Further Remarks on John Scott Russel and on the Early History of His Solitary Wave

The first recorded observation of a solitary wave is undoubtedly that made by Russell in the month of August 1834 as described in the quotation from his paper [1.3] given in Chap.1. Russell's fascination for the solitary wave was plain then: that it continued throughout his life is also clear-see for example the quotation from Russell's book of 1865 which appears on the facing page. Russell's career following his discovery of the solitary wave was not uneventful. We mention some points in it. In 1832-1833 he held only the temporary appointment at created by the death of the Professor, Dr. Leslie. He did not apply for permanent appointment to the Chair: Brewster was a candidate and the Chair went finally to J.D. Forbes. In 1838 he did apply for the vacant Chair of Mathematics, but despite a good reference from Hamilton who described Russell as 'a person of active and inventive genius' [1.5], he failed to get it. It is interesting to spe• culate whether theory would have developed some 100 years earlier if Russell had got either of these Chairs. Forbes's interests were certainly very different from Russell's although he has left us permanent work. His reputation now rests primarily on his early work on glaciers (especially that on the Mer de Glace which he was the first to map [1.10,11]). But we believe he also introduced "Forbes's Bar" used in the measurement of the thermal conductivity of metals. The anomalous thermal behavior of one-dimensional anharmonic lattices predicted from numerical studies by FERMI, PASTA and ULAM in 1955 (the FPU problem [1.12]) stimulated the investigations which led Kruskal to the inverse scattering method for solving the KdV, whilst the soliton solutions of that equation may cause some of the difficul• ties in deriving from microscopic theory the Fourier law of heat conduction [1.13].1 Russelll had invented a steam trolley in the years before 1834; in that year the Scottish Steam Carriage Company was formed with the proposal to run a regular service between and Edinburgh. It had a short life [1.5]. Nevertheless [1.5] it seems to have been in consequence of his association with this Company

1Professor Peierls has pointed out to us that even close to the melting point the magnitudes of excitations in real three-dimensional systems are so small that solitonlike contributions can surely be neglected. 374 that Russell received the invitation2 from the Union Canal Company to investigate the Canal's prospects for steam navigation and so see the first soliton. Subse• quently Russell made his reputation as a naval architect. His "wave-line" hulls were designed to minimise wave making resistance due to the generation of solitary wave type bow waves. In 1865 he published the first major work on naval architec• ture [1.14]. And in this he describes his early experiments on the production of low waves and the way they move water at a speed independent of that of the ship [1.5,14]. 8y 1853, however, Russell had been engaged by Brunel to construct the great iron ship the "Great Eastern" [1.5], a direct or indirect cause of many mis• fortunes. He entered the first of his periods of bankruptcy before that ship could be launched, became embroiled in the controversy surrounding the failure of a steam valve during the trials of that vessel in the English Channel in 1859 and the death of five seamen at which time Brunel also died, whilst in 1867 he was forced to re• sign (perhaps by Brunel 's family [1.5]) from the Institution of Civil Engineers of which he was a founder member following a charge of unprofessional conduct associ• ated with a second financial failure whilst acting to purchase guns for the North in the American Civil War. In the last years of his life Russell completed the book The Wave of Translation in the Oceans of Water, Air and Ether published posthumously in 1882 [1.15]. That book contains again reproduced the British Association's 'Report on Waves' (1844) [1.3] where the first observation of the solitary wave is reported. It contains a number of curious speculations on the structure of matter; and it applies the for• mula (1.7) to compute from the velocity of sound the depth of the atmosphere (5 miles) and from the velocity of light the depth of the universe (5x 10 17 miles)! Russell's point for the former was that, for distortionless transmission, sound must be carried by his solitary waves. This it is the velocity c of (1.7) which is to be interpreted as the sound speed [and not cs = I9h in (1.6) or (1.8)]. The calculated depth of 5 miles proves to be the actual equivalent depth at uniform density. However, for the size of the universe, 5x 10 17 miles is out by at least five orders of magnitude; and in any case it appears to rely on a value of g re• duced arbitrarily by Russell by a factor 10-5. Nevertheless these early speculations perhaps begin already to hint at the current significance of in modern . Russell's 'Report on Waves' [1.3] apparently stimulated work by DE BOUSSINESQ [1.9] and RAYLEIGH [1.16]. Both authors derived the sech form of the solitary wave and the formula (1.7) for its speed. Rayleigh also gave reasons why it breaks for

