# Nonlinear fractional waves at elastic interfaces

## TL;DR

This paper derives a nonlinear fractional wave equation for surface waves at elastic interfaces, capturing experimental features like amplitude-dependent propagation and phase transition effects without adjustable parameters.

## Contribution

It introduces a novel nonlinear fractional differential equation model for interface surface waves, linking experimental observations to intrinsic membrane properties.

## Key findings

- Propagation length increases abruptly at a threshold amplitude
- Wave velocity is around 40 cm/s and slightly increases with amplitude
- Model reproduces experimental features without adjustable parameters

## Abstract

We derive the nonlinear fractional surface wave equation that governs compression waves at an interface that is coupled to a viscous bulk medium. The fractional character of the differential equation comes from the fact that the effective thickness of the bulk layer that is coupled to the interface is frequency dependent. The nonlinearity arises from the nonlinear dependence of the interface compressibility on the local compression, which is obtained from experimental measurements and reflects a phase transition at the interface. Numerical solutions of our nonlinear fractional theory reproduce several experimental key features of surface waves in phospholipid monolayers at the air-water interface without freely adjustable fitting parameters. In particular, the propagation length of the surface wave abruptly increases at a threshold excitation amplitude. The wave velocity is found to be of the order of 40 cm/s both in experiments and theory and slightly increases as a function of the excitation amplitude. Nonlinear acoustic switching effects in membranes are thus shown to arise purely based on intrinsic membrane properties, namely the presence of compressibility nonlinearities that accompany phase transitions at the interface.