Mark Ablowitz
University of Colorado Boulder, CO·Mathematics & Statistics
Waves are everywhere: light lets us see, sound lets us hear, and water waves move ships and carry tidal energy. For small waves, mathematics describes behavior well and exact formulas are often available.
- Fields
Mathematics & Statistics
Physics
- Topics
nonlinear waves
solitons
inverse scattering transform
- Grants
- 1 known grant, through Aug 2027
What they work on
- 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.
- Construct and analyze soliton solutions, stable localized waves with special interaction properties, of fractional nonlinear wave equations relevant to settings such as amorphous materials.
- Study the properties of multidimensional lump solitons, waves that decay in all directions, for equations arising in quantum mechanics and optics.
- Develop insight into large-amplitude wave propagation and wave energy transfer for phenomena such as tsunamis, rogue waves, high-power lasers, and shock or blast waves.
More about the research
Large-amplitude or high-power waves are different. They are nonlinear, meaning the wave's own size changes how it travels, and very few such equations can be solved exactly. This project develops mathematical theory for nonlinear wave propagation and for how energy moves between parts of a wave field, with applications in mind including tsunamis, rogue (freak) ocean waves, high-power lasers, and sharp pressure jumps such as shock and blast waves.
Why it matters
Exact solutions and analytical understanding of large-amplitude waves give scientists and engineers a reliable reference for phenomena that are otherwise accessible only through simulation. The results could inform how researchers model tsunamis and rogue ocean waves, high-power laser pulses, shock and blast waves, and wave behavior in disordered materials.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Dionyssios Mantzavinos
University of Kansas, KS·Mathematics & Statistics
Develop analytical tools to establish existence, uniqueness and stability of solutions to nonlinear partial differential equations posed in bounded domains in two or more spatial dimensions.
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.
Avraham Soffer
Rutgers University-New Brunswick, NJ·Mathematics & Statistics
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.
The school
University of Colorado Boulder: 36 labs on PI Pages, $718M a year on research, 447 research doctorates a year.
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