Summary
A kinetic interpretation of the conservation laws of a class of completely integrable nonlinear evolution equations is obtained by introducing a distribution function depending on the spectral parameter of the inverse spectral transform of the equations. The formalism so introduced leads to a possible thermodynamical description of nonlinear and dispersive phenomena and has been applied in order to investigate the properties of the Madelung fluid associated to a nonlinear Schrödinger equation.
Riassunto
Attraverso una funzione di distribuzione dipendente dal parametro spettrale si dà una interpretazione cinetica delle leggi di conservazione di una classe di equazioni d’evoluzione non lineari, completamente integrabili. Il formalismo introdotto, che conduce naturalmente ad una descrizione termodinamica dei fenomeni dispersivi e non lineari, è stato impiegato nello studio delle proprietà del fluido di Madelung associato ad una equazione di Schrödinger non lineare.
Резюме
Предлагается кинетическая интерпретация законов сохранения для класса полностью интегрируемых нелинейных уравнений эволюции, вводя функцию распределения, зависящую от спектрального параметра обратного спектрального преобразования уравнений. Предложенный формализм приводит к возможному термодинамическому описаний нелинейных и диссипативных явлений и приименяется для исследования жидкости Маделунга, связанной с нелинейным уравнением Шредингера.
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References
E. Fermi, J. Pasta andS. Ulam: Los Alamos report LA-1940 (1955);E. Fermi:Note e Memorie, Vol.2, (Roma-Chicago, Ill., 1965), p. 978.
Preliminary results were presented at theSecond and Third Italian Informal Meeting on Nonlinear Phenomena (Firenze, May 1983-Amantea, May 1984).
See, for instance,B. Mielnik:Commun. Math. Phys.,37, 221 (1974);R. Haag andU. Bannier:Commun. Math. Phys.,60, 1 (1978);T. W. Kibble:Commun. Math. Phys.,64, 73 (1978);F. M. Giusto, T. A. Minelli andA. Pascolini:Physica D,11, 227 (1984).
SeeD. Bohm:Hidden Variables in the Quantum theory, inQuantum Theory, Vol.3, edited byD. R. Bates (New York, N. Y., 1962).
E. Nelson:Phys. Rev.,150, 1079 (1966);P. Guerret andJ. P. Vigier:Found. Phys.,12, 1057 (1982). The connection between the kinetic and the stochastic interpretation of NSE will be discussed byT. A. Minelli andL. Morato in a fortheoming paper.
A. S. Davydov:Physica D,3, 1 (1981).
V. E. Zakharov andL. A. Taktadzhyan:Theor. Math. Phys.,38, 17 (1979).
V. E. Zakharov andA. B. Shabat:Sov. Phys. JETP,34, 62 (1972).
M. J. Ablowitz, D. J. Kaup, A. C. Newell andH. Segur:Stud. Appl. Math.,53, 249 (1974).
SeeC. Cercinani:Theory and Applications of the Boltzmann Equation (Edinburgh, 1975), p. 84.
A. C. Newell:The Inverse Scattering Transform, inSolitons, edited byB. K. Bullough andP. J. Caudrey (Berlin, 1980). See alsoH. Flaschka andA. C. Newell:Lect. Notes Phys.,38, 355 (1974).
R. K. Dodd andR. W. Bullough:Phys. Scr.,20, 514 (1979). See also,R. K. Dodd, J. C. Eilbeck, J. D. Gibbon andH. C. Morris:Solitons and Nonlinear Wave Equations (London, 1982), Chapt. VI.
M. Wadati, H. Sanuki andK. Konno:Prog. Theor. Phys.,53, 419 (1975).
For an up-to-date review of the Madelung fluid seeE. A. Spiegel:Physica D,1, 236 (1980);R. W. Hasse:Z. Phys. B,37, 83 (1980), and references quoted therein. See alsoT. F. Nonnenmacher, G. Dukek andG. Baumann:Lett. Nuovo Cimento,36, 453 (1983);T. F. Nonnenmacher andJ. D. F. Nonnenmacher:Lett. Nuovo Cimento,37, 241 (1983).
SeeC. Truesdell andR. G. Muncaster:Fundamentals of Maxwell’s Kinetic Theory (New York, N.Y., 1980).
For the classical kinetic interpretation of caloric entropy besides ref. (15) SeeC. Truesdell andR. G. Muncaster:Fundamentals of Maxwell’s Kinetic Theory (New York, N.Y., 1980), seeH. J. Kreuzer:Nonequilibrium Thermodynamics (Oxford, 1981).
An information entropy for the Schrödinger equation was introduced inI. Bialynicki-Birula andJ. Mycielski:Commun. Math. Phys.,44, 129 (1975). See alsoE. R. Caianiello:Lett. Nuovo Cimento,32, 65 (1981).
The present calculations have been performed by integrating the NSE with a finite-difference method which will be discussed byC. Buttazzoni, F. Degan andA. Pascolini in a forthcoming paper. For other recent numerical experiments on the Madelung fluid seeC. Dewdney andH. Hiley:Found. Phys.,12, 27 (1982).
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Minelli, T.A., Pascolini, A. Kinetic and thermodynamical interpretation of the conservation laws of a nonlinear Schrödinger equation and other completely integrable systems. Nuov Cim B 85, 1–16 (1985). https://doi.org/10.1007/BF02721517
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DOI: https://doi.org/10.1007/BF02721517