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 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

Definition of ‘Soliton’ 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 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

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

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

“…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.

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 1 http: //www-gap. dcs. st-and. ac. uk/~history/Mathematicians/Russell_Scott. html

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

Korteweg-de Vries (Kd. V) Equation Rescaling: Kd. V Equation: Nonlinear Term Dispersion Term (Steepen) (Flatten)

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

Solitary Wave Solutions 1. Assume traveling wave of the form: 2. Kd. V 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 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:

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

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

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

2. Linearize Kd. V

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

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 Kd. V equation is isospectral to obtain spectrum for all t - Lax pair {L, A}:

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

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

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

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

Decay of 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: - Temporal solitons (optical pulses) Lasers: - Spatial solitons (coherent beams of light) - BEC solitons (coherent beams of atoms)

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

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

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

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 http: //cua. mit. edu/ketterle_group/

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

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

Many Faces of Solitons Quantum Field Theory - Quantum solitons - Monopoles - Instantons General Relativity - Bartnik-Mc. Kinnon 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

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 Kd. V 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. html Alkali Gases @ Mit Home page: http: //cua. mit. edu/ketterle_group/ www. rowan. edu/math/nguyen/soliton/