An Ablowitz-Ladik Integrable Lattice Hierarchy with Multiple Potentials
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
AN ABLOWITZ-LADIK INTEGRABLE LATTICE HIERARCHY WITH MULTIPLE POTENTIALS∗
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Wen-Xiu MA (
School of Mathematics, South China University of Technology, Guangzhou 510640, China Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa E-mail : [email protected] Abstract Within the zero curvature formulation, a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type. The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity. When the involved two potential vectors are scalar, all the resulting integrable lattice equations are reduced to the standard AblowitzLadik hierarchy. Key words
Integrable lattice; discrete spectral problem; symmetry and conserved functional
2010 MR Subject Classification
1
35Q51; 35Q58; 37K10; 37K40
Introduction
The inverse scattering transform is one of the powerful methods to solve Cauchy problems for nonlinear integrable equations, and it can also explain various geometrical and algebraic integrable properties of nonlinear equations such as Hamiltonian structures and Virasoro algebras [1, 2]. The starting point of the direct and inverse scattering theory is a pair of matrix spectral problems (or a Lax pair [3]), which has a significant role in solving nonlinear integrable equations. Other powerful theories on integrable equations, for example, Riemann-Hilbert problems [2], R-matrix theory [4], and Sato τ -function theory [5], also underline the importance of matrix spectral problems or Lax pairs. It is important to search for nonlinear integrable equations from zero curvature equations, that is, compatibility conditions of pairs of matrix spectral problems, in both continuous and ∗ Received
November 11, 2018. The work was supported in part by NSF (DMS-1664561), NSFC (11975145 and 11972291), the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17KJB110020), and Emphasis Foundation of Special Science Research on Subject Frontiers of CUMT (2017XKZD11).
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W.X. Ma: ABLOWITZ-LADIK HIERARCHY WITH MULTIPLE POTENTIALS
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discrete cases. Among the well-known integrable equations are the KP-type continuous and discrete equations. The related KP theories are well developed, and the residue technique is systematically used to present sufficiently many conserved quantities required in showing integrability of the KP equations. For non-KP-type equations, the working techniques are varied [6]. The structures t
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