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Fourier Analysis and Partial Differential Equations Lab
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The lab studies nonlinear partial differential equations describing wave phenomena using Fourier analysis.
AboutAbout the lab
Verifying the equations of waves through mathematics.
Prof. Kato studies nonlinear partial differential equations.
Nonlinear partial differential equations appear as models of a wide range of natural phenomena.
The professor examines mathematical well-posedness, such as whether a solution exists and whether it is uniquely determined.
In particular, equations describing wave phenomena are studied with methods from functional analysis and harmonic analysis (including Fourier analysis).
Key terms
- Partial differential equation
- An equation that describes how a quantity determined by several variables, such as time and position, changes
- Fourier analysis
- A method of studying a function by decomposing it into a superposition of waves of various frequencies
- Manifold
- A shape, like a sphere, that can be regarded as flat space near any of its points
ResearchResearch themes
The lab studies dispersive equations such as the nonlinear Schrödinger equation, as well as nonlinear wave equations. After considering the existence and uniqueness of solutions, it investigates their smoothness and asymptotic behavior (how they behave as time passes).
*Illustrative imageNonlinear Dispersive and Wave Equations
The lab treats dispersive equations such as the nonlinear Schrödinger equation, and wave equations, using methods of functional analysis. Because it is difficult to find solutions explicitly, the lab first considers the existence and uniqueness of solutions, and then investigates their smoothness and asymptotic behavior.
*Illustrative imageEquations with a Geometric Background
The lab is interested in wave maps and in Schrödinger maps, also called dispersive flows of harmonic maps. These equations involve problems spanning geometry and analysis; for example, the properties of the target manifold are expected to affect the global-in-time behavior of solutions.
MembersFaculty
- Associate ProfessorJun Kato
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: “Ocean waves water” Jon Sullivan / Public domain (cropped)
- Nonlinear Dispersive and Wave Equations: “Wave packet (no dispersion) instance” User:Dicklyon / CC BY-SA 3.0 (cropped)
- Equations with a Geometric Background: “Surface of revolution illustration violett” W like wiki / CC BY-SA 4.0 (cropped)
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