2See especially Russell's Edinb. Roy. Soc. Trans. XIV (1840) paper listed below. There seems to have been no contract, but the Union Canal Company paid the ex• penses-according to their report of 27th January, 1835: 'Report on the practical results of experiments on Canal Navigation (Canal Office, Edinburgh). 375 k~h as had been found experimentally by Russell. Rayleigh reviewed the concept of the solitary wave which Russell had introduced in his 'Report on Waves', noted that with lengths 6 or 8 times the canal depth it could apparently be treated by the theory of long waves, noted nevertheless Russell's observations of different be• haviors for positive and negative waves, quoted Airy's objection, namely that his (Airy's) theory of shallow waves of great lengths admits both 'positive and negative waves' ("We are not disposed to recognise this wave as deserving of the epithets 'great' or 'primary' ... "), quoted Stoke's counter opinion, and then by the impli• cations of his analysis came down firmly in favour of Russell. By the time KORTEWEG and DE VRIES [1.7] derived their equation, (1.6), the sech 2 sol itary wave was lOwe 11 known" and thei r paper was primarily concerned with re• futing AIRY's opinion [1.8] that long waves in a canal must necessarily change their form. They showed that Stoke's theory of long waves [1.17] gave the first two terms of the solution of their KdV equation and that whilst sinusoidal waves become steeper in the front when advancing, other waves behave differently. They showed n = k sech2 px was stationary (in a moving coordinate system) if k = 4op2: for k> 4op2 the waves steepen in front; for k < 4op2 they steepen behind. In our notation a = ~ h3 - yhp-1 g-1. KORTEWEG and DE VRIES [1.7] quoted DE BOUSSINESQ [1.9], RAYLEIGH [1.16] and ST. VENANT [1.18] as establishing the theory of the solitary wave but noted that (in 1895) treatises by Lamb and Basset still assert that Airy was correct in his opinion. They said that even RAYLEIGH and McCOWAN [1.19] do not directly refute Lamb's and Basset's assertions "It is the desire to settle this question definitively which has lead us into the somewhat tedious calculations which are to be found at the end of this paper." The KdV equation it• self noted as 'very important' nevertheless gets rather less discussion. In surveying the history of the solitary wave one should certainly also mention the paper actually entitled 'On the solitary wave' by J. McCOWAN [1.19] and which is quoted by KORTEWEG and DE VRIES [1.7]. ~1cCOWAN isolated and analysed the error of STOKES [1.17] who in 1847 anyway had concluded that the degradation of a wave is an essential characteristic of its mbtion (and not due to friction therefore). He examined and confirmed the approximations of DE BOUSSINESQ [1.9] and RAYLEIGH [1.16] for the sech 2 form of the sol itary wave, derived (1.7) exactly, and noted the agreement with Russell's deductions and their experimental confirmation by BAZIN [1.20]. He also gave an approximate theory of the breaking of waves passing into shallower water. It is remarkable that despite the work of KOR~EWEG and DE VRIES [1.7] four years later so little other work followed. Indeed the signifi• cance of the soundly based early work [1.3] of Russell's is only now being re• cognised as the wide range of application of the concepts of the solitary wave and soliton becomes properly appreciated. The references quoted in this short article appear in the list of references to Chap.1. The reader may care to have other reference to the published scientific 376 work of Russell. The following,. no doubt still incomplete, list was compiled by J.C. Eilbeck from the library of the Royal Society of Edinburgh:

Russell's Published Scientific Work

Russell, John Scott: Notice of the reduction of an anomalous fact in Hydrodynamics, and of a new law of the Resistance of Fluids to the motion of floating bodies. Brit. Assoc. Rep. 1834, pp.531-534 On the motion of floating bodies. Brit. Assoc. Rep. 1835 (pt.2), p.16 On the solid of least resistance. Brit. Assoc. Rep. 1835 (pt.2), p.107-108 On the mechanism of the waves, in relation to steam navigation. Brit. Assoc. Rep. 1837 (pt.2), pp.130-131 On the fallacies of the Rotatory Steam Engine. 1837. Edinb. New Phil. Journ. XXIV., 1838, pp.35-64; Edinb. Trans. Scot. Soc. Arts, I., 1841, pp.172-202; Dingler, Poly tech. Journ. LXVII., 1838, pp.332-355; LXXVIII., 1840, pp.4-18 On the economical proportion of power to tonnage in steam vessels. Brit. Assoc. Rep. 1839 (pt.2), pp.124-125 On the temperature of most effective condensation in steam vessels. Brit. Assoc. Rep. 1840 (pt.2), pp.186-187 On the most economical and effective proportion of engine power to the tonnage of the hull in steam vessels. Brit. Assoc. Rep. 1840 (pt.2), pp.188-190 Description of a Polyphotal Lamp and Reflector of single curvature employed in steam vessels, canal-boats, & c. Edinb. New Phil. Journ. XXVIII., 1840, pp.193-196 Experimental researches into the laws of certain hydrodynamical phenomena that ac• company the motion of floating bodies, and have not previously been reduced into conformity with the known laws of the Resistance of Fluids. 1837. Edinb. Roy. Soc. Trans. XIV., 1840, pp.47-109 On the vibration of Suspension Bridges and other structures, and the means of pre• venting injury from this cause. 1839. Edinb. Trans. Scot. Soc. Arts, I., 1841, pp.304-313; Edinb. New Phil. Journ. XXVI., 1839, pp.386-396 Elementary considerations of some principles in the construction of buildings de• signed to accommodate spectators and auditors. 1838. Edinb. Trans. Scot. Arts, I., 1841, pp.314-318; Edinb. New Phil. Journ. XXVII., 1839, pp.131-136 Report of a Committee on the Form of Ships. Brit. Assoc. Rep. 1841, pp.325-326; 1842, pp.104-105 Supplementary Report of a Committee on Waves. Brit. Assoc. Rep. 1842 (pt.2), pp.19-21 On the indicator of speed of steam vessels. Brit. Assoc. Rep. 1842 (pt.2), p.109 On the abnormal tides of the Firth of Forth. Brit. Assoc. Rep. 1842 (pt.2), pp.115-116 Report of a series of observations on the tides of the Firth of Forth and the east coast of Scotland. Brit. Assoc. Rep. 1843, pp.110-112 Notice of a report of the Committee on the Form of Ships. Brit. Assoc. Rep. 1843, pp.112-115 On the application of our knowledge of the laws of sound to the construction of buildings. Brit. Assoc. Rep. 1843 (pt.2), pp.96-98; Majocchi, Ann. Fis. Chim. XXVIII., 1847, pp121-123 Description of a Marine Salinometer to indicate the density of brine in the boilers of marine steam-egines. Edinb. New Phil. Journ. XXXIV., 1843, pp.278-285 Report on Waves. Brit. Assoc. Rep. 1844, pp.311-390. (reference 1.3) On the tides of the east coast of Scotland. Brit. Assoc. Rep. 1844 (pt.2), p6 On the nature of the Sound-wave. Brit. Assoc. Rep. 1844 (pt.2), p.11 On the resistance of railway trains. Brit. Assoc. Rep. 1844 (pt.2), p.96 Account of a cheap and portable self-registering Tide-Gauge, invented by John WOOD. 1844. Edinb. New Phil. Journ. XXXVIII., 1845, pp.71-76 On the terrestrial mechanism of the Tides. Edinb. Roy. Soc. Proc. I., 1845. pp.179- 182 377

