Nonlinear dispersive wave equations model a variety of phenomena in fluid mechanics, plasma physics, turbulence, and optics. Analyzing and simulating such equations numerically allows us to make predictions pertaining to the corresponding application. For example, it would be desirable to address the following questions arising in applications: Will signals in optical devices, used for the transmission and storage of data, be stable, or how is energy transferred in turbulent systems? This project will advance our knowledge in two ways. First, it will seek to prove, in a mathematically rigorous sense, properties of the equations, an example of which is the stability of optical pulses. Second, it will develop and justify numerical algorithms for simulating the equations on computers. Systematic analysis of the algorithms ensures that the computer simulations truly reflect the equations. Finally, both the analysis and simulation of the equations will determine the regime of validity of these equations as models for the physical applications. A supplemental benefit of this research is that the numerical algorithms it develops will be applicable to a broad class of equations and physical problems.
The equations under investigation in this project are of the nonlinear Schrödinger type, including the derivative nonlinear Schrödinger equation and the Gross-Pitaevskii equation. Major analytical questions this work will address include the existence and stability of singular solutions and nonlinear bound states. These problems will be addressed through asymptotic analysis, variational methods, and spectral theory. Singularity formation will also be investigated through simulation, using adaptive meshing methodology, such as iterative grid redistribution. Numerical algorithms for simulating the time dependent problems will be implemented by operator splitting methods, treating the nonlinear and linear dispersive parts of the equation separately. This numerical analysis will be pursued with the intention of constructing efficient energy preserving methods suitable for long time simulation. An algorithm for computing nonlinear bound states will be studied in a semi-discrete form, so as to be insensitive to spatial discretization. The bound state algorithms will assist in the construction of initial conditions, which can then be studied using the time dependent algorithms.
