Assignment schedule:

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Homework: Typically there will be TWO homeworks per week. The HW will be posted here and must be handed in on the due date as soon as you enter the classroom before the lecture starts. In some occasions the HW will be posted a few days earlier so you are encouraged to start on it EARLY! Late HW will not be accepted.

 

Assignments: [bottom]

Lecture#
 Topics: Notes: Homework: Due:
01
 Mon/24/Jul
LINEAR WAVES
Introduction +
Waves on a string +
Linear wave equation +
Superposition principle +
D'Alambert solution +
Initial conditions +
Dissipation +
Dispersion +
Plane wave solutions +
Dispersion relation +
Play with integrating code +
Coordinate transformations +
Adimensionalization +
Classification of linear 2nd order PDEs

NONLINEAR WAVES
Method of characteristics +
Quasilinear PDEs +
Wave breaking
My_notes#1 Debnath Chap#1
  • Go over Matlab [PDF] tutorial [For students without previous knowledge on Matlab (if you are an "expert" in Matlab just send me an email indicating so)]:
    Type in all instructions in Matlab (follow the whole tutorial) and submit to me by email a diary of it (to do the diary: do "diary LastName_Waves.txt" as the first instruction and this will save all your inputs in that file)

  • HW#1: [pdf]
We/26/Jul
02
 Tu/25/Jul
Korteweg-de Vries (KdV) equation
Water waves +
Euler Eqs. +
Boussinesq → KdV +
KdV : scale invariance +
KdV : galilean invariance +
KdV soliton +
Elementary sols to KdV +
Cnoidal waves in KdV +
Similarity sols. in KdV +
Rational sols. in KdV
Exact 2-soliton sol of KdV +
My_notes#2
Shen_Chapter#3
   
03
 We/26/Jul
Constants of motion of KdV +
Center of mass for KdV +
Two soliton collisision in KdV +
More conservation laws of KdV +
Bäcklund transform for KdV
My_notes#2 KdV_Backlund_Summary
Mo/31/Jul
04
 Th/27/Jul
Nonlinear Schrödinger (NLS) equation
Envelope waves (group and carrier velocities) +
NLS from envelope wave eq. +
Solitons in NLS +
Focusing → bright solitons +
Defocusing → dark solitons
My_notes#3
05
 Fr/28/Jul
Galilean boost for NLS +
More general dark solitons +
Modulational Instability for NLS
My_notes#3
We/02/Aug
06
 Mo/31/Jul
Intro to BECs:
* Repulsive/Attractive, +
* Low vs high atom numbers, +
* Thomas-Fermi approx, +
* ground state, chemical potential +
Avoiding modulation instability:
See: Kevrekidis et al. 70 (2004) 023602.
Conservartion laws for NLS
My_notes#3
   
07
 Tu/01/Aug
Variational principles +
Variational Approximation +
See: Anderson PRA 27 (1983) 3135
[Notes on Anderson's paper (courtesy of Julia Rossi)]
and: Malomed Prog. Opt. 43 (2002) 71
Perturbed Variational Approximation:
Gain/Loss in NLS +
My_notes#4
Fr/04/Aug
08
 We/02/Aug
Soliton-Soliton interactions:
See: Gerdjikov et al., PRE 55 (1997) 6039.
See: Karpman+Solovev, Physica D 3 (1981) 487.
and Carretero+Promislow PRA 66 (2002) 033610.

Extensive review on dark solitons and its applications, see:
Kivshar+Luther-Davis, Phys. Repts. 298 (1998) 81.
Frantzeskakis, J. Phys. A 43 (2010) 2130011.

Perturbation theory for dark solitons:
See: Kivshar PRE 49 (1994) 1657.

3D -> 2D dynamical reduction of GPE.
My_notes#4
GPE_3D_to_2D
   
09
 Th/03/Aug
Transverse stability for dark solitons:
Kivshar+Luther-Davis, Phys. Repts. 298 (1998) 81.

NLS vortices.
Vortex drift in inhomogeneous backgrounds:
Kivshar+ et al., Opt. Comm. 152 (1998) 198.
Vortex-vortex interactions +
Effective ODEs for vortices in BECs +
Small vortex clusters in BEC (talk)
My_notes#4
 
 
10
 Fr/04/Aug
Steady states for PDEs:
* Newton's method +
* Numerical stability for steady states +
Examples: bright/dark in 1D +
Examples: vortices in 2D
My_notes#4
Fr/25/Aug