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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 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 Model of Long Shallow Water Waves D.J. Korteweg and G. de Vries (1895) 3 g 1 2 2 1 2 2 t 2 l x 2 3 3 x - surface elevation above equilibrium l - depth of water T - surface tension - density of water g - force due to gravity - small arbitrary constant 1 3 Tl l 3 g Korteweg-de Vries (KdV) Equation Rescaling: 3 g x 2 t t, x , 2 u 2 l 3 u KdV Equation: ut ut 6uu x u xxx 0 t u ux Nonlinear Term Dispersion Term x ut 6uux 0 ut uxxx 0 (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: u ( x, t ) U ( z ), z x ct 2. KdV reduces to an integrable equation: dU dU d 3U c 6U 3 0 dz dz dz 3. Cnoidal waves (periodic): U ( z ) a cn 2 bz , k 4. Solitary waves (one-solitons): - Assume wavelength approaches infinity c c U ( z ) sech 2 z 2 2 u ( x, t ) 2k 2sech 2 k ( x 4k 2t ) ) , c 4k 2 -u x Other Soliton Equations Sine-Gordon Equation: uxx utt sin u - Superconductors (Josephson tunneling effect) - Relativistic field theories Nonlinear Schroedinger (NLS) Equation: iut u u u xx 0 2 - 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: n x n x f ( x) a0 an cos bn sin n 1 L L 4 1 1 f ( x) sin x sin 3 x sin 5 x ... 3 5 http://mathworld.wolfram.com/FourierSeriesSquareWave.html Solving Linear PDEs by Fourier Series u (0, t ) u ( L, t ) 0 1. Heat equation: ut c u xx , 2 u ( x, 0) f ( x) 2. Separate variables: xx k vt ckv 3. Determine modes: cn 2 n t n ( x) sin x, v(t ) e L , n 1, 2,3,... L cn 2 n x t 4. Solution: u ( x, t ) an sin e L n 1 L 2 L n x an f ( x) sin dx L 0 L Solving Nonlinear PDEs by Inverse Scattering 1. KdV equation: ut 6uux uxxx 0, u( x,0) is reflectionless 2. Linearize KdV: xx u( x, t ) 0 3. Determine spectrum: {n , n } (discrete) 4. Solution by inverse scattering: N u ( x, t ) 4 knn2 ( x, t ), kn n n 1 2. Linearize KdV KdV: ut 6uu x u xxx 0 Miura transformation: u v 2 vx mKdV: vt 6v 2 vx vxxx 0 (Burger type) x Cole-Hopf transformation: v Schroedinger's equation: xx u ( x, t ) 0 (linear) Schroedinger’s Equation (time-independent) xx [u( x,0) ] 0 Potential Eigenvalue Eigenfunction (t=0) (mode) Scattering Problem: - Given a potential u , determine the spectrum { , }. Inverse Scattering Problem: - Given a spectrum { , }, determine the potential u. 3. Determine Spectrum (a) Solve the scattering problem at t = 0 to obtain reflection-less spectrum: {0 1 2 ... N } (eigenvalues) {1 , 2 ,..., N } (eigenfunctions) {c1 , c2 ,..., cN } (normalizing constants) (b) Use the fact that the KdV equation is isospectral to obtain spectrum for all t L [ L, A] t - Lax pair {L, A}: 0 t A t 4. Solution by Inverse Scattering (a) Solve GLM integral equation (1955): B( x, t ) c e2 8 kn t k n x 3 n K ( x, y, t ) B( x y, t ) B( x z, t ) K ( z , y, t )dz 0 x xx (u ) 0 u ( x, t ) 2 K ( x, x, t ) x (b) N-Solitons ([GGKM], [WT], 1970): 2 u ( x, t ) 2 2 log det( I A) x Soliton matrix: cm cn km m kn n A e , n x 4kn2t (moving frame) km kn One-soliton (N=1): 2 c12 2 k11 u ( x, t ) 2 2 log 1 e x 2k1 2k12sech 2 k1 1 Two-solitons (N=2): 2 2 c12 2 k11 c2 2 k2 2 u ( x, t ) 2 2 log 1 e e x 2k1 2k 2 k1 k2 c12c2 2 k11 2 k2 2 2 2 e k1 k2 4k1k2 Unique Properties of Solitons Signature phase-shift due to collision Infinitely many conservation laws u ( x, t )dx 4 kn (conservation of mass) n 1 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: - Temporal solitons (optical pulses) Lasers: - Spatial solitons (coherent beams of light) - BEC solitons (coherent beams of atoms) Hieu Nguyen: Temporal solitons involve weak nonlinearity Optical Phenomena whereas spatial solitons involve strong nonlinearity Refraction Diffraction Coherent Light NLS Equation it xx 0 2 Dispersion/diffraction term Nonlinear term One-solitons: i [ ( x t ) / 2 ( 2 2 / 4) t ] ( x, t ) 2 sech[ ( x t )]e 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 1 it xx 2 2 Atom-atom interaction External potential Molecular Lasers Cold molecules - Bound states between two atoms (Feshbach resonance) Molecular laser equations: 1 it xx a am * 2 2 2 (atoms) 1 i t xx m am ( ) 2 (molecules) 4 2 2 2 Joint work with Hong Y. Ling (Rowan University) Many Faces of Solitons Quantum Field Theory - Quantum solitons - Monopoles - Instantons General Relativity - 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 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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