Notice of the remarkable mathematical properties of a certain parallelogram. Edinb. Roy. Soc. Proc. I., 1845, pp.187-188 On the law which connects the elastic force of vapour with its temperature. Edinb. Roy. Soc. Proc. I., 1845, pp.227-231 On the law which governs the resistance to motion of railway trains at high velo• cities. Brit. Assoc. Rep. 1846 (pt.2), pp.109-111 On the practical forms of breakwaters, sea walls, and other engineering works ex• posed to the action of the waves. Civ. Eng. Instit. Proc. VI., 1847, pp.135- 143 On the practical forms of engineering works exposed to the action of the waves of the sea, and on the advantages and disadvantages of certain forms of construc• tion for breakwaters and sea-walls. Franklin Inst. Journ. XIV., 1847, pp.13-15 On certain effects produced on sound by the rapid motion of the observer. Brit. Assoc. Rep. 1848 (pt.2), pp.37-38 On recent applications of the wave-principle to the practical construction of steam-vessels. Brit. Assoc. Rep. 1849 (pt.2), pp.30-33 On wave-line ships and yachts. Roy. Inst. Proc. I., 1851-1854, pp.115-119 On the progress of naval architecture and steam navigation, including a notice of the large ship of the Eastern Steam Navigation Company. Brit. Assoc. Rep. 1854 (pt.2), pp.160-161 Mechanical structure of the Great Eastern steamship. Brit. Assoc. Rep. 1857 (pt.2), pp.195-198 The Wave-line principle of ship construction. Naval Architects' Insttt. Trans. I., 1860, pp.184-211; II., 1861, pp.230-245 Disturbing forces of locomotive engines. Civ. Eng. Instit. Proc. XXII., 1862-1863, pp.107-108 On the rolling of ships, as influenced by their forms and by the disposition of their weights. Naval Architects' Instit. Trans. IV., 1863, pp.219-231 Postscript to Mr. FROUDE's remarks on Rolling. Naval Architects' Instit. Trans. IV., 1863, pp.276-283 Russett, John Scott and (Sir) John Robinson: Report on Waves. Brit. Assoc. Rep. 1837, p.417-496; 1840, pp.441-443 Russett, John Scott: On gun-cotton. Quarterly Journ. Sci. I., 1864, pp.401-412 On the mechanical nature and uses of gun-cotton. 1864. Roy. Instit. Proc. IV., 1866, pp.292-299 For biography see Naval Architects Trans. 23, 1882, pp.258-261; Roy. Soc. Proc., 34, 1883, pp.xv-xyi On the true nature of the wave of translation, and the part it plays in removing the water out of the way of a ship with least resistance. Naval Architects Trans., 20, 1879, pp.59-84 On the true nature of the resistance of armour to shot. Naval Architects Trans., 21, 1880, pp.69-92 On storm stability as distinguished from smooth-water stiffness. 1879. United Service Instit. Journ., 23, 1880, pp.821-849 The wave of translation and the work it does as the carrier wave of sound. Roy. Soc. Proc., 32, 1881, pp.382-383 For biography see also Inst. Civ. Engin. Proc. 87 (1886), pp.427-440

Recent Biography John Scott Russell-A Great Victorian Engineer and Naval Architect. G.S. Emmerson, 1977 (John Murray, London) (Ref. [1.5]) 378

Some Other References of General Interest J. Roy. Soc. Arts 115 (Feb. 1967) pp.204-208 (March 1967) pp.299-302 (on John Scott Russell and , by G.P. Mabon) "College Courant" Glasgow, Martinmas 1958, pp.28-37 and Whitsun 1959, pp.98-109 (on John Scott Russell and the 'Great Eastern' by A.M. Robb) The Naval Archtect, January 1978, pp.30-35 (a review of John Scott Russell by G.S. Emmerson) Harpers and Queen, October 1979, pp.258-259 (a poetic mention of John Scott Russell by Ian Hamilton Finlay) Note Added in Proof (Chapter 1)

