The spontaneous breaking of time-translation symmetry has led to the discovery of a new
phase of matter: the discrete time crystal. Discrete time crystals exhibit rigid subharmonic …

A liquid free surface in partially filled containers can experience a wide spectrum of motions
such as planar, non-planar, rotational, quasi-periodic, chaotic, and disintegration. Civil …

Pattern formation in physical systems is one of the major research frontiers of mathematics.
A central theme of this book is that many instances of pattern formation can be understood …

Patterns in natural systems abound, from the stripes on a zebra to ripples in a riverbed. In
many of these systems, the appearance of an ordered state is not unexpected as the …

We present an experimental investigation of superlattice patterns generated on the surface
of a fluid via parametric forcing with two commensurate frequencies. The spatiotemporal …

We report the observation of a square superlattice pattern in a dielectric barrier discharge
system. The correlation measurements indicate that the square superlattice pattern is an …

Liquid parametric sloshing, known also as Faraday waves, has been a long standing subject
of interest. The development of the theory of Faraday waves has witnessed a number of …

The relative stability of two-dimensional soft quasicrystals in systems with two length scales
is examined using a recently developed projection method, which provides a unified …

We investigate both analytically and numerically the dynamics of Rydberg-dressed Bose-
Einstein condensates subjected to periodic modulation of the nonlocal repulsive interactions …

The Faraday wave experiment is a classic example of a system driven by parametric forcing,
and it produces a wide range of complex patterns, including superlattice patterns and …