Subharmonic modulational instabilities

Benjamin and Feir in 1967 demonstrated that Stokes waves in deep water are unstable against longitudinal sideband perturbations, and this instability leads to the transformation of an initial periodic wave train into a chain of wave packets. Nowadays, this phenomenon is known as the Benjamin–Feir or modulational instability (MI), with a more general definition. In particular, it may be the subharmonic instability, when periodic wave trains are unstable against perturbations whose spatial period is a multiple of the underlying wave-train’s period. MI is well known as a ubiquitous effect occurring in diverse fields, such as water waves, plasmas, optics and photonics, and Bose–Einstein condensates (BECs). One can examine the initial (i.e., linear) stage of the MI development by the linearization of the governing equations for small perturbations around the initial (unperturbed) periodic wave train. The linearization ceases to be valid when the growing amplitude of the perturbations becomes comparable to the amplitude of the unperturbed periodic wave trains, which makes investigation of the nonlinear stage of the MI-driven dynamics necessary. It is well known that the nonlinear evolution of MI leads to the formation of localized structures, such as solitons, breathers and rogue waves (RWs). Another essential type of the instability is the high-frequency MI (HFMI), which also originates from the water-wave theory and has been extended to other fields of physics. Similar to the classical MI (alias the low-frequency MI (LFMI)), HFMI may also be viewed as a subharmonic instability of periodic wave trains, but with a very small growth rate and narrow intervals of the Floquet exponents in which HFMI occurs. In this review, we mainly focus on the linear and nonlinear dynamics of periodic wave trains in integrable and nearly integrable systems, including the linear and nonlinear stage of the MI development and investigation of the subharmonic MI. First, we review findings concerning HFMI in fluid dynamics. Subsequently, we address outcomes of HFMI in various physical fields produced by the universal nonlinear Schrödinger (NLS) equation, complemented by a spatially periodic potential. Second, we review discoveries regarding MI of cnoidal waves in nearly integrable and non-integrable systems, drawing insights from the consideration of the long-wave-short-wave resonance equation and universal ${\varphi }^4$ equation. In addition to that, we summarize analytical results pertaining to MI and modulational stability of cnoidal waves in some integrable systems. Third, we review results for the nonlinear stage of the MI development, by introducing exact correspondence between MI and the formation of localized waves, including RWs with ultra-high peak amplitudes in the baseband-MI regime, and the formation of RWs in the zero-wavenumber-gain MI regime, in the framework of the three-component Gross–Pitaevskii equations, Bers–Kaup–Reiman system, Lugiato–Lefever equations and nonlinear Dirac equations.