For greater clarity of presentation we here expand the comment on the geometrical argument used by LUND [1.142]. The metric tensor g determines intrinsic proper• llV 2 ties of V and fixes the 'first fundamental form' ds = g dyll dyv. The curvature n ~ ~ llV tensor LllV = axll/ayv. Xn+1 (where Xn+1 is the normalised vector Xn+1) describes extrinsic properties of Vn relative to E and fixes the 'second fundamental form' L yll yV. Given g and L the Gauss-Weingarten equations have a unique solution llV ~ llV llV for X, Xn+1 (for given initial values at a point) if and only if the Gauss-Codazzi system are satisfied. To reach the covariant form of (1.21) Lund chooses the first fundamental form ds 2 = Edx 2 + 2FdXdt + Gdt2 with F = 0 and E = cos 2e, G = sin2e. He then chooses the condition L22 - L11 = 2 sine cose on the curvature tensor. The reason for this is that this together with the conditions on the metric represent the equations of motion of a relativistic string in a given external field-so this is where the physics enters the problem. The Gauss-Codazzi equations now relate B = ~(L11 + L22 ) and L12 to the two fields e and A. Thus the physics selects one particular class of surfaces. At first sight these are formally surfaces of constant negative Gaussian curvature K = -1, since the Gauss-Weingarten system (1.122) formally takes the form (1.123) and is a generalised AKNS system, whilst it is proved from (1.123) follow• ing that all AKNS systems represent surfaces with K = -1. This proof, however, ap• peals to Gauss's fundamental theorem that the Gaussian curvature K is an intrinsic property of the surface: the AKNS system has K = -1 with respect to the particular metric (1.131). But Lund embeds his surface choosing an (in general different) metric, together with the condition on the extrinsic curvature tensor, L . In con• sequence Lund's surfaces have curvature K = (L 11 L22 -L122)/(EG - F2), a~~, if L11 - L22 = -2 sine cose, F = 0 and EG = sin2e cos 2e, then K = -1 if and only if B2 = L122. The two descriptions coincide if q is real in (1.122). For then A = 0, and B = L12 (= 0). From (1.121) this is the sine-Gordon equation whose real surfaces represent surfaces of constant negative Gaussian curvature K = -1. Note that the proof that all AKNS systems represent surfaces with curvature K = -1 with respect to the metric (1.131) is formal in so far as the AKNS scattering problem involves two fields q and r or, for example, two complex fields q and q . The surface is therefore complex in general or more complicated, but it is a real surface with 380

K = -1 for real fields r = aq (a = constant). the case which includes the sine• Gordon equation. Note that the work of LAMB [1.145] and LAKSHMANAN [1.146] referred to relate integrable systems like the s-G and NLS to the motion of strings. It is also pos• sible to relate the general AKNS system to the motion of a string and then com• plete the connection with the geometry of the moving string with the geometry of surfaces following from (1.123) and with the vanishing curvature condition (1.125) (cf. M. Lakshmanan: Private communication and to be published). Additional 'References with Titles

In order to bring the several reference lists up to date the following references have been added in proof by some of the authors.

D. Anker, N.S. Freedman: Proc. Roy. Soc. London A360, 529 (1978) A.A. Belavin, V.E. Zakharov: Multidimensional method of the inverse scattering problem and duality equations for the Yang-Mills fields. Pis'ma Zh. Eksp. Teor. Fiz. 25(12), 603-607 (20 June 1977) A.A. Belavin, V.E. Zakharov: Yang-Mills equations as inverse scattering problem. Phys. Lett. 73B, 63 (1978) V.A. Belinsky, V.E. Zakharov: Integration of the Einstein equations by the inverse scattering problem technique and the calculation of the exact soliton solutions. Zh. Eksp. Teor. Fiz. 75, 1953-1971 (December 1978) V.A. Belinsky, V.E. Zakharov: Stationary gravitational solitons with axial sym• metry. Zh. Eksp. Teor. Fiz. 77, 3-19 (1979) S.V. Manakov, V.E. Zakharov, L.A. Bordag, A.R. Its, V.B. Matveev: Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction. Phys. Lett. 63A, 205 (1977) V.E. Zakharov, A.V. Mikhailov: Relativistically invariant Two-dimensional models of field theory which are integrable by means of the inverse problem method. Zh. Eksp. Teor. Fiz. 74, 1953-1958 (June 1978) V.E. Zakharov, L.A. Takhtajan: Equivalence of nonlinear Schrodinger equation and Heisenberg ferromagnet equation. Theor. Math. Phys. 38(1), 26 (1979) V.E. Zakharov, S.V. Manakov: Soliton theory. SOY. Sci. Rev.Sect.A 1, 133 (1979) K. Konno, M. Wadati: Simple derivation of Backlund transformation from Riccati form of inverse method. Prog. Theor. Phys. 53, 1652 (1975) M. Wadati, M. Toda: Backlund transformation for the exponential lattice. J. Phys. Soc. Jpn. 39, 1196 (1975) M. Toda, M. Wadati; Canonical transformations for the exponential lattice. J. Phys. Soc. Jpn. 39, 1204 (1975) M. Wadati: A remarkable transformation in nonlinear lattice problem. J. Phys. Soc. Jpn. 40, 1517 (1976) M. Wadati: On the exact solution of the Korteweg-de Vries equation. Sci. of Light (Tokyo) 25, 37 (1976) Y. Kodama, M. Wadati: Canonical transformation for sine-Gordon equation. Prog. Theor. Phys. 56, 342 (1976) Y. Kodama, M. Wadati: Wave propagation in nonlinear lattice. III. J. Phys. Soc. Jpn. 41, 1499 (1976) Y. Kodama, M. Wadati: Theory of canonical transformations for nonlinear evolution equations. I. Prog. Theor. Phys. 56, 1740 (1976) M. Wadati, M. Watanabe: Conservation laws of Volterra system and nonlinear self• dual network equation. Prog. Theor. Phys. 57, 808 (1977) M. Wadati, H. Sanuki, K. Konno, Y.H. Ichikawa: Circular polarized nonlinear Alfven waves. Rocky Mount. Math. J. 8, 323 (1978) Y.H. Ichikawa, M. Wadati: "Solitons in Plasmas and Other Dispersive ~1edia-Dawn of Nonlinear Physics", in Festschrift in honor of Professor T.Y. Wu, ed. by S. Fujita (Gordon and Breach, London 1978) p.137 M. Wadati: Invariances and conservation laws of the Korteweg-de Vries equation. Stud. Appl. Math. 59, 153 (1978) M. Wadati: "Infinitesimal Transformations and Conservation Laws, Field Theoretic Approach to the Theory of Soliton", in Research Notes in Mathematics, Vol.26, ed. by F. Calogero (Pitman, London 1978) p.33 382

