Spring Semester 2019



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 pass/fail to regular grade or vice versa.
March 3, 2019 (all students) - Last day to make registration changes without the Dean's signature.
March 4, 2019 (all students) - All Change of Schedule forms to ADD or CHANGE classes require not only the instructor's signature indicating permission, but also the Dean's signature. By policy, permission from the Dean to make a registration change at this time requires an extraordinary reason.
March 26, 2019 (all students) - Last day for students to withdraw from a class online through UAccess. Last day for students to change to/from audit with only instructor approval. Last day for instructor to administratively drop students.

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


Legendre Polynomials


Bessel Functions


Homework 1

Due January 31


Homework 2

Due February 28


Homework 3

Due April 2


Practice test for Exam 1

Date of Exam: March 14


Homework 4

Due April 16


Homework 5

Due May 7


Practice test for Exam 2

Date of Exam: April 30


Practice Test for Final Exam






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) 



Lecture 1


Lecture 2


Lecture 3


Lecture 4


Lecture 5


Lecture 6


Dressing Method


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


Sato Theory


Lecture 7


Lecture 8


Lecture 9


Lecture 10


Lecture 11


Lecture 12



Lecture 14 (1)


Lecture 14 (2)


Lecture 15


Lecture 16


Lecture 17











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