×

Computational synergetics and mathematical innovation. (English) Zbl 0489.65043


MSC:

65Lxx Numerical methods for ordinary differential equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ulam, S.M., Introduction to “studies of nonlinear problems”, () · Zbl 0086.24101
[2] Dyson, F.J.; Dyson, F.J., The mathematical sciences, A collection of essays, Sci. amer., 24, 129, (1969), MIT Press Cambridge, Mass, Also in · Zbl 0172.01404
[3] Dyson, F.J., The mathematical sciences, A collection of essays, (1969), MIT Press Cambridge, Mass, indicates that mathematicians and philosophers agree that philosophizing is best left to philosophers and historians · Zbl 0172.01404
[4] Ulam, S.M., A collection of mathematical problems, (1960), Wiley-Interscience New York · Zbl 0086.24101
[5] Zabusky, N.J., A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, (), 223-256 · Zbl 0183.18104
[6] Davis, P.J.; Anderson, J.A., Nonanalytic aspects of mathematics and their implication for research and education, SIAM rev., 21, 112-127, (1979) · Zbl 0399.03004
[7] Goldstine, H.H., The computer from Pascal to von Neumann, (1972), Princeton Univ. Press Princeton, N. J · Zbl 0245.01017
[8] Richardson, L.F., Weather prediction by numerical processes, (1966), Dover New York
[9] Goldstine, H.H.; von Neumann, J., On the principles of large scale computing machines, (), 1-32
[10] Goldstine, H.H.; von Neumann, J.; Goldstine, H.H.; Goldstine, H.H.; Goldstine, H.H., The NORC and problems in high-speed computing, (), 238-247, a speech at the first public showing of the I.B.M. Naval Ordnance Calculator (1954) · Zbl 0061.29706
[11] Ulam’s, S.M., The interaction of mathematics and computing, (), 93-99, remarks in von Neumann
[12] Courant, R.; Robbins, H., What is mathematics, () · JFM 67.0001.05
[13] Hahn, H., The crisis in intuition, (), 1956-1976
[14] Heisenberg, W.; Heisenberg, W., Nonlinear problems in physics, (), Physics today, 20, 27, (1967), Also published in
[15] Debye, P., Vorträge über die kinetische theorie der materie und der elektrizität, (1914), Leipzig, Germany
[16] Peierls, R.E., Quantum theory of solids, (), 140 · Zbl 0068.23207
[17] Fermi, E.; Pasta, J.R.; Ulam, S.M.; Fermi, E.; Pasta, J.R.; Ulam, S.M., Studies on nonlinear problems, I, (), 978-988, Also, · Zbl 0353.70028
[18] Zabusky, N.J., Phenomena associated with the oscillations of a nonlinear model string (the problem of Fermi, pasta, and Ulam), (), 99-133
[19] Arnold, V.I., Mathematical methods of classical mechanics, (1978), Springer-Verlag Berlin, New York, For a discussion of basic ideas see · Zbl 0386.70001
[20] Kruskal, M.D., Some unsolved mathematical problems, (January, 1960), A seminar at the Princeton Plasma Physics Laboratory
[21] Kruskal, M.D., The birth of the soliton, (), 1-8
[22] Ornstein, L.S.; Zernike, F., Contributions to the kinetic theory of solids, II. the unimpeded spreading of heat even in the case of deviations from Hooke’s law, (), 1295
[23] Zabusky, N.J., Exact solution for the vibrations of a nonlinear continuous model string, J. math. phys., 3, 1028-1039, (1962) · Zbl 0118.41802
[24] Lax, P.D., Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. math. phys., 5, 611, (1964) · Zbl 0135.15101
[25] Kruskal, M.D., Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic, J. math. phys., 3, 806-828, (1962) · Zbl 0113.21201
[26] Kruskal, M.D.; Zabusky, N.J., Stroboscopic perturbation procedure for treating a class of nonlinear wave equations, J. math. phys., 5, 231-244, (1964)
[27] Korteweg, D.J.; de Vries, G., On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Philos. mag., 39, 422-443, (1985) · JFM 26.0881.02
[28] Gardner, C.S.; Morikawa, G.K., Similarity in the asymptotic behavior of collision-free hydromagnetic waves and water waves, Courant institute of mathematical sciences rept. NYO 9082, (May, 1960)
