Rogue waves in fluid dynamics and optical waveguides are unexpectedly large displacements from a background state, and occur in the nonlinear Schr\"odinger equation with positive linear dispersion in the regime of positive cubic nonlinearity. Rogue waves of a derivative nonlinear Schr\"odinger equation are calculated in this work as a long-wave limit of a breather (a pulsating mode), and can occur in the regime of negative cubic nonlinearity if a sufficiently strong self-steepening nonlinearity is also present. This critical magnitude is shown to be precisely the threshold for the onset of modulation instabilities of the background plane wave, providing a strong piece of evidence regarding the connection between a rogue wave and modulation instability. The maximum amplitude of the rogue wave is three times that of the background plane wave, a result identical to that of the Peregrine breather in the classical nonlinear Schr\"odinger equation model. This amplification ratio and the resulting spectral broadening arising from modulation instability correlate with recent experimental results of water waves. Numerical simulations in the regime of marginal stability are described.

Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schrödinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics, plasmas, etc., exhibits RWs only in the regime of modulation instability (MI) of the background. For a system of multiple waveguides, the governing coupled NLSEs can produce regimes of MI and RWs, even if each component has dispersion and cubic nonlinearity of opposite signs. A similar effect is demonstrated here for a system of coupled derivative NLSEs (DNLSEs) where the special feature is the nonlinear self-steepening of narrow pulses. More precisely, these additional regimes of MI and RWs for coupled DNLSEs depend on the mismatch in group velocities between the components, and the parameters for cubic nonlinearity and self-steepening. RWs considered in this paper differ from those of the NLSEs in terms of the amplification ratio and criteria of existence. Applications to optics and plasma physics are discussed.

Quintic derivative nonlinear Schr\"odinger equations arise in various physical contexts, notably in the study of hydrodynamic wave packets and media with negative refractive index. A procedure to isolate propagating wave patterns in such nonlinear Schr\"odinger equations is proposed which is based on two integrals of motion. As an illustration of the method, a ``gray'' solitary pulse, a ``dark'' localized mode with nonzero minimum in intensity on a continuous-wave background is identified.

Modulation instability, breather formation, and the Fermi–Pasta–Ulam–Tsingou recurrence (FPUT) phenomena are studied in this article. Physically, such nonlinear systems arise when the medium is slightly anisotropic, e.g., optical fibers with weak birefringence where the slowly varying pulse envelopes are governed by these coherently coupled Schrödinger equations. The Darboux transformation is used to calculate a class of breathers where the carrier envelope depends on the transverse coordinate of the Schrödinger equations. A “cascading mechanism” is utilized to elucidate the initial stages of FPUT. More precisely, higher order nonlinear terms that are exponentially small initially can grow rapidly. A breather is formed when the linear mode and higher order ones attain roughly the same magnitude. The conditions for generating various breathers and connections with modulation instability are elucidated. The growth phase then subsides and the cycle is repeated, leading to FPUT. Unequal initial conditions for the two waveguides produce symmetry breaking, with “eye-shaped” breathers in one waveguide and “four-petal” modes in the other. An analytical formula for the time or distance of breather formation for a two-waveguide system is proposed, based on the disturbance amplitude and instability growth rate. Excellent agreement with numerical simulations is achieved. Furthermore, the roles of modulation instability for FPUT are elucidated with illustrative case studies. In particular, depending on whether the second harmonic falls within the unstable band, FPUT patterns with one single or two distinct wavelength(s) are observed. For applications to temporal optical waveguides, the present formulation can predict the distance along a weakly birefringent fiber needed to observe FPUT.

Interactions between surface gravity waves and a floating rigid body are complex, as waves may reflect from, break on, and be transmitted behind the body. Studies of these phenomena are critically important in improving the safety and functional efficiency of offshore structures. Here, the wave attenuation performance and motions of a type of floating breakwater (FB) are studied through numerical and experimental approaches. A numerical wave tank (NWT) is developed based on the software OpenFOAM and properties of wave channel from a laboratory. In the NWT, the air–water interface is captured by the volume of fluid method. The motions of FB are tracked by the six degrees of freedom model. A mooring system model is developed to simulate the constraints of the FB. Large eddy simulation turbulence modeling is implemented for the wave breaking processes. A model FB with a scale of 1:20 is tested in both the experimental and numerical wave channel. Wave heights at the back/front of the FB and the constraint forces of the mooring wires are measured. The numerical models are validated by comparing the results with experimental measurements. The variations of transmission/reflection coefficients, energy dissipation rate, and maximum mooring force are calculated. Changes of the response amplitude operators with the ratio of FB width to wavelength ($B/L_w$ ) and wave steepness are analyzed. The wave transmission coefficient will drop below 0.8 if the value of $B/L_w$ is larger than 0.3, but will go over 0.95 if $B/L_w$ is less than 0.1. Wave steepness has a large influence on FB motions and the mooring system. The effect of Stokes drift is observed by the shift of position of the FB.

A resonance between long and short waves will occur if the phase velocity of the long wave matches the group velocity of the short wave. In this paper, a system with two distinct packets of short waves in resonance with a common long wave is studied. Breather solutions are calculated by the Hirota bilinear method, and rogue wave modes (unexpectedly large displacements from an otherwise calm background state) are obtained from the breathers through a long wave limit. The location and magnitude of the maximum displacement are determined quantitatively. Remarkably this coupling enables a rogue wave to attain a larger magnitude than that in a configuration with just one single short wave component. Furthermore, as the wavenumber varies, a transition from an elevation rogue wave to a depression rogue wave is possible. This transformation of the wave profile is elucidated in terms of the properties of the carrier envelope. The connection with the modulation instability of the background plane wave is investigated. Some numerical simulations are performed to demonstrate both the robust nature and unstable behavior for these rogue waves, depending on the parameters of the system. Dynamics and properties of rogue waves with three or more short wave components are also considered.