One may alternatively explore what happens to the mathematical, computational (or even physical 21) setting at, or past, the moment of collapse see, e.g., the relevant chapter of ref. One may try to avoid collapse (e.g., by imposing space 15, 16, 17 or time modulations 12, 13, 18, 19) or by identifying the higher order effects that preclude collapse in physical experiments 20. The contexts range from scaling 1, 2, to focusing in prototypical dispersive equations such as the Korteweg−de Vries (KdV) equation 3 and most notably the nonlinear Schrödinger (NLS) equation 4, 5, 6 on the one hand, and from droplets in thin films 7, 8 and flow in porous media 9, 10 to the roughening of crystal surfaces 11 and integrate-and-fire neuronal models 12, 13, 14 on the other. A convenient and ubiquitous illustrative example lies in studying self-similar solutions that collapse in finite time, a topic of widespread interest in both the mathematical and the physical literature. In the actual phenomena modeled, an approach toward infinity is often an indication of model breakdown. This causes the computation to become exceedingly difficult, or to simply fail returning from the neighborhood of infinity to more meaningful computational regimes is a notoriously hard task, often difficult to justify. Mathematical models of physical, biological, as well as socio-economic phenomena and computations based on these models are often observed to approach infinity. This methodology could be useful toward a systematic approach to bypassing infinity and thus going beyond it in a broader range of evolution equation models. Along the path of our analysis, we present a regularization process via complexification and explore its impact on the dynamics we also discuss a set of compactification transformations and their intuitive implications. For Partial Differential Equations, the crossing of infinity may persist for finite time, necessitating the introduction of buffer zones, within which an appropriate transformation is adaptively identified. In our Ordinary Differential Equation examples the crossing of infinity occurs instantaneously. We suggest how, upon suitable transformations, it may become possible to go beyond infinity with the solution becoming again well behaved and the computations continuing normally. When mathematical and computational dynamic models reach infinity in finite time, extending analysis and numerics beyond it becomes a notorious challenge.
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