For the focusing Ablowitz–Ladik equation, the double- and triple-pole solutions are derived from its multi-soliton solutions via some limit technique. Also, the asymptotic analysis is performed for such two multi-pole solutions (MPSs) by considering the balance between exponential and algebraic terms. Like the continuous nonlinear Schrödinger equation, the discrete MPSs describe the elastic interactions of multiple solitons with the same amplitudes. But in contrast to the common multi-soliton solutions, most asymptotic solitons in the MPSs are localized in the curves of the nt plane, and thus they have the time-dependent velocities. In addition, the solitons’ relative distances grow logarithmically with
, while the separation acceleration magnitudes decrease exponentially with their distance.
This paper is concerned about certain generalization of Laurent biorthogonal polynomials together with the corresponding related integrable lattices. On one hand, a generalization for Laurent biorthogonal polynomials is proposed and its recurrence relation and Christoffel transformation are derived. On the other hand, it turns out the compatibility condition between the recurrence relation and the Christoffel transformation for the generalized Laurent biorthogonal polynomials yields an extension of the fully discrete relativistic Toda lattice. And also, it is shown that isospectral deformations of the generalized Laurent biorthogonal polynomials lead to two different generalizations of the continuous-time relativistic Toda lattice, one of which can reduce to the Narita–Itoh–Bogoyavlensky lattice.
During production of the article unfortunately mistakes have been introduced in Eqs. 36, 37, 55, 56 and the one on page 17, second last line. The original article has been corrected.
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