Nonlinear Waves

New York University, New York NY

Investigators

Abstract

Dispersive waves occur in a variety of physical systems such as nonlinear optics, atmosphere and ocean waves, and plasmas. Often these waves exhibit resonant interactions, i.e., interactions of waves that oscillate at the same frequency. Resonant interactions are of paramount importance in the long-time behavior of the waves present in the relevant physical system. Resonances can cause the wave amplitude to grow or can lead to genuine nonlinear behavior of solutions. It quantifies the long-time effects of the nonlinearities present in the problem. In addition to resonance, dispersion (waves moving into different parts of space) plays a central role in the long-time behavior of solutions. Dispersion often tames the effects of resonances by pushing resonant waves into different regions in space. It provides a mechanism to control the amplitude of resonant waves. Thus, together with dispersion, resonances form the backbone of the analytical tools which have been developed to study stability of nonlinear waves. The development of the space-time resonance method is a corner stone in such an endeavor. It brings existing methods together in a new setting that extends their applicability far beyond what was previously possible. This method is still in development and has been extended far beyond its initial inception. Nevertheless, there is lots of progress that needs to be made. This proposal is concerned with the study of propagation, interaction, and asymptotic behavior of small amplitude nonlinear waves. The PI proposes to study the asymptotic behavior of small solutions of four dispersive and hyperbolic problems. The aim of the research is to develop new methods to determine the asymptotic behavior of such solutions. The PI, in collaboration with others, plans to consider problems when either the energy or decay estimates are weak, i.e., non-integrable, or when the nonlinearity is resonant. One should note here that there are techniques which have already been developed to deal with resonant nonlinearity, and although these techniques should prove to be very useful for the proposed research, new methods need to be developed to deal with additional difficulties. The problems that the PI will investigate are 1) the Klein-Gordon-Zakharov system; 2) Gravity-Capillary Surface Waves; 3) the asymptotic behavior of solutions to systems where the nonlinearities cause long-range effects. In addition to these projects, the PI will investigate resonant dynamics and infinite volume approximation, where the final goal is to derive a reduced system of equations for the effective dynamics of gravity waves in a 2-dimensional fluid on a large domain of size. The question that the PI wants to address can be roughly stated as follows: Given a system whose domain is a large box of size and with periodic boundary conditions, is there a simple system of equations that describe the dynamics of small solutions for a very long time? The time is, of course, assumed to be longer than what it takes for a wave packet to reach the boundary of the box. Solutions to the proposed problems will require the development of new methods and techniques and the refinement of existing ones.