Breather Solutions to a Two-dimensional Nonlinear Schr\"odinger Equation with Non-local Derivatives

We consider the nonlinear Schr\"odinger equation with non-local derivatives in a two-dimensional periodic domain. For certain orders of derivatives, we find a new type of breather solution dominating the field evolution at low nonlinearity levels. With the increase of nonlinearity, the breathers break down, giving way to wave turbulence (or Rayleigh-Jeans) spectra. Phase-space trajectories associated with the breather solutions are found to be close to that of the linear system, revealing a connection between the breather solution and Kolmogorov-Arnold-Moser (KAM) theory

On the time scales of spectral evolution of nonlinear waves

As presented in Annenkov & Shrira (2009), when a surface gravity wave field is subjected to an abrupt perturbation of external forcing, its spectrum evolves on a ``fast'' dynamic time scale of $O(\varepsilon^{-2})$, with $\varepsilon$ a measure of wave steepness. This observation poses a challenge to wave turbulence theory that predicts an evolution with a kinetic time scale of $O(\varepsilon^{-4})$. We revisit this unresolved problem by studying the same situation in the context of a one-dimensional Majda-McLaughlin-Tabak (MMT) equation with gravity wave dispersion relation. Our results show that the kinetic and dynamic time scales can both be realised, with the former and latter occurring for weaker and stronger forcing perturbations, respectively. The transition between the two regimes corresponds to a critical forcing perturbation, with which the spectral evolution time scale drops to the same order as the linear wave period (of some representative mode). Such fast spectral evolution is mainly induced by a far-from-stationary state after a sufficiently strong forcing perturbation is applied. We further develop a set-based interaction analysis to show that the inertial-range modal evolution in the studied cases is dominated by their (mostly non-local) interactions with the low-wavenumber ``condensate'' induced by the forcing perturbation. The results obtained in this work should be considered to provide significant insight into the original gravity wave problem

Energy transfer for solutions to the nonlinear Schr\"odinger equation on irrational tori

We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schr\"odinger (NLS) initial value problem on 2D irrational tori. Moreover we complement the analytic study with numerical experimentation. As a biproduct of our investigation we also prove that the quasi-resonant part of the NLS initial value problem we consider, in both the focusing and defocusing case, is globally well-posed for initial data of finite mass

Numerical Studies of Wave Turbulence in Finite Domains

Nonlinear wave systems are ubiquitous in nature, and when many incoherent dispersive waves interact, there is the potential for wave turbulence (WT). Just as in flow turbulence, systems in WT exhibit inter-scale energy cascades, power-law inertial-range spectra, and even intermittency. Unlike in flow turbulence, however, a natural analytical closure for field statistics has been developed. By closing the hierarchy of moment equations that determine field statistics, spectral evolution can be expressed as a Boltzmann-like Wave Kinetic Equation (WKE). The WKE and its supporting closure make formal predictions for the steady power-law inertial-range spectra (known as the Kolmogorov-Zakharov (KZ) spectra), the energy cascade strength and direction, and much more. In addition to being of great theoretical interest, the WKE has been widely employed as a reduced-order model for spectral evolution in practical applications such as global ocean wave forecasting models. The WT closure and the WKE are derived in the large-domain and infinitesimal wave amplitude limit (together, the kinetic limit), and they describe the average effect of the wave-wave interactions that drive spectral evolution. When a wave system is realized on a finite domain with finite wave amplitude, this assumption of the kinetic limit does not hold. As a result, WKE predictions such as the KZ spectrum become questionable. Numerical and physical experiments in bounded domains often describe steeper spectra and weaker energy cascades than theory predicts. In extreme cases, coherent structures can form that even lead to the breakdown of the kinetic wave description. While recent theory for predicting finite-size effects is in fairly good agreement with observations, it is a largely qualitative model built on kinematic relationships, considering finite-size effects by comparing Fourier domain discreteness to nonlinear broadening of the dispersion relation. For a given domain size, this theory predicts that finite-size effects will dominate when nonlinear broadening becomes much smaller than characteristic Fourier-space frequency spacing. In this dissertation, we work towards a more quantitative, dynamics-based understanding of finite-size effects through numerical studies of the Majda-McLaughlin-Tabak (MMT) model. First, we explore a limitation of the aforementioned kinematic model: we show that weakly nonlinear wave dynamics in a finite-domain are shaped by the structure of the Discrete Resonant Manifold of wave-wave interactions, which in some cases can support WKE-like dynamics even when nonlinear broadening goes to zero. Next, we explore the properties of the energy cascade in a bounded domain as nonlinear broadening goes to zero. In addition to showing the importance of quasi-resonant interactions to kinetic behavior, we develop an interaction-based energy flux decomposition that allows for a direct, dynamical measurement of nonlinear broadening and a novel and effective study of the WT closure. This tool is then used to study WT in the kinetic limit for a one-dimensional model, where we show numerically that, as the domain is made larger and nonlinearity is made weaker, the error of the WT closure is reduced for a statistically steady WT field. A final study explores a novel, almost-periodic coherent structure in the two-dimensional MMT model that emerges when nonlinearity is weak, where we draw a possible connection to Kolmogorov-Arnold-Moser Theory. We conclude with discussion.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/177880/1/ahrabski_1.pd