Solutions of Several Coupled Discrete Models in terms of Lame Polynomials of Arbitrary Order

TL;DR

This paper presents exact quasiperiodic solutions for several coupled discrete models in physics using Lamé polynomials of arbitrary order, revealing a connection to Chebyshev polynomials of Jacobi elliptic functions.

Contribution

It provides a comprehensive set of solutions for multiple coupled discrete models using Lamé polynomials, highlighting their relation to Chebyshev polynomials.

Findings

Exact quasiperiodic solutions for coupled models

Lamé polynomial coefficients relate to Chebyshev polynomials

Applicable to models like Salerno, Ablowitz-Ladik, φ^4, φ^6

Abstract

Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lam\'e polynomials of arbitrary order. The models discussed are (i) coupled Salerno model, (ii) coupled Ablowitz-Ladik model, (iii) coupled $\phi^4$ model, and (iv) coupled $\phi^6$ model. In all these cases we show that the coefficients of the Lam\'e polynomials are such that the Lam\'e polynomials can be reexpressed in terms of Chebyshev polynomials of the relevant Jacobi elliptic function.