# On Hamiltonian systems integrable in elliptic functions that describe waves over underwater banks and ridges

## TL;DR

This paper analyzes 4-dimensional Hamiltonian systems modeling water waves over underwater features, demonstrating their exact integrability using elliptic functions and solutions to complex transcendental equations involving elliptic integrals.

## Contribution

It introduces a class of Hamiltonian systems that are exactly integrable in elliptic functions, providing explicit solutions for wave dynamics over underwater structures.

## Key findings

- Systems are exactly integrable in elliptic functions.
- Solutions involve transcendental equations with elliptic integrals.
- Provides a mathematical framework for wave modeling over underwater features.

## Abstract

We discuss the 4-dimensional Hamiltonian systems that describe waves over underwater banks and ridges. The systems are exactly integrable in terms of elliptic functions and of solutions to nontrivial transcendental equations involving the elliptic integrals (Weierstrass' $\zeta$-function).