Assignment schedule:
THESE WEBPAGES DESCRIBE HOW THIS COURSE WILL BE RUN.
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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]
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Lecture#
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Topics: | Notes: | Homework: | Due: |
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01
Mon/06/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 |
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Tu/07/Jul
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02
Tu/07/Jul |
Dispersion relation + Play with integrating code + Coordinate transformations + Adimensionalization + Classification of linear 2nd order PDEs |
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Fr/10/Jul
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03
We/08/Jul |
NONLINEAR WAVES
Method of characteristics + Quasilinear PDEs + Wave breaking Korteweg-de Vries (KdV) equation Water waves + Euler Eqs. + Boussinesq → KdV + KdV : scale invariance + KdV : galilean invariance + KdV soliton |
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Mo/13/Jul
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04
Th/09/Jul |
Elementary sols to KdV + Cnoidal waves in KdV + Similarity sols. in KdV + Rational sols. in KdV Exact 2-soliton sol of KdV + Constants of motion of KdV |
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We/15/Jul
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05
Fr/10/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 |
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06
Mo/13/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 |
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07
Tu/14/Jul |
Galilean boost for NLS + More general dark solitons + Modulational Instability for NLS |
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Fr/17/Jul
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08
We/15/Jul |
Modulational Instability for NLS + 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 |
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09
Th/16/Jul |
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 + |
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10
Fr/17/Jul |
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. |
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We/22/Jul
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11
Mo/20/Jul |
Perturbation theory for dark solitons: See: Kivshar PRE 49 (1994) 1657. Extensive review on soliton perturbation theory: See: Kivshar+Malomed, Rev. Mod. Phys. 61 (1989) 763. 3D -> 2D dynamical reduction of GPE. Ring dark soliton dynamics: See: Theocharis et al., Phys. Rev. Lett. 90 (2003) 120403. | |||
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12
Tu/21/Jul |
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 + Steady states for PDEs: Newton's method + Numerical stability for steady states + Examples: bright/dark in 1D + Examples: vortices in 2D |
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13
We/22/Jul |
Nonlinear lattices: FPU + Toda Lattice + Intrinsic localized modes (ILMs), breathers + Map approach: See: R. Carretero pub#59. |
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