# On the soliton solutions of a family of Tzitzeica equations

## TL;DR

This paper investigates soliton solutions of a family of Tzitzeica equations using dressing and Hirota methods, classifying solutions by eigenvalues and discussing their singularities and transformations.

## Contribution

It introduces two types of soliton solutions derived via dressing method, linking eigenvalues to soliton types, and outlines construction of multi-soliton solutions for Tzitzeica equations.

## Key findings

- Two types of soliton solutions identified based on eigenvalues.
- Construction method for general multi-soliton solutions provided.
- Discussion of singularities and variable transformations included.

## Abstract

We analyze several types of soliton solutions to a family of Tzitzeica equations. To this end we use two methods for deriving the soliton solutions: the dressing method and Hirota method. The dressing method allows us to derive two types of soliton solutions. The first type corresponds to a set of 6 symmetrically situated discrete eigenvalues of the Lax operator $L$; to each soliton of the second type one relates a set of 12 discrete eigenvalues of $L$. We also outline how one can construct general $N$ soliton solution containing $N_1$ solitons of first type and $N_2$ solitons of second type, $N=N_1+N_2$. The possible singularities of the solitons and the effects of change of variables that relate the different members of Tzitzeica family equations are briefly discussed. All equations allow quasi-regular as well as singular soliton solutions.