A generalized integrable lattice hierarchy associated with the Toda and modified Toda lattice equations: Hamiltonian representation, soliton solutions

• Published in: Wave motion Vol. 103; p. 102727
• Main Authors: Fan, Fang-Cheng, Wen, Xiao-Yong
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.2021; Elsevier BV
• Subjects: Discrete element method; Eigenvectors; Exact solutions; Hierarchies; Integrable lattice hierarchy; Lattice theory; Modified Toda lattice equation; Nonlinear equations; Parameters; Propagation; Representations; Schrodinger equation; Solitary waves; Soliton solutions; Studies; Tu-scheme; Wave propagation; Modified Toda lattice equation; Soliton solutions; Integrable lattice hierarchy; Tu-scheme
• ISSN: 0165-2125; 1878-433X
• DOI: 10.1016/j.wavemoti.2021.102727

In this paper, we give an integrable four-field lattice hierarchy associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. The Hamiltonian representation of the four-field lattice hierarchy are constructed by using the trace identity. We consider reductions of our hierarchy, and find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, and some new integrable three-field lattice hierarchies. In addition, the modified Toda lattice equation is studied, the figures of one-, two-soliton and periodic solutions with properly parameters are shown graphically to illustrate the propagation of solitary waves, and we find that the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The elastic and inelastic interactions among the two-soliton for the modified Toda lattice equation are also discussed. Furthermore, we generate the relationships between the structures of exact solutions and parameters, and present soliton solutions with free parameters. The results in this paper might be helpful for interpreting certain physical phenomena. •Construct a four-field integrable lattice hierarchy and its Hamiltonian representation.•Discuss reductions of the hierarchy in detail.•Present one-, two-soliton and periodic solutions and show the elastic and inelastic interactions.•Generate the relationships between the structures of exact solutions and parameters and obtain soliton solutions with free parameters.