M. Wadati, K. Konno, Y.H. Ichikawa: A generalization of inverse scattering method. J. Phys. Soc. Jpn. 46, 1965 (1979) R. Hirota, M. Wadati: A functional integral representation of the soliton solution. J. Phys. Soc. Jpn. 47, 1385 (1979) M. Wadati, K. Konno, Y.H. Ichikawa: New integrable nonlinear equations. J. Phys. Soc. Jpn. 47, 1698 (1979) Y.H. Ichikawa, K. Konno, M. Wadati, H. Sanuki: Spiky soliton in circular polarized Alfven wave. J. Phys. Soc. Jpn. 48(1) (1980) M. Wadati, K. Sawada: New representations of the soliton solution for the Korteweg• de Vries equation. J. Phys. Soc. Jpn. 48(1) (1980) M. Wadati, K. Sawada: Application of the trace method to the modified Korteweg-de Vries equation. J. Phys. Soc. Jpn. 48(1) (1980) K. Fukushima, M. Wadati, T. Kotera, K. Sa~lada, Y. Narahara: Experimental and theoretical study of the recurrence phenomena in nonlinear transmission line. J. Phys. Soc. Jpn. (to be published) T. Shimizu, M. Wadati: A new integrable nonlinear evolution equation. Prog. Theor. Phys. (to be published) A. Oegasperis: "Solitons, Boomerons and Trappons", in Nonlinear Evolution Equations Solvable by the Spectral Transform, Research Notes in Mathematics, Vol.26, ed. by F. Calogero (Pitman, London 1978) A. Oegasperis: "Spectral Transform and Solvability of Nonlinear Evolution Equations", in Nonlinear Problems in Theoretical Physics, Lectures Notes in Physics, Vol.98, ed. by A.F. Ranada (Springer, Berlin, Heidelberg, New York 1979) F. Calogero, A. Oegasperis: Reduction technique for matrix nonlinear evolution equations. J. Maths. Phys. (in press, 1980) Subject Index

Abelian variety 333,336 Benjamin-Ono equation 53 Abel map 333 Bilinear differential equations 157, Abel substitution 331 161,162 Absorber (attenuator) 109 Binding energy 352 nondegenerate 109 Bloch eigenfunctions (or Floquet functions) 327 Action-angle variables 31,94,186,198, 340,356 Bloch equations 113,121 Admissible operator 328 Bloch-Bloembergen equations 276 Akheizer function 334 Bloch-Maxwell equations see Maxwell• Bloch equations AKNS formalism 293 see also Zakharov-Shabat scattering Boomerons 13,301,311 problem Bose or Fermi gas 264 Amplifier 109 Boson-fermion duality see Operator Anisotropic magnetic liquids 109 democracy Area theorem (McCall-Hahn) 73,99,115, Boson fields 358 204 Bound states 9,352 Attenuator see Absorber Boussinesq equation 4,51,157,165,172 Breaking of waves 375 Backlund transformations 1,13,49,66, Breather solutions 11,211 76,153,157,171,301,319 of the sine-Gordon equation 7 auto 14,111 Long lived breather-like state 124 in bilinear form 167 Burgers equation 15,52 superposition principle 15 theorem of permutability 76 Canonical creation and destruction jet bundle formulation of 23 operators 364 Backward scattering model (of Canonical variables 348 Luttinger) 38 Capillary surface waves 3 Baker-Hausdorf formula 364,367 Carrier wave 9 Bare or undressed operators 248,253 Cartan's theory of exterior forms 32 Bargmann potential 74 Cauchy (- Goursat) problem 257,283 Baroclinic waves 181 Causal representation 101 Beer's law 203,204 Characteristics 1 Beltrami's surface of revolution 24 Charge density wave 109 Benjamin-Feir instability 25,179 Chemical potential 359 384

