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SOLITONS From Canal Water Waves to Molecular Lasers

SOLITONS From Canal Water Waves to Molecular Lasers. Hieu D. Nguyen Rowan University. IEEE Night 5-20-03. from SIAM News , Volume 31, Number 2, 1998 Making Waves: Solitons and Their Practical Applications. "A Bright Idea“ Economist (11/27/99) Vol. 353, No. 8147, P. 84

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SOLITONS From Canal Water Waves to Molecular Lasers

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  1. SOLITONS From Canal Water Waves to Molecular Lasers Hieu D. Nguyen Rowan University IEEE Night 5-20-03

  2. from SIAM News, Volume 31, Number 2, 1998 Making Waves: Solitons and Their Practical Applications "A Bright Idea“Economist (11/27/99) Vol. 353, No. 8147, P. 84 Solitons, waves that move at a constant shape and speed, can be used for fiber-optic-based data transmissions… From the Academy Mathematical frontiers in optical solitons Proceedings NAS, November 6, 2001 Number 588, May 9, 2002 Bright Solitons in a Bose-Einstein Condensate Solitons may be the wave of the future Scientists in two labs coax very cold atoms to move in trains 05/20/2002 The Dallas Morning News

  3. Definition of ‘Soliton’ One entry found for soliton. Main Entry: sol·i·tonPronunciation: 'sä-l&-"tänFunction: nounEtymology: solitary + 2-onDate: 1965: a solitary wave (as in a gaseous plasma) that propagates with little loss of energy and retains its shape and speed after colliding with another such wave http://www.m-w.com/cgi-bin/dictionary

  4. Solitary Waves John Scott Russell (1808-1882) • Scottish engineer at Edinburgh • Committee on Waves: BAAC Union Canal at Hermiston, Scotland http://www.ma.hw.ac.uk/~chris/scott_russell.html

  5. Great Wave of Translation “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind,rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed…” - J. Scott Russell

  6. “…I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.” “Report on Waves” - Report of the fourteenth meeting of the British Association for the Advancement of Science, York, September 1844 (London 1845), pp 311-390, Plates XLVII-LVII.

  7. Copperplate etching by J. Scott Russell depicting the 30-foot tank he built in his back garden in 1834

  8. Controversy Over Russell’s Work1 George Airy: • Unconvinced of the Great Wave of Translation • Consequence of linear wave theory G. G. Stokes: - Doubted that the solitary wave could propagate without change in form Boussinesq (1871) and Rayleigh (1876); - Gave a correct nonlinear approximation theory 1http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Russell_Scott.html

  9. Model of Long Shallow Water Waves D.J. Korteweg and G. de Vries (1895) • surface elevation above equilibrium • depth of water • surface tension • density of water • force due to gravity • small arbitrary constant

  10. Korteweg-de Vries (KdV) Equation Rescaling: KdV Equation: Nonlinear Term Dispersion Term (Steepen) (Flatten)

  11. Stable Solutions Profile of solution curve: • Unchanging in shape • Bounded • Localized Do such solutions exist? Steepen + Flatten = Stable

  12. Solitary Wave Solutions 1. Assume traveling wave of the form: 2. KdV reduces to an integrable equation: 3. Cnoidal waves (periodic):

  13. 4. Solitary waves (one-solitons): - Assume wavelength approaches infinity

  14. Other Soliton Equations Sine-Gordon Equation: • Superconductors (Josephson tunneling effect) • Relativistic field theories Nonlinear Schroedinger (NLS) Equation: • Fiber optic transmission systems • Lasers

  15. N-Solitons Zabusky and Kruskal (1965): • Partitions of energy modes in crystal lattices • Solitary waves pass through each other • Coined the term ‘soliton’ (particle-like behavior) Two-soliton collision:

  16. Inverse Scattering “Nonlinear” Fourier Transform: Space-time domain Frequency domain Fourier Series: http://mathworld.wolfram.com/FourierSeriesSquareWave.html

  17. Solving Linear PDEs by Fourier Series 1. Heat equation: 2. Separate variables: 3. Determine modes: 4. Solution:

  18. Solving Nonlinear PDEs by Inverse Scattering 1. KdV equation: 2. Linearize KdV: 3. Determine spectrum: (discrete) 4. Solution by inverse scattering:

  19. 2. Linearize KdV

  20. Schroedinger’s Equation (time-independent) Potential (t=0) Eigenvalue (mode) Eigenfunction Scattering Problem: Inverse Scattering Problem:

  21. 3. Determine Spectrum (a) Solve the scattering problem at t = 0 to obtain reflection-less spectrum: (eigenvalues) (eigenfunctions) (normalizing constants) (b) Use the fact that the KdV equation is isospectral to obtain spectrum for all t - Lax pair {L, A}:

  22. 4. Solution by Inverse Scattering (a) Solve GLM integral equation (1955): (b) N-Solitons ([GGKM], [WT], 1970):

  23. Soliton matrix: One-soliton (N=1): Two-solitons (N=2):

  24. Unique Properties of Solitons Signature phase-shift due to collision Infinitely many conservation laws (conservation of mass)

  25. Other Methods of Solution Hirota bilinear method Backlund transformations Wronskian technique Zakharov-Shabat dressing method

  26. Decay of Solitons Solitons as particles: - Do solitons pass through or bounce off each other? Linear collision: Nonlinear collision: • Each particle decays upon collision • Exchange of particle identities • Creation of ghost particle pair

  27. Applications of Solitons Optical Communications: - Temporal solitons (optical pulses) Lasers: • Spatial solitons (coherent beams of light) • BEC solitons (coherent beams of atoms)

  28. Hieu Nguyen: Temporal solitons involve weak nonlinearity whereas spatial solitons involve strong nonlinearity Optical Phenomena Refraction Diffraction Coherent Light

  29. NLS Equation Dispersion/diffraction term Nonlinear term One-solitons: Envelope Oscillation

  30. Temporal Solitons (1980) Chromatic dispersion: - Pulse broadening effect Before After Self-phase modulation - Pulse narrowing effect Before After

  31. Spatial Solitons Diffraction - Beam broadening effect: Self-focusing intensive refraction (Kerr effect) - Beam narrowing effect

  32. BEC (1995) Cold atoms • Coherent matter waves • Dilute alkali gases http://cua.mit.edu/ketterle_group/

  33. Atom Lasers Atom beam: Gross-Pitaevskii equation: - Quantum field theory Atom-atom interaction External potential

  34. Molecular Lasers Cold molecules - Bound states between two atoms (Feshbach resonance) Molecular laser equations: (atoms) (molecules) Joint work with Hong Y. Ling (Rowan University)

  35. Many Faces of Solitons Quantum Field Theory • Quantum solitons • Monopoles • Instantons General Relativity • Bartnik-McKinnon solitons (black holes) Biochemistry - Davydov solitons (protein energy transport)

  36. Future of Solitons "Anywhere you find waves you find solitons." -Randall Hulet, Rice University, on creating solitons in Bose-Einstein condensates, Dallas Morning News, May 20, 2002

  37. Recreation of the Wave of Translation (1995) Scott Russell Aqueduct on the Union Canal near Heriot-Watt University, 12 July 1995

  38. References C. Gardner, J. Greene, M. Kruskal, R. Miura, Korteweg-de Vries equation and generalizations. VI. Methods for exact solution, Comm. Pure and Appl. Math. 27 (1974), pp. 97-133 R. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Review 18 (1976), No. 3, 412-459. A. Snyder and F.Ladouceur, Light Guiding Light, Optics and Photonics News, February, 1999, p. 35 P. D. Drummond, K. V. Kheruntsyan and H. He, Coherent Molecular Solitons in Bose-Einstein Condensates, Physical Review Letters 81 (1998), No. 15, 3055-3058 B. Seaman and H. Y. Ling, Feshbach Resonance and Coherent Molecular Beam Generation in a Matter Waveguide, preprint (2003). H. D. Nguyen, Decay of KdV Solitons, SIAM J. Applied Math. 63 (2003), No. 3, 874-888. M. Wadati and M. Toda, The exact N-soliton solution of the Korteweg-de Vries equation, J. Phys. Soc. Japan 32 (1972), no. 5, 1403-1411. Solitons Home Page: http://www.ma.hw.ac.uk/solitons/ Light Bullet Home Page: http://people.deas.harvard.edu/~jones/solitons/solitons.html Alkali Gases @ Mit Home page: http://cua.mit.edu/ketterle_group/ www.rowan.edu/math/nguyen/soliton/

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