[29] Su, C.H.; Gardner, C.S., Korteweg-devries equation and generalizations. III. derivation of the Korteweg-devries equation and bugers equation, J. math. phys., 10, 536-539, (1969) · Zbl 0283.35020
[30] Miles, J.W., The Korteweg-devries equation: an historical essay, J. fluid mech., 106, 131-147, (1981) · Zbl 0468.76003
[31] Ursell, F., The long wave paradow in the theory of gravity waves, (), 685-694 · Zbl 0052.43107
[32] Hopf, E., The partial differential equation ut + uux = μuxx, Comm. pure appl. math., 3, 201, (1950)
[33] Cole, J.D., On a quasilinear equation occurring in aerodynamics, Quart. appl. math., 9, 225, (1951) · Zbl 0043.09902
[34] Kruskal, M.D., Asymptotology in numerical computation; progress and plans on the Fermi-pasta-Ulam problem, ()
[35] Tuck, J.L.; Menzel, M.T., The superperiod of the nonlinear weighted string (FPU) problem, Adv. in math., 9, 339-407, (1972)
[36] Zakharov, V.E., On stochastization of one-dimensional chains of nonlinear oscillators, Sov. phys. JETP, 38, 108-110, (1974)
[37] Zabusky, N.J.; Kruskal, M.D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. rev. lett., 15, 240, (1965) · Zbl 1201.35174
[38] Toda, M., Studies of a nonlinear lattice, Phys. rep., 18, 1-124, (1975)
[39] Abe, K.; Abe, T., Recurrence of initial states of the Korteweg-de Vries equation, Phys. fluids, 22, 1644-1646, (1979) · Zbl 0408.76010
[40] Lax, P.D., Integrals of nonlinear equations of evolution and solitary waves, Comm. pure appl. math., 21, 467-490, (1968) · Zbl 0162.41103
[41] Tappert, F.D., Numerical solutions of the Korteweg-de Vries equations and its generalizations by the split-step Fourier method, (), 215-216
[42] Zabusky, N.J., Nonlinear lattice dynamics and energy sharing, J. phys. soc. Japan, 26, 196, (1969), suppl.
[43] Zabusky, N.J.; Zabusky, N.J., A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, (), 244 · Zbl 0183.18104
[44] Whitham, G.B., Non-linear dispersive waves, (), 238-261 · Zbl 0125.44202
[45] Miura, R.M., Korteweg-de Vries equation and generalizations, I. A remarkable explicit nonlinear transformation, J. math. phys., 9, 1202-1204, (1968) · Zbl 0283.35018
[46] Miura, R.M.; Gardner, C.S.; Kruskal, M.D., Korteweg-de Vries equations and generalizations. II. existence of conservation laws and constants of motion, J. math. phys., 9, 1204-1209, (1968) · Zbl 0283.35019
[47] Kruskal, M.D.; Miura, R.M.; Gardner, C.S.; Zabusky, N.J., Korteweg-de Vries equation. V. uniqueness and nonexistence of polynomial conservation laws, J. math. phys., 11, 952-960, (1970) · Zbl 0283.35022
[48] Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M., Method for solving the Korteweg-de Vries equation, Phys. rev. lett., 19, 1095-1097, (1967) · Zbl 1061.35520
[49] Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M., Korteweg-de Vries equation and generalizations. VI. method for exact solution, Comm. pure appl. math., 27, 97-133, (1974) · Zbl 0291.35012
[50] Zabusky, N.J., Solitons and bound states of the time-independent Schrödinger equation, Phys. rev., 168, 124, (1968)
[51] Toda, M., Vibration of a chain with nonlinear interaction, J. phys. soc. Japan, 22, 431-436, (1967)
[52] Toda, M., Problems in nonlinear dynamics, Rocky mountain J. math., 8, 197-209, (1970) · Zbl 0406.35057
[53] Ford, J.; Waters, J., Computer studies of energy sharing and ergodicity for nonlinear oscillator systems, J. math. phys., 4, 1293-1306, (1963)
[54] Toda, M., Mechanics and statistical mechanics of nonlinear chains, J. phys. soc. Japan suppl., 26, 235-237, (1969)
[55] Saito, N.; Ooyama, N.; Aizawa, Y.; Hirooka, H., Progr. theor. phys. suppl., 45, 201, (1970)
[56] {\scJ. Ford}, letter to N. J. Zabusky dated 20 October 1980.
