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Applied Mathematics Lab (Nobu Kishimoto)
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The lab studies the behavior of solutions to nonlinear wave equations.
AboutAbout the lab
Uncovering how solutions to wave equations behave
The lab studies nonlinear partial differential equations, in particular evolution equations known as dispersive equations.
Dispersive equations include the nonlinear Schrödinger equation and the KdV equation.
Using methods from harmonic analysis and real analysis, the lab investigates the well-posedness of initial value problems (existence and uniqueness of solutions, and stability with respect to changes in the initial data).
The lab also studies global-in-time properties of solutions, such as asymptotics toward linear solutions and finite-time blow-up.
Key terms
- Initial value problem
- A problem in which the state at an initial time is given and the subsequent time evolution is determined from the equation
- Well-posedness
- The property that a solution exists and is unique, and that changing the initial data slightly changes the solution only slightly
- Smoothing effect
- The effect by which a solution becomes smoother over time than its initial state
ResearchResearch themes
Using methods from harmonic analysis and real analysis, the lab studies the well-posedness of initial value problems and global-in-time properties of solutions for nonlinear partial differential equations, in particular evolution equations known as dispersive equations. In recent years, it has investigated how nonlinearity affects the solvability of equations and the behavior of their solutions under periodic boundary conditions.
*Illustrative imageSmoothing effects from dispersion and well-posedness
Using methods such as the Fourier restriction norm method, the lab studies the well-posedness of a variety of equations, the breakdown of well-posedness for non-smooth initial data, and general frameworks for proving uniqueness of solutions.
*Illustrative imagePeriodic boundary conditions and resonance
For a nonlinear dispersive equation that models plasma, the lab showed that solutions become smooth over time even when the initial data are not smooth.
*Illustrative imageMethodology for analyzing nonlinearity
Interested in the "parabolic" nature inherent in the resonant interactions of dispersive equations, the lab aims for a unified understanding of methods such as gauge transformations, modified energy, and normal form transformations.
MembersFaculty
- LecturerNobu Kishimoto
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Rogue waves breaking behavior at different crossing angles, McAllister 2019” McAllister et al 2019 / CC BY 4.0 (cropped)
- Smoothing effects from dispersion and well-posedness: “Realistic rogue wave model” Юрій Бельчинський / CC BY 4.0 (cropped)
- Periodic boundary conditions and resonance: “Filaments inside a plasma tube” Bravemimoza / CC BY 4.0 (cropped)
- Methodology for analyzing nonlinearity: “Fourier Series-Square wave 3 H (no scale)” Saisundar.s / CC BY-SA 4.0 (cropped)
- Related lab (Ozawa Lab): “Cayley graph of Alternating Group A5” Kathryn Cramer / CC BY-SA 4.0 (cropped)
- Related lab (Sehnem Lab): “Hyperbolic tiling 7 7-2” Double sharp (talk), based on this picture by Jim McNeill, but with the remaining overlapping heptagons left as outlines instead of deleted. A green background to the tiling has also been added. Created using Tyler and Paint. / Public domain (cropped)
- Related lab (Applied Mathematics Division (Kawamura) Lab): “Turing Machine” Porao / Public domain (cropped)
- Related lab (Isono Lab): “Bernoullishift” Andreas Mandl / CC BY-SA 3.0 (cropped)
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