# Solitary Wave Resolution: Long-Time Asymptotics for Near-Integrable Systems

This project lies within a set of fundamental questions concerning the qualitative features of solutions to nonlinear Hamiltonian partial differential or difference equations (PDEs or P$\Delta$Es, respectively). It is well known that in these types of equations several different phenomena can appear over long times, among them blow-up, scattering to the free evolution, or the emergence of stable nonlinear structures, such as solitary waves and breather solutions. A detailed description of any such phenomenon often depends on the precise structure of the equation in question. However, there is the common belief, usually referred to as the **solitary wave resolution conjecture**, that (in the absence of finite time blow-up) generic solutions can be decomposed at large times into a sum of solitary waves plus a dispersive tail (i.e., radiation).

For Hamiltonian equations that are **completely integrable**, the mechanism underlying this resolution (called **soliton-resolution** in this case) is rather well-understood, and can be rigorously obtained using the Deift-Zhou method of nonlinear steepest descent for oscillatory Riemann-Hilbert problems.

## Goal

To extend the analytical methods which use Riemann-Hilbert problem formulation of associated inverse scattering transforms to certain (non-integrable) perturbations of completely integrable PDEs or P∆Es which admits solitary wave solutions, and hence obtain rigorous long-time asymptotics for these non-integrable systems.

This project involves collaboration with Irina Nenciu.