2016) 815–832. doi: 10.1007/s00605-015-0802-z
   Web of Science ®Google Scholar
• S.W. Liu and Y. Cheng, Sato’s Bäcklund transformations, additional symmetries and ASvM formula for the discrete KP hierarchy, J. Phys. A: Math. Theor. 43 (2010) No.135202 (11pp).
   Web of Science ®Google Scholar
• I. Merola, O. Ragnisco and G.Z. Tu, A novel hierarchy of integrable lattices, Inverse Problems 10 (1994) 1315–1334. doi: 10.1088/0266-5611/10/6/009
   Web of Science ®Google Scholar
• A.V. Mikhailov, Formal diagonalisation of the Lax-Darboux scheme and conservation laws of integrable partial differential, differential-difference and partial difference equations, (2013) http://www.newton.ac.uk/files/seminar/20130711140014301-153657.pdf.
   Google Scholar
• W. Oevel, Darboux theorems and Wronskian formulas for integrable systems I: Constrainted KP flows, Physica A 195 (1993) 533–576. doi: 10.1016/0378-4371(93)90174-3
   Web of Science ®Google Scholar
• Z.Y. Qin, A generalized Ablowitz-Ladik hierarchy, multi-Hamiltonian structure and Darboux transformation, J. Math. Phys. 49 (2008) No.063505 (14pp).
   Web of Science ®Google Scholar
• O. Ragnisco and G.Z. Tu, A new hierarchy of integrable discrete systems, preprint (Dept. of Phys., University of Rome 1989).
   Google Scholar
• S.K. Vel and K.M. Tamizhmani, Lax pairs, symmetries and conservation laws of a differential-difference equation — Sato’s approach, Chaos, Solitons & Fractals 8 (1997) 917–931. doi: 10.1016/S0960-0779(96)00142-7
   Web of Science ®Google Scholar
• H.X. Yang, X.X. Xu and H.Y. Ding, New hierarchies of integrable positive and negative lattice models and Darboux transformation, Chaos, Solitons & Fractals 26 (2005) 1091–1103. doi: 10.1016/j.chaos.2005.02.011
   Web of Science ®Google Scholar
• V.E. Zakharnov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulating waves in nonlinear media, Sov. Phys. JETP. 34 (1972) 62–69.
   Google Scholar
• D.J. Zhang, T.K. Ning, J.B. Bi and D.Y. Chen, New symmetries for the Ablowitz-Ladik hierarchies, Phys. Lett. A 359 (2006) 458–466. doi: 10.1016/j.physleta.2006.06.077
   Web of Science ®Google Scholar
• D.J. Zhang and S.T. Chen, Symmetries for the Ablowitz-Ladik hierarchy: Part I. Four-potential case, Stud. Appl. Math. 125 (2010) 393–418. doi: 10.1111/j.1467-9590.2010.00493.x
   Web of Science ®Google Scholar
• D.J. Zhang and S.T. Chen, Symmetries for the Ablowitz-Ladik hierarchy: Part II. Integrable discrete nonlinear Schrödinger equations and discrete AKNS hierarchy, Stud. Appl. Math. 125 (2010) 419–443. doi: 10.1111/j.1467-9590.2010.00494.x
   Web of Science ®Google Scholar
• H.W. Zhang, G.Z. Tu, W. Oevel and B. Fuchssteiner, Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure, J. Math. Phys. 32 (1991) 1908–1918. doi: 10.1063/1.529205
   Web of Science ®Google Scholar