Course details
Semester
 Spring 2022
 Monday, January 31st to Friday, March 25th; Monday, April 25th to Friday, May 6th
Hours
 Live lecture hours
 20
 Recorded lecture hours
 0
 Total advised study hours
 80
Timetable
 Wednesdays
 12:05  12:55 (UK)
 Thursdays
 13:05  13:55 (UK)
Description
Prerequisites
Syllabus
 Introduction and general overview (2 hours)
 Wave motion, linear and nonlinear dispersive waves, nondispersive waves, shocks.
 Canonical linear and nonlinear wave equations, integrability and inverse scattering transform (IST), asymptotic and perturbation methods.
 Dispersive wave models: derivation techniques and basic properties (4 hours)
 FermiPastaUlam (FPU) problem, ZabuskyKruskal model and Boussinesq equation, derivation of the Korteweg  de Vries (KdV) equation, travelling waves, phaseplane analysis, solitons and cnoidal waves.
 FrenkelKontorova model, sineGordon equation, travelling waves, phaseplane 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 (threewave and fourwave interactions, second harmonic generation, longshort wave resonance). Phaseplane analysis for travelling waves (threewave interactions).
 Inverse scattering transform (IST) and applications (4 hours)
 KdV equation: Lax pair, discrete and continuous spectrum of the timeindependent Schrödinger operator, direct and inverse scattering problems, initialvalue problem by the inverse scattering transform (scheme). Reflectionless potentials and Nsoliton solutions. Example: deltafunction initial condition. Infinity of conservation laws. Hamiltonian structures. KdV hierarchy.
 AKNS scheme, linear problem, inverse scattering transform (scheme) for the focusing NLS equation, Nsoliton solutions.
 Nearintegrable equations: perturbed and higherorder KdV equations (waves in variable environment), asymptotic integrability, Gardner equation.
 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. RankineHugoniot conditions. Lax entropy condition.
 Structure of the viscous shock wave, Burgers equation, ColeHopf transformation, Taylor's shock profile, Nwave.
 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. GurevichPitaevskii problem.
 Dispersive shock waves in the defocusing Nonlinear Schroedinger equation. Application to BoseEinstein condensates
London.
Lecturers

Professor Gennady El
 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 9th May 2022 at 00:00 and is due in before Monday 23rd May 2022 at 11:00.
Assessment for all MAGIC courses is via takehome 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 uptodate 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.
Files
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Lectures
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