Clifford algebras 39 65,67 Cnoidal wave 146 geometric 182 Commutation relations 122,123,358 relation 83,196,327 anti- 360,362 Dispersion relation, linearised 37 Complete integrability 340,344 Dispersion relations, systems with Hamiltonian systems 1,32,93,329 two 201 Completely integrable Hamiltonian Doppler broadening 118,203 systems, finite dimensional 16 Domain walls (magnetic) 108 infinite dimensional 1 D operators 159 Conjugate coordinates 186 properties of 160 Conservation laws 49,93,218 Dressing method 243,244 higher 65 multidimensional generalisation of infinite number of 180 249 Conserved densities 13,43 dressing "L-A" pairs 253 polynomial 16 dressing of operator families 280 transcendental 47 Dual transformation 144 infinity of 9,111 Conserved quantities (or constants of Eight vertex model 110,355,371 motion) 1,93,177,196,301,321 Electric field 65 infinite number of 184 Electromagnetic wave 168 locally 16 high intensity 179 globally 16 Energy 346 Consistency condition 232 of the breather 12 Constants of motion see Conserved Ernst equation 55 quantities Euler's equation for the free rotation Continuum limit 152,357,360 of a solid body 261 Cont.i nuum phys i cs 181 Exchange formula 168 Corpuscular wave 3 Explosive instability 227,261,270 Correlation functions 356 Exponential interaction 145 n-particle 362,366 Extended Bethe ansatz 371 Cosmology 55 Fermi liquid theory 355 Coupling constants 36,130 Fermi-Pasta-Ulam (FPU) problem 26,373 Crystal dislocation theory 5 Fermi sea 362 Fields continuum equations 361 Damping constants 119 model theories 24,355 Deep water gravity waves 179,227 nonlinear theories 1,23 Detuning parameter 89 Finite zoned operators 327,328 Differential geometry 1,13,76 Flows in involution 199 Dipole interaction 122 Fl uxons 5,110 Direct methods 7 Forbes's bar 373 Direct scattering (or spectral) transform 29 Fourier transform 177 Fredholm operator 253 Galilean invariance 41 Heisenberg operators 263 Galilean transformation 27 Helium, 3He A and B 109 Gap matrix 123 Hobart-Derrick theorem 54 Gauge field 49 Hodograph transformation 26 classical self-dual non-Abelian 54 Hopf-Cole transformation 33,52 Gauge transformation 42,43,359 Gauss-Codazzi equations 41 Instanton 54 Gaussian curvature 43 Integrable evolution equations 1 Gauss-Weingarten system of equations Integrable systems 30 41 Integrability conditions 14,35,42 Gel 'fand-Levitan-Marchenko integral equation 28,90,103,258,291,344 Integrability theorem of Frobenius 32 Gregarious waves 3 Interacting systems 67 Gross-Pitaevskii equation 25 Internal gravity waves 181 Inverse scattering method (or spectral Group On 39 transform) 1,7,9,29,65,111,150,243, Gyromagnetic ratio 122 301,304 as a canonical transformation 31,186 Hamiltonian 339 for the Korteweg-de Vries equation of Fermi lattice gas 359 170 of nearest neighbor spin inter• matrix generalization of 288 action 358 quanti sed 371 density of double sine-Gordon 116 Inverse problem 9 density of spin waves in 3He 122 Inverse matrix Schrodinger problem 301, Hamiltonian description 93 303,306 Hamiltonian dynamics 65 Involution 16,51,94 Hamiltonian structure (of a class of Ion-acoustic waves 66,82,179 evolution equations) 198 Irreversible flow 184 Hamiltonian formalism for Korteweg-de Ising model 110 Vries equation 331 two-dimensional 235,355 Hamiltonian flow (restricted) 51 Isospectral deformation 35,233 Hamiltonian systems 259,331 Isospectral time evolution 39 completely integrable see Complete integrability Hamiltonian's equations 340 Jacobi identities 232 Harmonic limit 147 Jacobi matrices 325 Heat conduction 154,181 Jacobi torus 333 Fourier law of 373 Jacobian varieties 53 in solids 179 Jordan-Weigner transformation 358 Thermal conductivity of metals 373 Josephson junctions 5 Heat equation 16,196 large area 110 Heisenberg's equations of motion 122, Jost functions 289 358 Heisenberg ferromagnet 53 386

Kadomtsev-Petviashvili equation 53, Linear unidirectional wave equation 196 165,172,182,228,265,336 Liouville's equation 14 Kerr effects 77 Loca 1ity 361 Kink 10 Local operators 325 anti- 10 Long internal gravity waves 179 solutions 7 Long waves in shallow water 279 solutions of the sine-Gordon equation 7 Long wave - short wave i nteracti on 182 kink-like solutions 6 Long wave transverse perturbations 228 28 kink 124 Lorentz invariance 41,355 41T-28 kink 124 of sine-Gordon equation 23 Klein-Gordon equation 180 Lorentzian linewidth 202,204 generalized 54 nonlinear 13,110 ~lagneti c shocks 130 Korteweg-de Vries (KdV) equation 1,4, Magnetic susceptibility 122 5,85,152,157,178,216,243,262,326,341, ~lany-body problems (solvable) 1,51 348,373,375 ~larchenko equation see Gel'fand• discrete analogue of 166,174 Levitan-Marchenko equation in matrix form 293 Massive Luttinger model 38,355,365 higher order family 157,164,166, Massive Thirring model 24,38,55,181, 173,228,326 232,355,365 Lax's hierarchy of 20,30 Matrix formalism 148,150 with cylindrical symmetry 55 Maxwell's equations 69,113,203 Maxwell-Bloch equations 68,181 ilL-Ali pairs 244 nonlinear 260 ~lerging 260 Lacu nas 328 (of secondary wave) "L-A-B" triad 270 Meson (breather as) 24 Lagrange multiplier 39 Miura transformation 16,27,171,180 Modified Korteweg-de Vries (MKdV) Lagrangian density 37 equation 16,27,180,196,210,228,233, for charge density waves 110 264 Langmuir turbulence 26 describing a weakly nonlinear Laser physics 55 1attice 169 exhibiting a shock wave type of Lattice dynamics 66 solution 170 nonlinear 143 Momentum 346 so 1iton 147 quasi- 327 Lax pair 30,35,49 Monodromy matrix 234 Lax representation 325 Leap frogging pulses 116 Na vapour 118 Levin's linearisation of matrix Riccati Nearest neighbor interaction 358 equations 39 Near-sonic Langmuir solitons 264 Lie algebras 43,184,232 Nonlinear differential-difference Lie invariance 41 equations 295 of sine-Gordon equation 23 387

