# Math 639 — Nonlinear Waves — Syllabus

Some details of the syllabus may change between semesters; consult your professor's syllabus!

## Stable Course Components

The course components described in this section are mostly professor independent, and should not greatly change from semester-to-semester.

### Catalog Description

Linear waves, dissipation, dispersion. Conservation laws. Water waves. KdV equation, solitary waves, cnoidal waves. Scattering and inverse scattering. Perturbation theory. Nonlinear Schroedinger equation, dark and bright solitons, vortex solutions. Variational techniques, modulational instability, stability.

### Prerequisites

1. One of:
1. Math 531 (Partial Differential Equations), or
2. Math 537 (Ordinary Differential Equations).

### Overview

The study of nonlinear systems has quietly and steadily revolutionized the realm of science over recent years. It is known that for nonlinear systems new structures emerge that have their features and peculiar ways of interacting. Examples of such structures abound in nature and include: vortices (like tornadoes or eddies in water tanks), solitons (bits of information used in optical fiber communications, water waves, tsunamis, humps of coherent matter waves, etc, ...), spirals (biological aggregates and chemical reactions). This course is intended as an introduction to the theory and of Nonlinear Waves and their applications. The course is intended for senior undergraduate and graduate students in Applied Mathematics, Computational Science, Engineering, Physics, Chemistry, Biology, etc. Examples from interdisciplinary areas will be covered. Most of the concepts and examples will be supplemented with hands-on computer codes.

### Student Learning Objectives

• Conservation Laws: We will derive the core conservation laws of fluid mechanics from both an Eulerian and Lagrangian perspective. Bernoulli's equation will be derived and basic physical intuition in fluids will be developed through its use to solve fluid flow problems. Scaling will be introduced, and you will rescale the core conservation equations in several different regimes and explore the impact of non-dimensional parameters.
• Vorticity: The impact of vorticity on flows will be determined through the use of point-vortex approximations. Connections with differential equations will be derived as well as core theoretical results such as Kelvin and Helmholtz's Theorems. The definitions of baroclinic and barotropic fluids will also be introduced.
• Potential Theory: The use of analytic function theory to determine inviscid, incompressible, irrotational flows will be presented in several different geometries. The formulas for finding drag and lift on a body will be derived and used to compute profiles for a given body.
• Fourier Analysis for PDE's: We will solve linear PDE's using Fourier Series/Transforms. The concepts of group and phase velocity and dispersion will be explored and used to show how energy is distributed a dispersive PDE.
• Free Boundary Value Problems: Techniques for succinctly deriving evolution equations in the context of oceanic-free surface flows will be developed. This will make use of core results in vector calculus, such as the Divergence Theorem.
• Multiple Scales Analysis: The use of multiple scale ansatzes to reduce complex nonlinear models to simpler ones over smaller scales will be developed and used to solve several problems in fluids.
• Deriving nonlinear wave equations: The Korteweg de-Vries and Nonlinear Schroedinger equations will be derived. Solutions will be derived and their properties and implications for oceanic flows will be examined.

## General Policies and Information

The information in this section applies to all courses offered by the department

### Students with Disabilities

If you are a student with a disability and believe you will need accommodations for this class, it is your responsibility to contact the Student Ability Success Center at (619) 594-6473. To avoid any delay in the receipt of your accommodations, you should contact Student Ability Success Center as soon as possible. Please note that accommodations are not retroactive, and that I cannot provide accommodations based upon disability until I have received an accommodation letter from Student Ability Success Center. Your cooperation is appreciated.

### Student Privacy and Intellectual Property

The Family Educational Rights and Privacy Act (FERPA) mandates the protection of student information, including contact information, grades, and graded assignments. I will not post grades or leave graded assignments in public places. Students will be notified at the time of an assignment if copies of student work will be retained beyond the end of the semester or used as examples for future students or the wider public. Students maintain intellectual property rights to work products they create as part of this course unless they are formally notified otherwise.

### Mathematics and Statistics Learning Center

The SDSU Math & Stat Learning Center is in the Love Library, Room LL-328. "The Math and Stats Learning Center is open to support students in all lower division math courses at SDSU. We have tutors available for walk-in help during all open hours. TAs for Math 141, 150, 151, and 252 also hold their office hours there. Please see the schedule of when the TAs for your class will be in the center by going to our website: mlc.sdsu.edu. The MLC is supported by your student success fee. We strongly encourage you to use this wonderful, free resource. Some students believe that they should not need to ask for help. But, research has shown that the average grade for students who attend the MLC is one half grade higher than those who don’t seek such support."

If you are enrolled in a class which does not have targeted support, the MLC can still serve as a great math study/meeting place; and if you are interested in becoming a tutor in the center, keep an eye on the center's webpage for hiring announcements.

### Cheating and Plagiarism

Students are generally encouraged to study together, and to work together to solve exercises. Finals, Midterms, Quizzes, Project, and other designated "individual work" activities must be completed without assistance. All violations will be reported to the Center for Student Rights and Responsibilities and will also result in score/grade reductions at the professor's discretion. Please review SDSU's full policy on academic honesty.

Examples of academic dishonesty include but are not limited to

• copying, in part or in whole, from another's test or other examination
• obtaining copies of a test, an examination, or other course material without the permission of the instructor
• collaborating with another or others in work to be presented without the permission of the instructor
• falsifying records, laboratory work, or other course data
• submitting work previously presented in another course, if contrary to the rules of the course
• altering or interfering with grading procedures
• assisting another student in any of the above
• using sources verbatim or paraphrasing without giving proper attribution (this can include phrases, sentences, paragraphs and/or pages of work)
• copying and pasting work from an online or offline source directly and calling it your own
• using information you find from an online or offline source without giving the author credit
• replacing words or phrases from another source and inserting your own words or phrases

### Religious Observances

According to the University Policy File, students should notify the instructors of affected courses of planned absences for religious observances by the end of the second week of classes.

### Medical-Related Absences

Students are instructed to contact their professor/instructor in the event they need to miss class, etc. due to an illness, injury or emergency. All decisions about the impact of an absence, as well as any arrangements for making up work, rest with the instructors. Student Health Services (SHS) does not provide medical excuses for short-term absences due to illness or injury. When a medical-related absence persists beyond five days, SHS will work with students to provide appropriate documentation. When a student is hospitalized or has a serious, ongoing illness or injury, SHS will, at the student's request and with the student’s consent, communicate with the student’s instructors via the Vice President for Student Affairs and may communicate with the student’s Assistant Dean and/or the Student Ability Success Center

### Kumeyaay Land Acknowledgement

For millennia, the Kumeyaay people have been a part of this land. This land has nourished, healed, protected and embraced them for many generations in a relationship of balance andharmony. As members of the San Diego State community we acknowledge this legacy. We promote this balance and harmony. We find inspiration from this land; the land of theKumeyaay.