[57] Ford, J.; Stoddard, S.D.; Turner, J.S., On the integrability of the Toda lattice, Progr. theor. phys., 50, 1547-1560, (1973)
[58] Henon, M., Integrals of the Toda lattice, Phys. rev. B, 9, 1921-1923, (1974) · Zbl 0942.37503
[59] Flaschka, H., The Toda lattice. existence of integrals, Phys. rev. B, 9, 1924-1925, (1974) · Zbl 0942.37504
[60] Perring, J.K.; Skyrme, T.H.R., A model unified theory, Nucl. phys., 31, 550-555, (1962) · Zbl 0106.20105
[61] Lamb, G.L., Analytical descriptions of ultra-short optical pulse propagation in a resonant medium, Rev. mod. phys., 43, 99-129, (1971)
[62] Frank, F.C.; van der Merwe, J.H.; Frank, F.C.; van der Merwe, J.H., One-dimensional dislocations. IV. dynamics, (), 261-268 · Zbl 0038.13702
[63] Seeger, A.; Donth, H.; Kochendorfer, A., Theorie der versetzungen in eindimensionalem atomreihen, Z. phys., 134, 173-193, (1953) · Zbl 0050.44808
[64] Abe, K.; Inoue, O., Fourier expansion solution of the Korteweg-de Vries equation, J. comput. phys., 34, 202-210, (1980) · Zbl 0443.76020
[65] Eilbeck, J.C., Numerical studies of solitons, (), 28-43 · Zbl 0325.65054
[66] Fornberg, B.; Whitham, G.B., A numerical and theoretical study of certain nonlinear wave phenomena, Philos. trans. roy. soc. London, 289, 373-404, (1978) · Zbl 0384.65049
[67] Ablowitz, M.J.; Kruskal, M.D.; Ladik, J.F., Solitary wave collisions, SIAM J. appl. math., 36, 428-437, (1979) · Zbl 0408.65075
[68] Abdulloev, K.O.; Bogolubsky, I.L.; Makhankov, V.G., Phys. lett., 56, 427, (1976)
[69] Bona, J.L.; Pritchard, W.G.; Scott, L.R., Solitary-wave interaction, Phys. fluids, 23, 438-441, (1980) · Zbl 0425.76019
[70] Santarelli, A.R., Numerical analysis of the regularized long wave equation: anelastic collision of solitary waves, Nuovo cimento B, 46, 179-180, (1978)
[71] Lewis, J.C.; Tjon, J.A., Resonant production of solitons in the RLW equation, Phys. lett. A, 73, 275-279, (1979)
[72] Zabusky, N.J., Solitons and energy transport in nonlinear lattices, Comp. phys. comm., 5, 1, (1973)
[73] {\scW. E. Ferguson, Jr., H. Flaschka and D. W. McLaughlin}, Nonlinear normal modes for the Toda chain, J. Comput. Phys., in press. · Zbl 0557.70028
[74] Christiansen, P.L.; Lomdahl, P.S.; Zabusky, N.J., Tunable oscillator using pulsons on large-area lossy Josephson junctions, Appl. phys. lett., 39, 170-172, (1981)
[75] Bogolyubsky, I.L.; Makhankov, V.G.; Bogolyubsky, I.L.; Makhankov, V.G.; Bogolyubsky, I.L.; Bogolyubsky, I.L.; Makhankov, V.G., Dynamics of spherically symmetrical pulson of large amplitude, Pisma zh. eksp. teor. fiz., JETP lett., JETP lett., JETP lett., 25, 107-110, (1977)
[76] Makhankov, V.G., Dynamics of classical solitons in non-integrable systems, Phys. lett. C, 35, 1-128, (1978)
[77] Christiansen, P.L.; Olsen, O.H., Ring-shaped quasisolitons of the two and threedimensional sine-Gordon equation, Phys. scripta, 20, 531-538, (1979) · Zbl 1063.81529
