# The Static Elliptic $N$-soliton Solutions of the KdV Equation

## TL;DR

This paper constructs static elliptic N-soliton solutions for the KdV equation using Bäcklund transformations, extending the known single-soliton elliptic solutions to infinitely many multi-soliton solutions.

## Contribution

It introduces a method to generate static elliptic N-soliton solutions for the KdV equation, a significant extension beyond the previously known single-soliton solutions.

## Key findings

- Successfully constructed static elliptic N-soliton solutions
- Extended the class of known elliptic solutions for KdV
- Demonstrated the use of Bäcklund transformations for multi-soliton solutions

## Abstract

Regarding $N$-soliton solutions, the trigonometric type, the hyperbolic type, and the exponential type solutions are well studied. While for the elliptic type solution, we know only the one-soliton solution so far. Using the commutative B\"{a}cklund transformation, we have succeeded in constructing the KdV static elliptic $N$-soliton solution, which means that we have constructed infinitely many solutions for the $\wp$-function type differential equation.