Avraham Soffer
Rutgers University-New Brunswick, NJ·Mathematics & Statistics
When a wave is left alone for a very long time, what does it eventually look like? That question sits at the heart of scattering theory, the branch of mathematical analysis that describes how waves spread out, break apart, or stay stuck in place.
- Fields
Mathematics & Statistics
Physics
- Topics
scattering theory
nonlinear Schrödinger equation
dispersive equations
- Grants
- 1 known grant, through Aug 2027
What they work on
- Characterize the large-time asymptotic behavior of solutions for a general class of nonlinear Schrödinger equations, the equations governing quantum waves and light in nonlinear media, and identify all possible asymptotic states.
- Prove that for spherically symmetric initial data with a space- and time-dependent perturbation, solutions decompose into a freely radiating wave plus a weakly localized component, and derive the properties of that localized part.
- Develop combined functional-analytic, computational and numerical tools tailored to these complex nonlinear equations, where general asymptotic results are currently scarce.
- Study, analytically and numerically, kinetic equations describing many-particle distributions in the case of multiple ergodic components, where the system does not relax to a single uniform state.
More about the research
The answer matters physically: a molecule hit by a laser pulse breaks into pieces, and knowing which pieces fly off is exactly a question about the long-time behavior of a wave equation. So is the gradual energy loss of a light pulse traveling far down an optical fiber, and so is the effect of time-varying noise on the stability of quantum and optical devices. This project aims to characterize the large-time behavior of solutions to complicated nonlinear equations of mathematical physics that model wave propagation in quantum systems and in nonlinear optics.
Why it matters
Knowing the possible end states of a nonlinear wave is what turns a model into a prediction: which fragments a laser-perturbed molecule releases, how much energy a light pulse loses over a long optical fiber, and whether a quantum or optical device stays stable under time-dependent noise. Mathematically, the work would supply general asymptotic results for nonlinear Schrödinger equations, a class where such results remain rare, and sharpen the analytical tools used across quantum mechanics and nonlinear optics.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Mark Ablowitz
University of Colorado Boulder, CO·Mathematics & Statistics
Extend the Inverse Scattering Transform, which turns nonlinear wave equations into solvable linear scattering problems, to new classes of equations including fractional nonlinear wave equations with non-integer-order derivatives.
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.
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.
The school
Rutgers University-New Brunswick: 57 labs on PI Pages, $837M a year on research, 713 research doctorates a year.
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