  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.
