Andrew Lawrie
University of Maryland, MD·Mathematics & Statistics
Waves are everywhere: electricity on a circuit board, light in an optical fiber, particles inside atoms, the gravitational field around a black hole. Yet the equations describing waves that interact with themselves are still poorly understood, and the basic question of what a wave does over very long times remains open.
- Fields
Mathematics & Statistics
Physics
- Topics
solitons
soliton resolution conjecture
nonlinear wave equations
- Grants
- 2 known grants, through Jun 2029
What they work on
- Work toward proving the soliton resolution conjecture, which predicts that generic solutions split into finitely many solitons plus radiation, without symmetry assumptions, beginning with the harmonic map heat flow in two dimensions.
- Give asymptotic descriptions of multi-soliton solutions and their collisions by reducing strongly interacting soliton dynamics to n-body problems for soliton positions, momenta and scales.
- Settle the unique continuation problem for singular nonlinear waves, asking whether a solution can be continued uniquely past the time it blows up.
- Analyze the kink stability problem for the classical phi-4 equation on the line, where the kink is the simplest topological soliton.
- Construct explicit counterexamples to the soliton resolution conjecture for the two-dimensional harmonic map heat flow and the sphere-valued wave maps equation, without symmetry assumptions, in settings where it is expected to hold.
More about the research
The research here asks when a wave spreads out and fades, when it settles into a persistent lump, and when it collapses into a singularity.
Why it matters
These are questions in pure analysis about the equations that model light, electricity, elementary particles and gravitation. Establishing when solutions disperse, persist as particle-like solitons, or collapse, and finding where the expected soliton resolution picture breaks down, sharpens the mathematical foundations used to reason about wave phenomena underlying technologies such as data transmission and about fundamental physics.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Wilhelm Schlag
Yale University, CT·Mathematics & Statistics
Determine whether the kink soliton of the phi-4 model is asymptotically stable, extending a long-range scattering theory for potentials that have a threshold resonance.
Gong Chen
Georgia Institute of Technology, GA·Mathematics & Statistics·Early-career
Determine whether multi-solitons, solutions made of several solitary waves at once, are stable and unique in nonlinear dispersive equations, using PDE, harmonic analysis and spectral theory.
Sung-Jin Oh
University of California-Berkeley, CA·Mathematics & Statistics·Early-career
Prove a forward-in-time scattering criterion for initially outgoing solutions of the energy-critical wave maps and hyperbolic Yang-Mills equations, as a step toward the soliton resolution conjecture in the one-soliton regime.
The school
University of Maryland: 49 labs on PI Pages, $1.5B a year on research, 602 research doctorates a year.
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