The Solitary Wave of Translation (Edinburgh, 1834)

2. Mathematics provides physical science with a language that connects cause and effect. The equations and functionsof mathematics provide a grammar and a vocabulary to understand and interpret the physical world. Mathematical language helps uncover hidden connections between seemingly disparate physical phenomena, and helps us to think about nature in new ways. In this lecture, we will illustrate the power of mathematical thinking by tracing the story of solitary waves, beginning with their discovery by John Scott Russell in 1834.

3. Making Waves with Mathematics The Solitary Wave of Translation (Edinburgh, 1834) The KdV Equation(Amsterdam, 1895) The Birth of Experimental Mathematics(Los Alamos, 1955) Solitary Waves and Quantum Waves(Princeton and New York, 1964-68) New Frontiers (1972-Present)

4. ÅkeHultkrantz (1920-2006), Swedish Anthropologist and scholar of the Saami, Shosone, and Arapaho peoples

5. The Solitary Wave of Translation John Scott Russell (1808-1882) This is a most beautiful and extraordinary phenomenon. The first day I saw it was the happiest day of my life. Nobody had ever had the good fortune to see it before or, at all events, to know what it meant. It is now known as the solitary wave of translation. John Scott Russell, 1865

6. Union Canal, Gyle, Edinburgh, Scotland “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.”

7. Plaque at Hermiston House, Union Canal “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.” John Scott Russell, Report on Waves, Fourteenth Meeting of the British Association for the Advancement of Science, September 1844

8. Russell re-created the solitary waves and discovered four facts about them: The shape of the wave is described by the hyperbolic secant function A larger solitary wave travels faster than a smaller one Solitary waves can cross each other “without change of any kind” (4) A large enough mass of water creates two or more independent solitary waves The Hyperbolic Secant Function

9. The Korteweg-De Vries Equation DiederikKorteweg (1848-1941) Gustav de Vries (1866-1934) We are not disposed to recognize this wave (discovered by Scott Russell) as deserving the epithets “great” or “primary,” and we conceive that ever since it was known that the theory of waves in shallow water was contained in the equation the theory of solitary waves has been perfectly well-known. George Biddle Airy, “Tides and Waves,” Encyclopedia Metropolitana, 1845

10. The Search for a Law of Motion A travelling wave in a narrow channel is described by a height function where is the distance along the channel is the elapsed time is the height of the wave above the undisplaced fluid surface

11. To understand Russell’s wave, one must find a law of motion that describes how the shape function changes with time, and correctly predicts the four properties of solitary waves observed by Russell.

12. Kortweg and de Vries revisited the equations of shallow water waves in a channel, assuming that the depth of the channel was small compared to the wave length. They assumed a wave moving to the right and in place of Airy’slinear equation for the shape function, they obtained the nonlinear Korteweg-de Vries Equation

13. What is the physical mechanism that generates solitary waves? • The rate of change of the shape function is determined by two different terms: • Aterm that tends to make waves disperse • A term that tends to make waves focus

14. If one looks for solutions to the KdV equation of the form one immediately finds a solution that describes a single solitary wave: explaining Russell’s observations about the wave shape and the relationship between wave amplitude and speed of propagation

15. When both dispersive and nonlinear terms are present, focussing can competes with dispersion to create stable, nondispersive waves. In special circumstances, it may be possible for the competition to “balance” and create a solitary wave which neither focuses nor disperses. But under what circumstances, and why?

16. The Birth of Experimental Mathematics Enrico Fermi (1901-1954) John Pasta (1918-1984) StanlisawUlam (1909-1984) Mary TsingouMenzel (1927- ) It is Fermi who had the genius to propose that computers could be used to study a problem or test a physical idea by simulation, instead of simply performing standard calculus. Thierry Dauxois

17. Fermi, Pasta and Ulam wanted to study how energy is distributed between normal modes of a one-dimensional crystal • The crystal consists of a lattice of collinear atoms each of which vibrates due to interaction with its two nearest neighbors • The system can be modeled as a system of masses coupled by springs

18. The spring mass system is linear if the force exerted on a given mass by its neighbor to the left or right is proportional to the compression or extension of the spring. A linear spring-mass system normal modes of vibration which, once started, continue repeating thesame motion over and over. FPUT believed that this would change if the force law for the springs was changed to a nonlinear law.

19. The FPTU model consists of: • balls on a line, connected by springs • is the displacement of the nth mass from its rest position • Newton’s law is applied to each mass to find the motion

20. Linear Versus Nonlinear Springs • In the linearspring model, each mass moves according to the law • In the nonlinearspring model, each mass moves according to the law

21. Fermi, Pasta, Tsingou, and Ulam computed the motion of the 64 nonlinear oscillators on a computer called the MANIAC I (Mathematical Analyzer, Numerator, Integrator, and Computer) based on ideas of John von Neumann MANIAC I

22. For the linear model, there are normal modes of oscillation which, once started, persist forever • Fermi, Pasta, Tsingou, and Ulam expected that in the nonlinearmodel, the energy of the system would be distributed over all accessible modes of vibration • Instead, they found that, if the system of nonlinear oscillators was “started” in a low mode, it would come back to that mode repeatedly, so that the nonlinear oscillators were behaving as if they were "really" linear oscillators

23. The FPTU Model and Solitary Waves • Kruskal and Zambusky (1965) sought to understand why the FPTU nonlinearoscillators exhibited periodic behavior. They computed a continuum limit of the system studied by FPTU. The continuum limit means: • Take the spacing h between oscillators to zero • Take the number N of oscillators to infinity • Consider to be a discrete "sampling" of a continuous function,

