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arXiv · 2110.14887

Construction of exact solutions to the Ruijsenaars-Toda lattice via generalized invariant manifolds

I. T. Habibullin·A. R. Khakimova·A. O. Smirnov
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Abstract

The article discusses a new method for constructing algebro-geometric solutions of nonlinear integrable lattices, based on the concept of a generalized invariant manifold (GIM). In contrast to the finite-gap integration method, instead of the eigenfunctions of the Lax operators, we use a joint solution of the linearized equation and GIM. This makes it possible to derive Dubrovin type equations not only in the time variable $t$, but also in the spatial discrete variable $n$. We illustrate the efficiency of the method using the Ruijsenaars-Toda lattice as an example, for which we have derived a real and bounded particular solution in the form of a kink, a periodic solution expressed in terms of Jacobi elliptic functions and a solution expressed through Weierstrass $\wp$-function.

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I. T. Habibullin, A. R. Khakimova, A. O. Smirnov. 2021-10-28. Construction of exact solutions to the Ruijsenaars-Toda lattice via generalized invariant manifolds. https://arxiv.org/abs/2110.14887

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