# MAGIC021: Nonlinear Waves

## Course details

A core MAGIC course

### Semester

Spring 2023
Monday, January 23rd to Friday, March 31st

### Hours

Live lecture hours
20
Recorded lecture hours
0
80

### Timetable

Wednesdays
11:05 - 11:55 (UK)
Wednesdays
13:05 - 13:55 (UK)

## Description

The aim of this module is to introduce students to the major ideas and techniques in the nonlinear wave theory (see the syllabus).

### Prerequisites

No specific requirements. Standard undergraduate courses in analysis, mathematical methods and partial differential equations are desirable.

### Syllabus

The aim of this module is to introduce major ideas and techniques of modern nonlinear wave theory in simple settings, with an emphasis on asymptotic methods for nonlinear dispersive PDEs and applied aspects of integrability and inverse scattering transform.

1. Introduction and general overview (2 hours)
• Wave motion, linear and nonlinear dispersive waves, non-dispersive waves, shocks.
•  Canonical linear and nonlinear wave equations, integrability and inverse scattering transform (IST), asymptotic and perturbation methods.
2. Dispersive wave models: derivation techniques and basic properties (4 hours)
• Fermi-Pasta-Ulam (FPU) problem, Zabusky-Kruskal model and Boussinesq equation, derivation of the Korteweg - de Vries (KdV) equation, travelling waves, phase-plane analysis, solitons and cnoidal waves.
•  Frenkel-Kontorova model, sine-Gordon equation, travelling waves, phase-plane analysis, Bäcklund transformations, kinks and breathers.
•  Nonlinear Schrödinger (NLS) equation, derivation, focusing and defocusing, criterion of modulational instability, bright and dark solitons, breathers.
•  Resonant wave interactions (three-wave and four-wave interactions, second harmonic generation, long-short wave resonance). Phase-plane analysis for travelling waves (three-wave interactions).
3. Inverse scattering transform (IST) and applications (4 hours)
• KdV equation: Lax pair, discrete and continuous spectrum of the time-independent Schrödinger operator, direct and inverse scattering problems, initial-value problem by the inverse scattering transform (scheme). Reflectionless potentials and N-soliton solutions. Example: delta-function initial condition. Infinity of conservation laws. Hamiltonian structures. KdV hierarchy.
•  AKNS scheme, linear problem, inverse scattering transform (scheme) for the focusing NLS equation, N-soliton solutions.
•  Near-integrable equations: perturbed and higher-order KdV equations (waves in variable environment), asymptotic integrability, Gardner equation.
4. Nonlinear hyperbolic waves and classical shocks (5 hours)
• Kinematic waves, solution via characteristics, hodograph transformation, Riemann invariants, gradient catastrophe.
•  Hyperbolic conservation laws, weak solutions and shock waves. Rankine-Hugoniot conditions. Lax entropy condition.
•  Structure of the viscous shock wave, Burgers equation, Cole-Hopf transformation, Taylor's shock profile, N-wave.
5. Dispersive hydrodynamics and modulation theory (5 hours)
•  Dispersive hydrodynamics: an overview.
•  Whitham's method of slow modulations (linear modulated waves, nonlinear WKB, averaging of conservation laws, Lagrangian formalism).
• Generalised hodograph transform and integrability of the Whitham equations. Connection with the inverse scattering transform.
•  Formation of a dispersive shock wave. Resolution of an initial discontinuity for the KdV equation. Gurevich-Pitaevskii problem.
• Dispersive shock waves in the defocusing Nonlinear Schroedinger equation. Application to Bose-Einstein condensates

Main references:

[1] Whitham, G.B. 1974
Linear and Nonlinear Waves, Wiley, New York.

[2] Ablowitz, M.J. & Segur, H. 1981
Solitons and the Inverse Scattering Transform, SIAM.

[3] Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. & Morris, H.C. 1982
Solitons and Nonlinear Waves Equations, Academic Press, Inc.

[4] Novikov, S.P., Manakov, S.V., Pitaevskii, L.P. & Zakharov, V.E. 1984
The Theory of Solitons: The Inverse Scattering Method, Consultants, New York.

[5] Newell, A.C. 1985
Solitons in Mathematics and Physics, SIAM.

[6] Drazin, P.G. & Johnson R.S. 1989
Solitons: an Introduction, Cambridge University Press,
London.

[7] Scott, A. 1999
Nonlinear Science: Emergence and Dynamics of Coherent Structures, Oxford University Press Inc., New York.

[8] Kamchatnov, A.M. 2000
Nonlinear Periodic Waves and Their Modulations-An Introductory Course, World Scientific, Singapore.

[9] Braun, O.M., Kivshar, Y.S. 2004
The Frenkel-Kontorova model. Concepts, methods, and applications. Springer, Berlin.

[10] Grimshaw, R. (ed.). 2005
Nonlinear Waves in Fluids: Recent Advances and Modern Applications. CISM Courses and Lectures, No. 483, Springer, Wien, New York.

[11] Grimshaw, R. (ed.) 2007
Solitary Waves in Fluids. Advances in Fluid Mechanics, Vol 47, WIT Press, UK.

[12] Ablowitz, M.J. 2011
Nonlinear Dispersive Waves. Cambridge University Press, UK.

## Lecturers

University
Northumbria University
Role
Main contact
• ### Dr Karima Khusnutdinova

University
Loughborough University

## Bibliography

No bibliography has been specified for this course.

## Assessment

The assessment for this course will be released on Monday 1st May 2023 at 00:00 and is due in before Monday 15th May 2023 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.