Nonlinear evolution equations 1 for the double sine-Gordon equation Nonlinear optics 187,267 131 equations 260 Pfaffian system 42 resonant 203,261 Phase locking 126 Nonlinear Schrodinger equation 1,5,81, Phase plane 110 177,179,196,220,263,341,344,356 Phase shift 8,178 higher analogues 326 Phase space 114,339 derivative 232 Phase transition theory 356 discrete 297 Phi-four (~-four) equations 6 m-component 295 Plasma 65 two-dimensional 25,182,227 cavitons, photon bubbles 26 Nonl inear self-dual network equation cold 179 297 physics 55,264 Nonlinear a-model 39 waves 181 Nonlinear superposition principle 301, Poisson brackets 17,32,37,94,199,332, 321 340 Nonlinear operator identity 301,323 Polarization 69 Normal modes 218 Poles (motion of) 50 Normal ordering 359 Principle of universal ity see Operator Nuclear magnetic resonance (NMR) 121 democracy ringing 129 Prolongation structures 23,31,32,43,49 satellite frequencies 126 Pul se Number of particles 346 area 11 breaku p 121 One-dimensional magnetic chains 356 coherent 1i ght- 66 One-dimensional many body problem 49 coherent propagation 177 ,203 One-dimensional conductors 356 through a resonant medium 184 One-forms 42 oppositely directed optical 121 "connection" 43 resonant short optical 5 Operator democracy 355,358,362 "sharp 1 i ne" 109 Operator equations 123 ultrashort 73 Optical filamentation 25,119,121 in a resonant 5-fold degenerate Order parameters 122,130 medium 107 velocity 72 zero-n (On) or breather 11,73,107, Pade approximant 158 187 Painleve equation (second) 234 2n or kink 187 Particle physics 55 wobbling 4n 107,117 Periodic boundary conditions 27 4.2n double peaked wobbling 119 Permanent profile (wave of) 3 Pulsons 55 Permitted zones, stability zones, or spectrum 328 Pumping wave 260,262,269 Perturbation theory 38 break down of 260 388

Quantisation 37,350 discrete version of matrix- 295 of sine-Gordon equation 12,21,24, Schrodinger operator 258,328 38,55,355 Secondary waves 260,262 sine-Gordon model 355 Self-adjoint 89 of double-sine-Gordon equation 110 Self-focussing 25,108,119,121 Self-induced transparency (SIT) 24,65, Radiation 111,117,180,205 66,71 Rank 18 degenerate 108,112 Rational solutions of the Korteweg-de Self-interaction of electromagnetic Vries equation 50 waves 263 Reduced Maxwell-Bloch equations (RMB) Self-similar solutions 234 33 Shallow water waves 146 Reduction (of sets of nonlinear equa ti ons) 258 model equations for 157,165,173 Reflection coefficient 28,132,302,305, Simple wave equation 6 319 Simple wave shock 27 matrix 291 Sine-Gordon equation 1,5,74,177,180, Relatively prime differential 209,228,276,340,349,353 operators 29 double 53,107 Renormalisation 357 higher analogues 326 mass 368 in light cone coordinates 342 Resolvent formula (generalized) 301, triple 109,118 322 Singular perturbation theory 53,111, Rest mass of breathers and kinks 12 177 ,217 Rest mass energies of kinks 124 Sinh-Gordon equation 21,35,180 Riccati equation 45 Shock wave solution 170 Riccati transformation 27,34 S-matrix 123,356,358 Riemann surface 234,245,328 crossing symmetry 38 hyperelliptic 330 factorisable 24 Riemann 8-function 51,335 unitarity 38 Rossby waves 181 Snake-like instability 227 Solid state physics 55 Scalar interacting fermion problem in in one dimension 355 one dimension 355 Solitary wave 1,3,6,373 Scale invariance (breaking of) 39 Great solitary wave 1,3 Scale transformation 4 Sol iton 147 Scattering data 9,28,89,132,151,183, collision property of 6 186,292,344 evolution of 188 composi te 126 Scattering lengths 98 definition of 6 Scattering matrix (full) 345 envelope 9 Scattering problem 9 hydrodynamic 65 instability of 264 Schrodinger eigenvalue equation 9,28, 68,177 ,183 ,212 N-soliton collision 11 matrix 301,302,306 N soliton solutions 51,150,152, 164,257 389

of the Korteweg-de Vries Theory of "pol ing" 110 equation 7 Thermal agitation 77 of the sine-Gordon equation 10 Thermal behaviour of one dimensional Multisoliton plane wave solutions 54 anharmonic lattices 373 Two-soliton bound state 91 Thermally excited breathers and kinks 130 Two-soliton solutions 147 Three wave interaction 181,342 origin of name 7 Toda lattice (equation) 53,98,143, Quantum solitons 355 157,166,173,181,297,343,350 Sound speed 4 periodic 154 Sound wave 3 discrete time 157,167,174 Spectrum Topological charges 23 conservation of 274 Topological quantum numbers 24 continuous 302,307 Topological solitons 117 discrete 302,304,306,307 Tori 94 time evolution of 307 Transfer matrix 355 of continuum limit equations 369 Translation (monodromy) matrix 327 moving eigenvalues 206 Transmission coefficient 28,302,304 not invariant 177 Trappons 13 of quantized Hamiltonian 352 Two-dimensional vortex model 21 of quantized sine-Gordon equation 355 Two-forms 42,86 Spins 5 Two-level atoms (quantum oscillators) Spinor field 360 65,87 Spin waves in the anisotropic magnetic five fold degenerate 112 liquids 3HeA and 3HeB 107 Spin-y, operators 358 Volterra factors 253 Spin-y, x-y-z model 355 Volterra integral operator 253 Spontaneous emission 116 Volterra systems 295 Squared eigenfunctions 133,177,183, 190,192,230 Stationary solutions 325 "Wave-line" hulls 374 Statistical mechanics 55,110,355,356 Wave on translation 3 Stimulated emission 71 Waves of vortex tubes 179 Stokes and anti-Stokes waves 262 Weakly nonlinear lattice 169 Structure equations 42 Weirstrass elliptic function 50 Sturm-Liouville equation 328 Weiner-Hopf decomposition 202 Superfluid number density 122 Wigner-Eckhart theorem 109 Superfluorescent emission 121 Wronskian 185,289,328 Surfaces of constant negative curva• ture (pseudo-spherical surfaces, sur• generalized 301,303 faces of Enneper) 13 , 23 ,42,43 SU(2) algebra 357 Zakharov-Shaba t-AKNS 2)( 2 sca tteri ng Symmetry - Q(J) and Q(3) 109 problem 18,32,48 Symplectic form 339 generalized 177,183,185 Symplectic structure 30 nth order 231 Inverse Source Problems M.Toda in Optics Theory of Nonlinear Lattices Editor: R P. Baltes 1980. (Springer Series in Solid-State Sciences, With a foreword by I-F. Moser Volume 20) 1978. 32 figures. XI, 204 pages ISBN 3-540-10224-8 (Topics in Current Physics, Volume 9) ISBN 3-540-09021-5 The mathematical methods for expressing wave pro• pagation in nonlinear systems are described rigor• Contents: ously and coherently in this volume, with main H. P. Baltes: Introduction. - H. A Ferwerda: The emphasis on the nonlinear lattice with exponential Phase Reconstruction Problem for Wave Amplitu• interaction between nearest' neighbours. This kind des und Coherence Functions. - B. J. Hoenders:The ofiattice, originally analysed by the author, has be• Uniqueness of Inverse Problems. - H. G. Schmidt• come the subject of wide and thorough investiga• Weinmar: Spatial Resolution of Subwave-Iength tions by many other researchers. Starting out with Sources from Optical Far-Zone Data. - H. P. Baltes; an historical exposition, the "soliton" or stable pulse J. Geist, A Walther: Radiometry and Coherence. - characteristics of nonlinear lattices is introduced to• A Zardecki: Statistical Features of Phase Screens getherwith the many quantities concerved, showing from Scattering Data. that the system is integrable. The method ofsolving the equations of motion under given initial condi• tions is described in detail, clarifying the so-called inverse scattering method for an infinite system and the inverse spectra1 method for a periodic lattice. Solitons and Condensed Finally, action and angle variables are given for the integration ofthe system following the general prin• Matter Physics ciple of analytical mechanics. The monograph is Proceedings ofthe Symposium on Nonlinear (Soli• supplemented with simple examples, relations to ton) Structure and Dynamics in Condensed Matter the well-known continuous. systems, and ma~y Oxford, England, June 27-29, 1978 appendices to make the text asseccible to students Editors: A R Bishop, T. Schneider and researchers in physics and related field. 1978. 120 figures, 4 tables. XI, 341 pages (Springer Series in Solid-State Sciences, Volume 8) ISBN 3c540-09138-6