[78] Christiansen, P.L.; Lomdahl, P.S., Numerical study of 2 + 1 dimensional sine-Gordon solitons, Physica 2D, 482-494, (1981) · Zbl 1194.65122
[79] Zalesak, S.T.; Ossakow, S.L., Nonlinear equatorial spread F. spatially large bubbles resulting from large horizontal scale initial perturbations, J. geophys. res., 85, 2131-2142, (1980)
[80] Petersen, G.; Bugdor, A.B., The computer language mathsy and applications to solid state physics, Comm. ACM, 23, 466-474, (1980)
[81] Siggia, E., Numerical study of small-scale intermittency in 3D turbulence, J. fluid mech., 107, 375, (1981) · Zbl 0476.76051
[82] Lax, P.D.; Lax, P.D., Periodic solutions of the KdV equation, (), Comm. pure appl. math., 28, 141-148, (1975) · Zbl 0295.35004
[83] Hyman, J.M., Time evolution of almost periodic solutions of the KdV equation, Rocky mountain math. J., 8, 95-104, (1978) · Zbl 0407.35066
[84] Gardner, C.S., Korteweg-de Vries equation and generalizations. IV. the Korteweg-de Vries equation as a Hamiltonian system, J. math. phys., 12, 1548, (1971) · Zbl 0283.35021
[85] Fadeev, L.D.; Zakharov, V.E.; Fadeev, L.D.; Zakharov, V.E., Korteweg-de Vries equation as a completely integrable Hamiltonian system, Funktional anal. priložen, Funct. anal. appl., 5, 280-27, (1971), [Russian] · Zbl 0257.35074
[86] Hirota, R., Phys. rev. lett., 27, 1192, (1971)
[87] Hirota, R., Exact solution of the sine-Gordon equation for multiple collisions of solitons, J. phys. soc. Japan, 33, 1459-1463, (1972)
[88] Zakharov, V.E.; Shabat, A.B.; Zakharov, V.E.; Shabat, A.B., Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media, Zh. eksp. teor. fiz., Soviet phys. JETP, 34, 62, (1972), [Russian]
[89] Kalantarov, V.K.; Ladyzhenskaya, O.A., On the origin of collapse for quasilinear equations of the parabolic and hyperbolic type, (), [Russian] · Zbl 0388.35039
[90] Boussinesq, M.J.; Boussinesq, M.J., Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesse sensiblement pareilles de la surface au fond, C. R. acad. sci. Paris, J. math. phys. (2), 17, 55-108, (1872) · JFM 04.0493.04
[91] Zakharov, V.E., On stochastization of one-dimensional chains of nonlinear oscillators, Sov. phys. JETP, 38, 108-110, (1974)
[92] Berryman, J.; Berryman, J.G., Reply to comments by R. Van dooren, Phys. fluids, Phys. fluids, 22, 1588-777, (1979)
[93] ()
[94] Wadati, M., The exact solution of the modified Korteweg-de Vries equation, J. phys. soc. Japan, 34, 168, (1972)
[95] Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H., The initial value solution for the sine-Gordon equation, Phys. rev. let., 30, 1262-1264, (1973)
[96] Bullough, R.K.; Caudrey, P.R., The soliton and its history, () · Zbl 0399.35023
[97] {\scS. Coleman}, Classical lumps and their quantum description, Lectures at the 1975 International School of Subnuclear Physics, “Ettore Majorana” (A. Zicchici, Ed.)
