# On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations

## TL;DR

This paper investigates self-similar solutions of the Airy shallow water equations, revealing their integrability, conserved quantities, and explicit solutions through a bi-Hamiltonian framework, enhancing understanding of their geometric structure.

## Contribution

It introduces a class of self-similar solutions linked to bi-Hamiltonian structures, providing explicit solutions and a geometric perspective on the Airy equations.

## Key findings

- Solutions reduce PDEs to finite ODE systems
- Existence of conserved quantities for these solutions
- Explicit solutions obtained via quadratures

## Abstract

Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schr\"odinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system which admits several conserved quantities, which allow to construct explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.