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.
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.