[98] Jackiw, R., Quantum meaning of classical field theory, Rev. mod. phys., 49, 681-706, (1977)
[99] Fadeev, L.D.; Korepin, V.E., Quantum theory of solitons, Phys rept., 42, (1978)
[100] Luther, A.H., Quantum solitons in statistical physics, ()
[101] Jackson, E.A., Nonlinearity and irreversibility in lattice dynamics, Rocky mountain math. J., 8, 127-196, (1978)
[102] Miura, K., The energy transport properties of one-dimensional anharmonic lattices, ()
[103] Jackson, E.A.; Pasta, J.R.; Waters, J.F., Thermal conductivity of one-dimensional lattices, J. comput. phys., 2, 207-227, (1968)
[104] Zabusky, N.J.; Deem, G.S., J. comput. phys., 2, 126, (1967)
[105] Payton, D.N.; Rich, M.; Visscher, V.M.; Payton, D.N.; Rich, M.; Visscher, V.M., Energy flow in disordered lattices, (), 160, 657-664, (1967), Also
[106] Rich, M.; Visscher, V.M.; Payton, D.N., Phys. rev. A, 4, 1682, (1971)
[107] Batteh, J.H.; Powell, J.D., Effects of solitary waves upon the shock profile in a three dimensional lattice, J. appl. phys., 51, 2050-2058, (1980)
[108] Holian, B.L.; Straub, G.K.; Straub, G.K.; Holian, B.L., Molecular dynamics of shock waves in one-dimensional chains. II. thermalization, Phys. rev. B, Phys. rev. B, 19, 4049-4055, (1979)
[109] Davydov, A.S.; Davydov, A.S.; Davydov, A.S., The role of solitons in the energy and electron transfer in one-dimensional molecular systems, Phys. scripta, Physica 3D, Physica 3D, 20, 1-22, (1981)
[110] Hyman, J.M.; McLaughlin, D.W.; Scott, A.C.; Hyman, J.M.; McLaughlin, D.W.; Scott, A.C., On Davydov’s alpha-helix solitons, Physica 3D, Physica 3D, 23-44, (1981) · Zbl 1194.37175
[111] Ulam, S.M., A collection of mathematical problems, (1960), Wiley-Interscience New York, Chap. VII, Sect. 4a · Zbl 0086.24101
[112] Metropolis, N.; Stein, M.L.; Stein, P.R., On finite limit sets for transformations on the unit interval, J. combin. theory, 15, 1, (1973) · Zbl 0259.26003
[113] May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467, (1976) · Zbl 1369.37088
[114] Feigenbaum, M.J.; Feigenbaum, M.J.; Feigenbaum, M.J., Universal behavior in nonlinear systems, Los alamos science, Los alamos science, Los alamos science, 15-27, (1980), Summer · Zbl 0533.58025
[115] Feigenbaum, M.J., Quantitative universality for a class of nonlinear transformations, J. statist. phys., 19, 25-52, (1978) · Zbl 0509.58037
[116] Feigenbaum, M.J., The onset spectrum of turbulence, Phys. lett. A, 74, 375-378, (1980)
[117] Gollub, J.P.; Benson, S.V., Many routes to turbulent convection, J. fluid mech., 100, 449-470, (1980)
[118] {\scM. J. Feigenbaum}, letter to N. J. Zabusky dated 7 August 1980.
