A class of solutions of the two-dimensional Toda lattice equation

TL;DR

This paper introduces a systematic method to generate solutions for the two-dimensional Toda lattice equation using known solutions of Laplace's equation, connecting Nakamura's solution as a special case.

Contribution

It presents a novel approach to construct Toda lattice solutions from Laplace solutions, expanding the toolkit for analyzing integrable systems.

Findings

Method successfully generates Toda solutions from Laplace solutions

Nakamura's solution is identified as a special case of this method

Provides a systematic framework for solution generation

Abstract

A method is proposed to systematically generate solutions of the two-dimensional Toda lattice equation in terms of previously known solutions $\phi\left(x,y\right)$ of the two-dimensional Laplace's equation. The two-dimensional solution of Nakamura's [J. Phys. Soc. Jpn. \textbf{52}, 380 (1983)] is shown to correspond to one particular choice of $\phi\left(x,y\right)$.