Integrability, discrete kink multi-soliton solutions on an inclined plane background and dynamics in the modified exponential Toda lattice equation
Tóm tắt
Under investigation in this paper is a discrete modified exponential Toda lattice equation which may describe vibration of particles in a lattice. Firstly, we construct an integrable lattice hierarchy associated with this equation from a
$$2\times 2$$
matrix spectral problem, and some related integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws of this hierarchy are discussed. Secondly, we present the discrete generalized
$$(m, N-m)$$
-fold Darboux transformation of the modified exponential Toda lattice equation on the basis of its known Lax representation. As applications of the obtained discrete generalized Darboux transformation, multi-kink-soliton solutions on an exponential surface background and an inclined plane background are obtained when
$$m=2N$$
, the discrete rational and semi-rational solutions are derived when
$$m=1$$
, and the mixed solutions of usual soliton solutions and rational solutions are given when
$$m=2$$
. Based on the asymptotic and graphic analysis, soliton elastic interaction phenomena and limit states related to rational solutions are discussed and analyzed. Furthermore, some mathematical features of rational solutions are summarized. Finally, numerical simulations are used to explore the dynamical behaviors of such soliton solutions which show the soliton evolutions are robust against a small noise. These results and properties given in this paper might be useful for understanding nonlinear lattice dynamics.
Tài liệu tham khảo
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