[119] Lorenz, E.N., On the prevalence of aperiodicty in simple systems, Global analysis, (1979)
[120] Lorenz, E.N., The problem of deducing the climate from the governing equations, Tellus, 16, 1-11, (1964)
[121] Saltzman, B., Finite amplitude free convection as an initial-value problem, I, J. atmos. sci., 19, 329-341, (1962)
[122] Freeman, N.C., Soliton interactions in two dimensions, Adv. appl. mech., 20, 1-37, (1980) · Zbl 0477.35077
[123] Kadomtsev, B.B.; Petviashvili, V.I., Sov. phys. dokl., 15, 539, (1970)
[124] Satsuma, J., J. phys. soc. Japan, 40, 286, (1976)
[125] Miles, J.W., J. fluid mech., 79, 157, (1977)
[126] Miles, J.W., J. fluid mech., 171, 171, (1977)
[127] Newell, A.C., Near-integrable systems, nonlinear tunnelling and solitons in slowly changing media, () · Zbl 0411.35008
[128] Yajima, N.; Satsuma, J., J. phys. soc. Japan, 44, 1711, (1978)
[129] Kako, F.; Yajima, N., Interaction of ion-acoustic solitons in two-dimensional space, J. phys. soc. Japan, 49, 2063-2071, (1980) · Zbl 1334.82062
[130] {\scF. D. Tappert}, private communication.
[131] Konno, K.; Suzuki, H., Self focussing of laser beam in nonlinear media, Phys. scripta, 20, 382-386, (1979)
[132] Bialynicki-Birula, I.; Mycielski, J., Gaussons: solitons of the logarithmic schrodinger equation, Phys. scripta, 20, 539-544, (1979) · Zbl 1063.81528
[133] Makhankov, V.G.; Kummer, G.; Shvachka, A.B., Many dimensional U(I) solitons, their interactions, resonances and bound states, Phys. scripta, 20, 454-461, (1979), and references therein · Zbl 1063.81546
[134] Makhankov, V.G., Computer and solitons, Phys. scripta, 20, 560-562, (1979)
[135] Deem, G.S.; Zabusky, N.J.; Deem, G.S.; Zabusky, N.J., Stationary V-states: interaction, recurrence and breaking, (), 40, 277-294, (1978), Also see the extended version
[136] Pierrehumbert, R.T., A family of steady translating vortex pairs with distributed vorticity, J. fluid mech., 99, 129-144, (1980) · Zbl 0473.76034
[137] {\scM. Landau and N. J. Zabusky}, Stationary solutions of the Euler equations in two-dimensions. Singly- and doubly-connected V-states, submitted for publication.
[138] Zabusky, N.J.; Hughes, M.H.; Roberts, K.V., Contour dynamics for the Euler equations in two dimensions, J. comput. phys., 30, 96, (1979) · Zbl 0405.76014
[139] Zabusky, N.J., Contour-dynamics: A boundary integral evolutionary method for incompressible dissipationless flows, (), 503-513 · Zbl 0455.76017
[140] {\scE. A. Overman II and N. J. Zabusky}, Interaction and scattering of dipolar V-states of the Euler equations in two dimensions, manuscript in preparation. · Zbl 0542.76032
[141] Making, M.; Kamimura, T.; Taniuti, T., Dynamics of two-dimensional vortices in a low-β plasma with convective motion, J. phys. soc. Japan, 50, 980-989, (1981)
[142] McWilliams, J.C.; Zabusky, N.J., Interactions of isolated vortices. I. modons colliding with modons, Geophys. astrophys. fluid dyn., (1981) · Zbl 0483.76030
[143] Fleierl, G.R.; Larichev, V.D.; McWilliams, J.C.; Reznick, G.M.; McWilliams, J.C., An application of equivalent modons to atmospheric blocking, Dyn. atmos. oceans, Dyn. atmos. oceans, 5, 43-46, (1980)
[144] Bona, J.L.; Pritchard, W.C.; Scott, L.R., An evaluation of a model equation for water waves, (1981), preprint · Zbl 0497.76023
[145] Zabusky, N.J.; Galvin, C.J., Shallow water waves, the Korteweg-de Vries equation and solitons, J. fluid mech., 47, 811, (1971)
[146] Hammack, J.L., A note on tsunamis: their generation and propagation in an Ocean of uniform depth, J. fluid mech., 60, 769, (1973) · Zbl 0273.76010
[147] Hammack, J.L.; Segur, H., The Korteweg-de Vries equation and water waves. 2. comparison with experiment, J. fluid mech., 65, 289, (1974) · Zbl 0373.76010
[148] Weidman, P.D.; Maxworthy, T., Experiments on strong interactions between solitary waves, J. fluid mech., 85, 417, (1978)
[149] Bona, J., Solitary waves and other phenomena associated with model equations for long waves, (), 77-112, (ISBN 83-01-03012-7)
[150] Chargaff, E., Preface to a grammar of biology, Science, 172, 637-642, (1971)
[151] Pauli, W., The influence of archetypal ideas on scientific theories of Kepler, (), 151-212
[152] See pp. 151-153.
