| Abstract: |
| In this talk, the finite-time blow-up in nonlinear dispersive PDEs will be discussed as a bifurcation problem. First, we will present a universal framework for the identification of self-similar waveforms as stationary solutions in a frame that co-explodes with the solution. This will allow us to perform a spectral analysis of these solutions in the co-exploding frame in order to infer their stability. As prototypical examples, we will consider the 1D Nonlinear Schroedinger (NLS) equation with power-law nonlinearity and generalized Korteweg-de Vries (gKdV) model. Self-similar collapsing branches emanate from the solitary branch at critical nonlinearity exponent therein. However, their stability analysis will reveal the emergence of unstable modes which in turn are intimately connected with symmetries of the systems in the original frame. We will show that these modes do not correspond to true instabilities but rather to neutral eigen-directions. Numerical results both at the existence and stability will be compared with normal forms and WKB results where an excellent agreement will be observed between the two. Then, time-permitting, we will depart from 1D settings and focus our considerations on the 2D NLS model with power-law nonlinearity. The accurate computation of states and their stability requires to use alternative techniques. Indeed, we will present a computational framework that have been developed in the open source software FreeFEM that combines domain-decomposition methods and mesh adaptivity tools. Novel results will be discussed together with the performance of the numerical codes. |
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