SOLITONS From Canal Water Waves to Molecular Lasers Hieu D. Nguyen Rowan University IEEE Night

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

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 Definition of ‘Soliton’

John Scott Russell ( ) Union Canal at Hermiston, Scotland Solitary Waves - Scottish engineer at Edinburgh - Committee on Waves: BAAC

“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

“…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 , Plates XLVII-LVII.

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

Controversy Over Russell’s Work 1 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

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

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

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

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

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

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

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:

Inverse Scattering “Nonlinear” Fourier Transform: Space-time domain Frequency domain Fourier Series:

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

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

2. Linearize KdV

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

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}:

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

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

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

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

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

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

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

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

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

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

BEC (1995) Cold atoms - Coherent matter waves - Dilute alkali gases

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

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

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

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

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

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 R. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Review 18 (1976), No. 3, 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, 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, M. Wadati and M. Toda, The exact N-soliton solution of the Korteweg-de Vries equation, J. Phys. Soc. Japan 32 (1972), no. 5, Solitons Home Page: Light Bullet Home Page: Alkali Mit Home page: References