Contents: Introduction. - Mathematical Aspects. - Statistical Mechanics and Solid-State Physics. - Summary. G. Eilenberger Solitons Mathematical Methods for Physicists 1980. (Springer Series in Solid-State Sciences, Volume 19) ISBN 3-540-10223-X This book was written in connection with a graduate• level course in theOretical physics. Main emphasis is placed on an introduction to in• verse scattering theory as applied to one-dimen• sional systems exhibiting solitons, as well as to the new mathematical concepts and methods developed for understanding them. Since the treatment is directed primarily at physicists, the mathematical background required is the same as that for courses Springer-Verlag in theoretical physics, namely an elementary know• ledge offunction theory, differential equations and Berlin operators in Hilbert space. This book offers readers interested in the applica• Heidelberg tions of soliton systems with a self-contained intro• duction to the subject, sparing them the necessity New York oftedious searches through original literature. Cavitation and Inhomo• Synergetics geneities in Underwater A Workshop Acoustics Proceedings of the International Workshop on Synergetics at Schloss Elmau, Bavaria, Proceedings of the First International Conference May 2-7, 1977 GOttingen, Fed. Rep. of Germany, July 9-11, 1979 Editor: H. Haken Editor: W. lauterborn Springer Series in Synergetics 1980. 192 figures, 6 tables. XI, 319 pages 1977. 136 figures. VIII, 274 pages (Springer Series in Electrophysics, Volume 4) ISBN 3-540-08483-5 ISBN 3-540-09939-5 Contents: Contents: General Concepts. - Bifurcation Theory. - Insta• Cavitation. - Sound Waves and Bubbles. - Bubble bilities in . - Instabilities in Astro• Spectrometry. - Particle Detection. - Inhomo• physics. - Solitons. - Nonequilibrium Phase Transi• geneities in Ocean Acoustics. - Index of Contri• tions in Chemical Reactions. - Chemical Waves and butors. Turbulence. - Morphogenesis. - Biological Systems. - General System.

Ocean Acoustics Editor: J.ADeSanto 1979. 109 figures, 5 tables. XI, 295 pages (Topics in Current Physics, Volume 8) ISBN 3-540-09148-3

Contents: 1. A. DeSanto: Introduction. - J. A. DeSanto:Theore• tical Methods in Ocean Acoustics. - F. R. DiNapoli, R. L. Deavenpon: Numerical Models of Under• water Acoustic Propagation. - J G. Zornig: Physical Modelling of Underwater Acoustics. - J. P. Dugan: Oceanography in Underwater Acoustics. - N. Bleistein, J K Cohen: Inverse Methods for Reflec• tor Mapping and Sound Speed Profiling. - R. P. Poner: Acoustic Probing of Space-Time Scales in the Ocean. - Subject Index.

Structural Stability in Physics Proceedings of Two International Symposia on Applications of Catastrophe Theory and Topologi• cal Concepts in Physics Ttibingen, Fed. Rep. of Germany, May 2-6 and December 11-14, 1978 Editors: W. Gtittinger, H. Eikemeier Springer Series in Synergetics 1979. 108 figures, 8 tables. VIII, 311 pages ISBN 3-540-09463-6

Contents: Springer-Verlag Introduction. - General Concepts. - Topological Berlin Aspects of Wave Motion. - Catastrophes in Infinite Dimensions. - Defects and Dislocations. - Statisti• Heidelberg cal Mechanics and Phase Transitions. - Solitons. - Dynamical Systems. - Index of Contributors. New York