Assignment schedule:
THESE WEBPAGES DESCRIBE HOW THIS COURSE WILL BE RUN.
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Homework: Typically there will be one homework per week. The HW will be posted here every Thursday (at the latest) and is due the following Thursday (Wednesday night) and must be submitted in Gradescope (GS) through Canvas. In some occasions the HW will be posted a few days before Thursday so you are encouraged to start on it earlier. Late HW will not be accepted.
Take-home exams: There will be two take home exams during the semester. They will replace the homework for that week. Take-home exams will be posted and due the same way the homework is. Late take-home exams will not be accepted.
Assignments: [bottom]
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Week #
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Topics: | Sec.: | Exercises: | Due: |
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01
23-25/Aug |
Linear waves Introduction + Waves on a string + Linear wave equation + Superposition principle + D'Alambert solution + Initial conditions + Dissipation + Dispersion + Plane wave solutions + Dispersion relation |
HW#01:
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Th 01 Sep
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02
30/Aug - 01/Sep |
Coordinate transformations + Adimensionalization + Classification of linear 2nd order PDEs + Method of characteristics + Quasilinear PDEs + Wave breaking + Water waves + Euler Eqs. |
HW#02:
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Th 08 Sep
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03
06-08/Sep |
KdV equation Boussinesq -> KdV + KdV : scale invariance + KdV : galilean invariance KdV soliton + Elementary sols to KdV + Cnoidal waves in KdV + |
HW#03:
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Th 15 Sep
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04
13-15/sep |
Similarity sols. in KdV + Rational sols. in KdV Exact 2-soliton sol of KdV + Constant of motion of KdV Center of mass for KdV + Two soliton collisision in KdV |
HW#04:
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Th 22 Sep
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05
20-22/Sep |
More conservation laws of KdV + Backlund transform for KdV |
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Th 29 Sep
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06
27-29/Sep |
NLS equation NLS from envelope wave eq. + Solitons in NLS + Focusing ⇒ bright solitons + Defocusing ⇒ dark solitons |
HW#05:
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Th 06 Oct
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07
04-06/Oct |
General dark solitons + Galilean boost + Conservartion laws for NLS + Modulational Instability for NLS |
HW#06:
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Th 13 Oct
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08
11-13/Oct |
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. Variational principles |
HW#07:
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Th 20 Oct
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09
18-20/Oct |
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 |
HW#08:
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Tu 01 Nov
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10
25-27/Oct |
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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11
01-03/Nov |
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. Numerical techniques for NLS * Steady states: Newton method * Stability: BdG equations * Integration: Finite differences and RK4 | |||
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12
08-10/Nov |
3D -> 2D dynamical reduction of GPE. Thomas-Fermi approximation Ring dark soliton dynamics: See: Theocharis et al., Phys. Rev. Lett. 90 (2003) 120403. Transverse stability for dark solitons: Kivshar+Luther-Davis, Phys. Repts. 298 (1998) 81. Adiabatic invariant approach : DS |
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13
15-17/Nov |
Adiabatic invariant approach : DS stripes
NLS vortices Vortex drift in inhomogeneous backgrounds: Kivshar+ et al., Opt. Comm. 152 (1998) 198. Vortex-vortex interactions + Numerical stability for steady states + Examples: bright/dark in 1D + Examples: vortices in 2D |
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14
22-24/Nov |
Thanksgiving break |
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15
29/Nov-01/Dec |
Effective ODEs for vortices in BECs + Talk on small clusters of vortices in BECs + Nonlinear lattices FPU + Toda Lattice |
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