24. Approaching The Continuum Limit

25. It was long known that the continuum limit for the linear model gives Airy’s wave equation • Kruskal and Zambusky found that the continuum limit of the FPTU nonlinear model gives the Korteweg-de Vries equation the same equation that describes Russell’s solitary wave! • They computed numerical solutions to the KdV equation and reproduced the periodic behavior of FPTU’s numerical experiment

26. Kruskal and Zabusky called the new solitary waves “solitons.” The mechanism produced them remained mysterious

27. Solitary Waves and Quantum Waves Clifford Gardner (1924- ) John Greene (1928-2007) Martin Kruskal (1925-2006) Robert Miura (1938- ) Peter Lax (1926- ) “Theories permit consciousness to `jump over its own shadow’, to leave behind the given, to represent the transcendent, yet, as is self-evident, only in symbols.” Hermann Weyl

28. Mathematicians suspected that some as yet undiscovered conservation laws were responsible for the existence of solitary waves • Conservation laws for the total mass and energy were known, and more conservation laws were found by hard calculation

29. Conservation Laws (Conservation of mass) (Conservation of Momentum) (Conservation of Energy)

30. “The next surge of momentum came with the arrival of Robert Miura who was asked by Kruskal to get his feet wet by searching for a conservation law at level seven. He found one and then quickly filled in the missing sixth. Eight and nine fell quickly… Miura was challenged to find the tenth. He did it during a two-week vacation in Canada (There is also a rumor that he was seen about this time in Mt. Sinai, carrying all ten).” Alan C. Newell, Solitons in Mathematics and Physics

31. The conservation laws pointed to a remarkable connection between two very different problems: (1) the initial value problem for the KdV equation (2) Schrödinger’s Equation for the “wave function” of a particle moving along a Straight line under the influence of a potential

32. A quantum mechanical particle in a one-dimensional potential well has two possible states of motion: (1) “Bound state” motion where the particle stays localized near the well(2) “Free motion” where the particle moves away from the well The wave function is largest in amplitude where the particle is most likely to be found

33. The rules of quantum mechanics imply that: Any potential well of finite depthcan haveat most finitely many bound states The deeper the well, the more bound states will occur

34. Using this connection, Gardner, Greene, Kruskal, and Miura (GGKM) discovered a remarkable method for solving the KdV equation using ideas from quantum mechanics

35. GGKM’s solution method connects two completely different problems:(1) The motion of waves in a shallow channel, and(2) The motion of a quantum-mechanical particle in one dimension by a precisely defined mathematical transformation. Their discovery explained Russell’s third and fourth observations about solitary waves from the KdVequation.

36. Recall those observations: (3) Solitary waves can cross each other “without change of any kind” (4) A sufficiently large mass of water creates two or more independent solitary waves

37. The set of all possible quantum states is called the spectrum of . There are at most finitely many bound states, described by the bound state energies. The following correspondence holds: Quantum Problem Water Wave Problem is the potential is the initial wave shape is the is the mass of thestrength of the potential wave Bound states of Solitons for KdV Moreover, the spectrum of is the same as the spectrum of

38. We now exploit the following facts: Fact 1: If evolves according to the KdV equation, the number of bound states (and hence, the number of solitons) stays fixed Fact 2: The shape of each soliton is determined by the corresponding bound state energy Fact 3: So long as there is at least one bound state in the quantum problem. The larger the integral is, the larger the number of bound states.

39. We can now explain Russell’s third and fourth observations using the connection with quantum Theory: (3) Solitary waves can cross each other “without change of any kind” The solitary waves correspond to quantum-mechanical bound states, which are determined by the spectrum and therefore do not change over time (4) A sufficiently large mass of water creates two or more independent solitary waves The larger the mass of water , the larger the number of quantum-mechanical bound states. Each bound state will correspond to a soliton.

40. Quiz Question Can the waveform shown below generate solitons? Hint: Remember that the Schrodinger equation has bound states only when

41. Peter Lax formulated the connection between the two problems in terms of a Lax Pair of operators, one that determines the scattering data and the other that determines how the scattering data evolve in time

42. Lax showed that the KdV equation can be described by a Lax Pair of operators and : (spectral problem) (time evolution) together with a law of motion that is equivalent to the originalKdV equation:

43. Lax’s law of motion is equivalent to the original KdV Equation, but expresses it in a different form. Key Fact: If a pair of operators obey Lax’s law of motion, then the spectrum of is automatically preserved, and soliton solutions are possible. Systems obeying Lax’s law of motion are called completely integrable.

44. Lax’s framework enabled mathematicians to look for other nonlinear wave equations that were completely integrable. In 1972, Zakharov and Shabat showed that the nonlinear Schrödinger equation could be solved by the inverse scattering methodand had solitonsolutions.

45. The nonlinear Schrödinger equation, in a slightly different guise, governs the conduction of pulses in optical fibers: where τ measures time from the pulse center and z measures length along the fiber. This equation admits “bright soliton” solutions.

46. The one-soliton solution takes the form The two-soliton solution takes the form

47. These “bright solitons” can be used to transmit data in optical fibers at high speed, with one soliton equal to one bit

48. New Frontiers All of the examples discussed so far concern waves in one dimension. What about two dimensions – e.g. surface waves in shallow water?

49. The Kadomtsev-Petviashvili Equation The solution represents unidirectional long waves propagating in shallow water. If does not depend on then the KP equation reduces to the KdV equation The KP equation admits line solitons

50. Line Soliton for the KP Equation