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Mathematical Sciences (Masaki) Lab
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Research on how solutions to equations describing “waves” behave.
AboutAbout the lab
Reading how solutions behave, from equations for waves
Prof. Masaki works mainly on nonlinear dispersive equations within the theory of partial differential equations.
Roughly speaking, dispersive equations are equations about “waves.”
The lab deals with the nonlinear Schrödinger equation, the nonlinear Klein-Gordon equation, the KdV equation, and others.
A major goal is to understand how nonlinearity affects the behavior of solutions after a long time.
Key terms
- Partial differential equation
- An equation that expresses how something changes with time and position, using derivatives in several variables
- Soliton
- A wave that keeps its shape because the effect that spreads the wave out and the effect that gathers it together balance each other
- Scattering solution
- A solution in which nonlinear effects weaken over time, so that it spreads out like a solution of a linear equation
ResearchResearch themes
The lab studies nonlinear dispersive equations, that is, equations for “waves.” The goal is to understand how nonlinearity affects the global-in-time behavior of solutions.
*Illustrative imageNonlinear Dispersive Equations
The lab works on the nonlinear Schrödinger equation, the nonlinear Klein-Gordon equation, and the (generalized) KdV equation, among others. A major goal of the research is to understand how nonlinearity affects the global-in-time behavior of solutions.
*Illustrative imageGlobal-in-Time Dynamics of Solutions
A soliton solution is a stationary wave produced by the balance between linear dispersion and nonlinear interaction. There are also scattering solutions, in which dispersion dominates, and solutions that blow up in finite time, in which nonlinear interaction dominates. The question is what structure these diverse solutions form together.
*Illustrative imageScattering Solutions and Classification of Equations
The lab also studies the effect of nonlinearity that appears in scattering solutions. Recently, it has also become interested in classifying systems of coupled nonlinear equations from the viewpoint of Hamiltonian structure and conservation laws.
MembersFaculty
- ProfessorSatoshi Masaki
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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: “Baltic Sea Wave (Cien Water)” Marek Nowocień / CC BY 2.5 (cropped)
- Nonlinear Dispersive Equations: “Asilomar State Beach (Breaking wave) 03” Tewy / CC BY 2.5 (cropped)
- Global-in-Time Dynamics of Solutions: “River Ribble bore” Frangle Plazma Goat / CC BY-SA 3.0 (cropped)
- Scattering Solutions and Classification of Equations: “Kugelstoßpendel, Dekoration” GeoTrinity / CC0 (cropped)
- Related lab (Analysis (Kubo) Lab): “Surface waves” Roger McLassus / CC BY-SA 3.0 (cropped)
- Related lab (Analysis Lab (Hamamuki)): “Real snowflake macro photo” Alexey Kljatov / CC BY-SA 4.0 (cropped)
- Related lab (Mathematical Sciences (Kuroda) Lab): “Newly fallen snow crystals accumulating on a window sill” W.carter / CC0 (cropped)
- Related lab (Geometry (Nozaki) Lab): “Trefoil knot cgi” Altapedia / CC BY-SA 4.0 (cropped)
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