**1. MATH 422-522
Advanced Applied Mathematics**

**Time and Place:** Tuesdays and Thursdays
– 12.30 - 1.45pm in PSYCH 205_{}

**Instructor:** Vladimir Zakharov, Math building, Rm 518

**Instructor Office Hours:** Friday 1.00 pm – 4.00 pm Math 518

**T.A.** To be announced.

**T.A. Office Hours:** To be announced.

**Textbook:** Donald A. McQuarrie, Mathematical Methods for Scientists
and Engineers, (University Science Books, 2003). Optional text: Advanced
Engineering Mathematics, Erwin Kreysiz (10-th
Edition, Wiley). Availability: purchased, library reserve with limited ebook access.

**Prerequisites:** (MATH 215 or MATH 410)
and MATH 223 and (MATH 254 or MATH 355 or MATH 250B)

**Final Examination:** Wednesday, May 8, 1.00pm – 3.00pm

**January 22, 2019 (undergrads)** and **February
5, 2019 (grads) **Last day to drop without a grade.

**February 6, 2019 (undergrad) **and **February 5 2019 (grads) **Last day to
change from pass/fail to regular grade or vice versa with only instructor
approval on a Change of Schedule form.
Last day to file Grade
Replacement Opportunity (GRO), deadline is 11:59 PM. Last day for department staff to add/drop in UAccess.**
February 7, 2019 (undergrad)** - Change of Schedule form with Instructor and
Dean's permission is required to change from

For more information please see Dates and
Deadlines link: https://registrar.arizona.edu/dates-and-deadlines/view-dates?field_display_term_value=191&=Apply

**Homework: **Homework
problems will be assigned every two weeks. Totally 6 homeworks
will be assigned. Homework problems will be available on webpage. Graduate
students will be assigned extra homework/class projects. Homeworks
are due one week after posting and should be handed in in class. The latest a
HW can be submitted is before 5pm on that day in Rm 518.

**Examinations:** Two one and a quarter hour tests will be
given and there will be a two-hour final examination. The final examination
will be based upon the entire course. Books and notes are allowed for these
examinations. The use of electronic calculators is permitted. Your final grade
for this course will be determined by the scores of all examinations and the
homework grades. The dates of each test will announced at least one week before
it is scheduled. The final exam counts 30%, each one-hour test counts 20%, and
the homeworks will contribute 30% to the total score.

If you have missed a
one-hour test** **through no fault of your own**,** the final grade for this course will
be determined by replacing the missing test scores with the score of the final
examination. If you miss the final examination and if you have a
well-documented excuse showing that for reasons beyond your control it has been
absolutely impossible for you to take the final examination, and if you had a
passing grade at the time of the final examination, a grade of I will be given
for the course. In all other cases a score of zero will be assigned for the
final examination and the course grade evaluated accordingly.

**Students with
Disabilities:**** If you anticipate barriers related to the format or requirements of
this course, please meet with me so that we can discuss ways to ensure your
full participation in the course. If you determine that disability-related
accommodations are necessary, please register with Disability Resources
(621-3268; drc.arizona.edu) and notify me of your eligibility for reasonable
accommodations. We can then plan how best to coordinate your accommodations.**

**Academic Integrity:** According to the Arizona Code
of Academic Integrity, “Integrity is expected of every student in
all academic work. The guiding principle of academic integrity is that a
student’s submitted work must be the student’s own.” Unless otherwise noted by
the instructor, work for all assignments in this course must be conducted
independently by each student. Co-authored work of any kind is unacceptable.
Misappropriation of exams before or after they are given will be considered
academics misconduct.

**Course Syllabus**

Prerequisites for this course include an introductory vector calculus and ordinary differential equations course. To refresh the material on vectors students may review Chapter 5 while Chapter 11 has a good review of ordinary differential equations.

**Week 1:** Vector
algebra. Vector functions. Frenet-Serret
equation. (5.3, 5.4)

**Week 2:** Vector
calculus. Gradient, divergence, and curl. Line and surface integrals. Green’s
and Stoke’s theorem (7.1 - 7.5)

**Week 3.** Complex
numbers and complex plane (4.1). Infinite series (2.2 – 2.6). Elementary
functions of complex variables. Euler formula (4.2 – 4.5).

