# Free surface lump wave dynamics of a saturated superfluid Helium film with nontrivial boundary condition at the substrate surface

## TL;DR

This paper investigates the dynamics of free surface waves in a saturated superfluid Helium film with a nontrivial boundary condition, deriving a forced KP-I equation and analyzing lump wave solutions affected by substrate leakage velocity.

## Contribution

It introduces a novel model for superfluid Helium film surface waves incorporating weak downward flow and derives exact and perturbative lump wave solutions.

## Key findings

- Surface tension critically influences wave evolution.
- Forced KP-I equation models the system dynamics.
- Leakage velocity affects lump wave solutions.

## Abstract

In this article, the free surface wave dynamics of a saturated superfluid Helium film is considered under the condition that there exists a very weak downward localized superfluid flow into the substrate. For saturated film, the effect of surface tension plays a decisive role in the surface wave evolution dynamics of the system. The free surface evolution is shown to be governed by forced Kadomtsev Petviashvili-I equation, with the forcing function depending on downward superfluid velocity at the substrate surface. Exact as well as perturbative free surface lump wave solutions of the (2+1) dimensional nonlinear evolution equation are obtained and the effect of the leakage velocity function on the lump wave solutions are shown.