[153] Scott, A.C.; Chu, F.Y.F.; McLaughlin, D.W., The soliton: A new concept in applied science, (), 1443-1483
[154] Toda, M., Studies of a nonlinear lattice, Phys. rep., 18, 1-124, (1975)
[155] Miura, R.M., The Korteweg-de Vries equation: A survey of results, SIAM rev., 18, 412-459, (1976) · Zbl 0333.35021
[156] Makhankov, V.G., Dynamics of classical solitons in nonintegrable systems, Phys. rep. C, 35, 1-128, (1978)
[157] Miles, J.W., Solitary waves, Ann. rev. fluid mech., 12, 11-43, (1980)
[158] Freeman, N.C., Soliton interactions in two-dimensions, Adv. appl. mech., 20, 1-37, (1980) · Zbl 0477.35077
[159] Drinfel’d, V.G.; Krichever, I.M.; Manin, Yu.I.; Novikov, S.P., Methods of algebraic geometry in contemporary mathematical physics, (), Sect. C · Zbl 0534.58001
[160] Chirikov, B.V., A universal instability of many-dimensional oscillator systems, Phys. rep., 52, 263-379, (1979)
[161] Helleman, R.H.G., Self-generated chaotic behavior in nonlinear mechanics, (), 165-233
[162] Makhankov, V.G., Computer experiments in soliton theory, (), 1-49
[163] Simonov, Yu.A.; Tjon, J.A., Inelastic effects in classical field-theoretical models with confinement, Ann. of phys., 129, 110-130, (1980)
[164] Karpman, V.I., Nonlinear waves in dispersive media, (), published by · Zbl 0972.35519
[165] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley-Interscience New York · Zbl 0373.76001
[166] Zakharov, V.E.; Manakov, S.V.; Novikov, S.P.; Pitaevskii, L.P., Theory of solitons. the method of the inverse problem, (), 320, English translation to be published · Zbl 0598.35003
[167] Lamb, G.L., Elements of soliton theory, (1980), Wiley-Interscience New York · Zbl 0445.35001
[168] Toda, M., Theory of nonlinear lattices, () · Zbl 0465.70014
[169] Calogero, F.; Degasperis, A., Spectral transform and solitons: tools to investigate and solve nonlinear evolution equations, (1981), North-Holland Amsterdam · Zbl 0485.35076
[170] Ablowitz, M.J.; Segur, H., Solitons and the inverse scattering transform, (1981), SIAM · Zbl 0299.35076
[171] Eilenberger, G., Solitons: mathematical methods for physicists, (1981), Springer New York · Zbl 0455.35001
[172] Burt, P.B., Quantum mechanics and nonlinear waves, (1981), Harwood/Academic Press New York
[173] Eckhaus, W.; van Harten, A., The inverse scattering transformation and the theory of solitons, () · Zbl 0463.35001
[174] {\scR. Dodd, C. Eilbeck, J. Gibbon, and H. Morris}, “Solitons and Nonlinear Wave Equations,” Academic Press, New York, in press. · Zbl 0496.35001
[175] ()
[176] ()
[177] ()
[178] ()
[179] ()
[180] ()
[181] ()
[182] ()
[183] ()
[184] ()
[185] (), 1-438
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.