**Week 4:** Review of
ODE’s (11.2 – 11.4). Ordinary and Singular Points and Method of Frobenius (12.2, 12.4).

**Week 5:** Special
functions – Legendre polynomials, Bessel functions (12.3 - 12.6)

**Week 6:** Sturm-Liouville theory, Eigenfunction
expansions (14.3 - 14.4)

**Week 7:** Fourier
Series – Sine cosine and complete (15.1 - 15.3)

**Week 8:**
Orthogonal Polynomials, generating functions (14.1 - 14.2).

**Week 9:** Functions
of a Complex variable, Cauchy-Riemann and complex integration (18.1 -18.3).

**Week 10:** Cauchy
integral formula, Taylor and Laurent series (18.4 - 18.5).

**Week 11:** Residue
theorem, evaluation of real definite integrals (18.6, 19.2).

**Week 12.** The
Laplace integral transform and its inversion (17.1, 17.2, 19.1).

**Week 13**: Fourier
transforms with application to generalized functions (17.5, 17.6).

**Week 14:** The
Laplace equation in two- and three-dimensional space (16.1, 16.2). The Heat
equation (16.5).

**Week 15.** The Wave
equation in one- and two-dimensional spaces (16.3, 16.4).

**Week 16.** The
electron in the Hydrogen atom (16.7).

**Lecture 8** February 5, 2019

Due January 31

Due February 28

Due April 2

Date of Exam: March 14

Due April 16

Due May 7

Date of Exam: April 30

**Spring Semester 2019
**

**2. MATH
488-588 *** **Solitons
in Mathematics and Physics*

The course is intended for the graduate students but will be available for the determined undergraduates with basic knowledge of ODE, linear algebra and complex analysis. During the second part of the course the students will be offered individual scientific projects, which potentially can become a foundation for scientific publication.

**Time and Place:** Tuesdays and Thursdays
– 2pm – 3.15pm in MLNG 310_{}

**Instructor:** Vladimir Zakharov, Math building, Rm 518

**Instructor Office Hours:** Friday 1.00 pm – 4.00 pm Math 518

**T.A.** To be announced.

**T.A. Office Hours:** To be announced.

**Prerequisites:** MATH 215, 254, 322 or
422.

**Textbook: ** Lecture notes will
be posted online before each lecture.

**Final Examination:** Monday, May 6, 3.30pm - 5.30pm

Prerequisites for the class are MATH 215, 254, 322 or 422.

**Course Content**

Theory of solitons is a relatively new and fast growing branch of mathematical
physics. Its development leads to progress in such areas of pure mathematics as
spectral theory of differential operators, complex algebraic geometry, and
classical theory of integrable systems. Solitons, the
nonlinear localized objects, play very important role in different areas of
physics: nonlinear optics, hydrodynamics, plasma theory, superfluidity,
and magnetism. Also, they are important for the theory of general relativity:
the black holes are solitons. It is remarkable that this broad variety of
physical phenomena, from microscopic to astronomic scale, can be described by
unified mathematical apparatus. The mathematical theory of solitons employs a
combination of several powerful methods: the method of inverse scattering
transform, the dressing method, the direct Hirota
method, the method of N-gap integration.

In this course we will discuss the basic elements of both mathematics and
physics of solitons, will make an accent on pure elementary algebraic methods
for construction of solitonic solutions, and will
develop the method of inverse scattering transform for the Schrodinger and
Dirac operators. The course will be organized as follows:

1. Basic integrable models in the nonlinear wave
dynamics and their interconnection. Lax representations. Gauge equivalence.
Simple solitonic solutions of basic equations: KdV, NLSE, N-wave, KP-1, KP-2, sine-Gordon equations (4
weeks)

2. Elementary methods for construction of multisolitonic equations (3 weeks)

3. Method of inverse scattering transform for the KdV equation. Riemann-Hilbert problem appears (3 weeks)

4. Method of inverse scattering transform for the Nonlinear Schrodinger equation (2 weeks)

5. Solitons in optical fibers. Solitons over unstable condensate (2 weeks)

6. Solitons in 2+1 dimensions (2 weeks)

7. Solitons on vortex line and in magnetics (1 week)

**Two-soliton
solution **https://www.youtube.com/watch?v=xwFMDod2Hms

V.E.Zakharov/Department of
Mathematics/Program in Applied Mathematics/University of Arizona/Tucson, AZ
85721