2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on advanced topics in Mathematics K
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Masahito Ohta / Michiaki Onodera
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E535
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 2Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The main subject of this course is the mathematical analysis of the nonlinear Klein-Gordon equation, which is one of the most fundamental nonlinear partial differential equations. In the first half, we will explain the conservation laws of the nonlinear Klein-Gordon equation, the well-posedness of the Cauchy problem in the energy space, and the existence of ground states for the stationary problem. In the second half, we will introduce and prove results on the stability and instablity of standing wave solutions of the nonlinear Klein-Gordon equation.
The goal of this course is to understand the basic knowledge and techniques required for the study of nonlinear partial differential equations through the mathematical analysis of the nonlinear Klein-Gordon equation.
Course description and aims
・To derive the conservation laws from the symmetry of the nonlinear Klein-Gordon equation.
・To understand the basics of the calculus of variations and prove the existence of a ground state.
・To understand the proof of theorems regarding the stability and instability of standing wave solutions.
Keywords
Nonlinear Klein-Gordon equation, Ground state, Standing wave, Stability
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The following topics will be covered in this order: ・Conservation laws of the nonlinear Klein-Gordon equation ・Well-posedness of Cauchy problem ・Existence of ground states for the stationary problem ・Stability of standing waves ・Instability of standing waves | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required.
Reference books, course materials, etc.
M. Ohta and G. Todorova, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. Math. Anal. 38 (2007), 1912-1931.
Evaluation methods and criteria
Assignments (100%)
Related courses
- None
Prerequisites
None Required