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

3. 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… 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 From the Academy Mathematical frontiers in optical solitons Proceedings NAS, November 6, 2001

4. One entry found for soliton. Main Entry: sol·i·ton Pronunciation: 'sä-l&-"tän Function: noun Etymology: solitary + 2 -on Date: 1965 : a solitary wave (as in a gaseous plasma) that propagatessolitary 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 Definition of ‘Soliton’

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

6. “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 Great Wave of Translation

8. “…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.

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

10. Controversy Over Russell’s Work 1 George Airy: 1 http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Russell_Scott.html - 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

11. 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

12. Nonlinear TermDispersion Term Korteweg-de Vries (KdV) Equation Rescaling: KdV Equation: (Steepen)(Flatten)

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

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

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

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

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

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

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

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

21. 2. Linearize KdV

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

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

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

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

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

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

28. 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

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

30. Optical Phenomena Hieu Nguyen: Temporal solitons involve weak nonlinearity whereas spatial solitons involve strong nonlinearity Hieu Nguyen: Temporal solitons involve weak nonlinearity whereas spatial solitons involve strong nonlinearity RefractionDiffraction Coherent Light

31. NLS Equation Envelope Oscillation One-solitons: Nonlinear termDispersion/diffraction term

32. Temporal Solitons (1980) Chromatic dispersion: BeforeAfter - Pulse broadening effect Self-phase modulation - Pulse narrowing effect BeforeAfter

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

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

35. Atom Lasers Gross-Pitaevskii equation: Atom-atom interactionExternal potential - Quantum field theory Atom beam:

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

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

38. 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

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

40. 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/http://www.ma.hw.ac.uk/solitons/ Light Bullet Home Page: http://people.deas.harvard.edu/~jones/solitons/solitons.htmlhttp://people.deas.harvard.edu/~jones/solitons/solitons.html Alkali Gases @ Mit Home page: http://cua.mit.edu/ketterle_group/http://cua.mit.edu/ketterle_group/ References www.rowan.edu/math/nguyen/soliton/