Harvey just finished what should be the last paper of his PhD studies: Cusp solitons mediated by a topological nonlinearity
Harvey's PhD project studied the intersection between topological data analysis (TDA) techniques and nonlinear and many-body quantum dynamics. His first paper devised a TDA-based pipeline for detecting the emergence of quantum chaos in a periodically-driven nonlinear Kerr cavity. He followed this up with a demonstration of many-body quantum scar detection using topology-based dimensional reduction.
These works, while very nice, were ultimately using TDA to recover known physics. We really want to find examples where TDA can unveil new physics. This is a hard problem. Where to look? And what counts as "new"?
The easier solution for us was to insert TDA "by hand" into a nonlinear model, and see what came out of it.
For our testbed we took the nonlinear Schrodinger equation, frequently used to model nonlinear waves in various platforms. In the usual nonlinear Schrodinger equation, the conserved energy is the Hamiltonian,
$$ H = \int dx \left[ \frac{1}{2} |\partial_x \psi |^2 - \frac{g}{2} |\psi|^4 \right] $$
The second term, responsible for the nonlinear dynamics, can be interpreted as an intensity-dependent potential of depth $\frac{g}{2}|\psi|^2$. We looked at what would happen if we replaced this term with a quantity obtained using TDA. When dealing with one-dimensional functions, such as intensity profiles $|\psi(x)|^2$, TDA frequently uses sublevel set persistent homology, characterizing shape in terms of the persistence of local maxima and minima. We used the total persistence of these features as an energy penalty term, leading to
$$ H^{\prime} = \int dx \left[ \frac{1}{2} |\partial_x \psi|^2 - \alpha \mathrm{sgn}( \partial_x |\psi|^2 ) (\partial_x |\psi|^2) \right] $$
Deriving the equations of motion, we found that this topological energy penalty gives rise to effective $\delta$ function potentials at the local maxima and minima of intensity, which act to enhance or suppress local maxima, depending on the sign of the nonlinear coefficient $\alpha$. We then studied the resulting nonlinear dynamics, including the focusing of Gaussian and flat-top beams.
We also uncovered some interesting connections to the physics of nonlocal nonlinear systems. Specifically, our "topological nonlinearity", when regularized, resembles a weakly nonlocal nonlinearity with a vanishing local part. Such nonlinearity leads to cusp solitons, as was previously studied in the context of plasma physics!
We hope to follow up this study with investigations of similar "topological" nonlinearities and potential experimental realizations. In the present work we speculated that similar nonlinearities may arise in the context of fluid-mediated nonlinearities and lattices undergoing Floquet modulation, but demonstrating such implementations explicitly remains an open problem for us.
