K. Murawski UMCS Lublin

1. Solitons around us K. Murawski UMCS Lublin

2. Outline • historical remarks - first observation of a soliton • definition of a soliton • classical evolutionary equations • IDs of solitons • solitons in solar coronal loops

3. Ubiquity of waves

4. First observation of Solitary Waves John Scott Russell (1808-1882) • Scottish engineer at Edinburgh Union Canal at Hermiston, Scotland

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

8. J. Scott Russell experimented in the 30-foot tank which he built in his back garden in 1834: Vph2 = g(h+h’)

9. ??? Oh no!!!

10. 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

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

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

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 (1-soliton): - Assume wavelength approaches infinity

16. M V(x) Fixed = Nonlinear Spring fixed Fermi-Pasta-Ulam problem Los Alamos, Summers 1953-4 Enrico Fermi, John Pasta, and Stan Ulam decided to use the world’s then most powerful computer, the MANIAC-1 (Mathematical Analyzer Numerical Integrator And Computer) to study the equipartition of energy expected from statistical mechanics in simplest classical model of a solid: a 1D chain of equal mass particles coupled by nonlinear* springs: *They knew linear springs could not produce equipartition V(x) = ½ kx2 + /3 x3 + /4 x4

17. What did FPU discover? Note only modes 1-5 • Only lowest few modes (from N=64) excited. • Recurrences

18. N-solitons Perring and Skyrme (1963) Zabusky and Kruskal (1965): • Derived KdV eq. for the FPU system • Solved numerically KdV eq. • Solitary waves pass through each other • Coined the term ‘soliton’ (particle-like behavior)

19. Solitons and solitary waves - definitions A solitary wave is a wave that retains its shape, despite dispersion and nonlinearities. A soliton is a pulse that can collide with another similar pulse and still retain its shape after the collision, again in the presence of both dispersion and nonlinearities.

20. Soliton collision: Vl = 3, Vs=1.5

21. Unique Properties of Solitons Signature phase-shift due to collision Infinitely many conservation laws, e.g. (conservation of mass)

22. mKdV solitons modified Korteweg-de Vries equation vt + vxxx + 6v2vx= 0

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

24. 2. Linearize KdV

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

26. 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}:

27. 4. Solution by Inverse Scattering (a) Solve Gelfand-Levitan-Marchenko integral equation (1955): (b) N-Solitons (1970):

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

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

30. Other Soliton Equations Sine-Gordon Equation: • Superconductors (Josephson tunneling effect) • Relativistic field theories Breather soliton Nonlinear Schroedinger (NLS) Equation: • optical fibers

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

32. Magnetic loops in solar corona (TRACE) Strong B dominates plasma

33. Thin flux tube approximation • The dynamics of long wavelength (λ»a) waves may be described by the thin flux tube equations (Roberts & Webb, 1979; Spruit & Roberts, 1983 ). V(z,t): longitudinal comp. of velocity

34. Model equations • Weakly nonlinear evolution of the waves is governed, in the cylindrical case, by the Leibovich-Roberts (LR) equation, viz.   • and, in the case of the slab geometry, by the Benjamin - Ono (BO) equation, viz • Roberts & Mangeney, 1982; Roberts, 1985

35. Algebraic soliton • The famous exact solution of the BO equation is the algebraic soliton, • Exact analytical solutions of the LR equation have not been found yet!!!

36. MHD (auto)solitons in magnetic structures • In presence of weak dissipation and active non-adiabaticity (e.g. when the plasma is weakly thermally unstable) equations LR and BO are modified to the extended LR or BO equations of the form   • B: nonlinear, A:non-adiabatic, δ:dissipative and D:dispersive coefficients. It has been shown that when all these mechanisms for the wave evolution balance each other, equation eLR has autowave and autosoliton solutions. • By definition, an autowave is a wave with the parameters (amplitude, wavelength and speed) independentof the initial excitation and prescribed by parameters of the medium only.

37. MHD (auto)solitons in magnetic structures • For example, BO solitons with different initial amplitudes evolve to an autosoliton. If the soliton amplitude is less than the autosoliton amplitude, it is amplified, if greater it decays: Ampflication dominates for larger  and dissipation for shorter . Solitons with a small amplitude have larger length and are smoother than high amplitude solitons, which are shorter and steeper. Therefore, small amplitude solitons are subject to amplification rather than dissipation, while high amplitude solitons are subject to dissipation. • The phenomenon of the autosoliton (and, in a more general case, autowaves) is an example of self-organization of MHD systems.

38. Solitons, Strait of Gibraltar These subsurface internal waves occur at depths of about 100 m. A top layer of warm, relatively fresh water from the Atlantic Ocean flows eastward into the Mediterranean Sea. In return, a lower, colder, saltier layer of water flows westward into the North Atlantic ocean. A density boundary separates the layers at about 100 m depth.

39. Andaman Sea Solitons Oceanic Solitons (Vance Brand Waves) are nonlinear, localized waves, that move in groups of six. They manifest as large internal waves, and move at a speed of 8 KPH. They were first recorded at depths of 120m by sensors on Oil Rigs in the Andaman Sea. Until that time Scientists denied their very existence…based on the fact that “There was no record of any such phenomenon.”

40. 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

41. 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. H. D. Nguyen, Decay of KdV Solitons, SIAM J. Applied Math. 63 (2003), No. 3, 874-888. A. Snyder and F.Ladouceur, Light Guiding Light, Optics and Photonics News, February, 1999, p. 35 B. Seaman and H. Y. Ling, Feshbach Resonance and Coherent Molecular Beam Generation in a Matter Waveguide, preprint (2003). 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