Summary
The spectral representation of the first conserved densities of the nonlinear Schrödinger equation is used in order to obtain a thermodynamical description of the Madelung fluid associated with its soliton solution. In this framework, the processes associated with soliton evolution under slow variations of the parameters characterizing the system are discussed and their properties tested in numerical experiments.
Riassunto
La rappresentazione spettrale delle prime densità conservate dell’equazione non lineare di Schrödinger viene utilizzata per ottenere una descrizione termodinamica del fluido di Madelung associato alle sue soluzioni solitoniche. Nell’ambito di questo schema i processi associati ad un’evoluzione lenta di un solitone sottoposto a lente variazioni dei suoi parametri caratteristici vengono esaminati e le loro proprietà verificate mediante esperimenti numerici.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. A. Minelli andA. Pascolini:Nuovo Cimento B,85, 1 (1985).
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).
For a review of Madelung fluid seeE. A. Spiegel:Physica (Utrecht) D,1, 236 (1980);R. W. Hasse:Z. Phys. B,37, 83 (1980), and references therein quoted.
For the breaking of the conservation laws consequent to a perturbation of the original evolution equation and for the corresponding slow evolution of the eigenvalues of the other motion constants seeV. I. Karpman andE. Maslov:JETP,46, 281 (1977);D. J. Kaup andA. C. Newell:Proc. R. Soc. London, Ser. A,361, 413 (1978);S. Watanabe, M. Miyakawa andN. Yajima:J. Phys. Soc. Jpn.,46, 1653 (1979);M. R. Gupta andJ. Ray:J. Math. Phys. (N. Y.),22, 2180 (1981);P. Lochak:J. Math. Phys. (N. Y.),25, 2472 (1984);M. N. Bussac, P. Lochak, C. Meunier andA. Heron-Gourdin:Physica (Utrecht) D,17, 313 (1985);R. K. Bullough, A. P. Fordy andS. V. Manakov:Phys. Lett. A,91, 98 (1982).
The deformation of the solitons of the NSE due to the slow variations of the nonlinearity has been discussed inR. Grimshaw:Proc. R. Soc. London, Ser. A,368, 377 (1979).
The behaviour of the solitons of the nonlinear Schrödinger equations under forcing fields has been extensively studied. SeeI. Bialynicki-Birula andJ. Mycielski:Ann. Phys. (N. Y.),100, 62 (1976);H. H. Chen andC. S. Liu:Phys. Rev. Lett.,37, 693 (1976);Phys. Fluids,21, 377 (1977);P. D. Gianino andB. Bendow:Phys. Fluids,23, 220 (1979);T. A. Minelli andA. Pascolini:Lett. Nuovo Cimento,27, 413 (1980);R. W. Hasse:Phys. Rev. A,25, 583 (1982);Q. P. J. Eboli andJ. Marques:Phys. Rev. B,28, 689 (1983);T. A. Minelli andA. Pascolini:Physica (Utrecht) D,11, 227 (1984);G. Burdet andM. Perrin: preprint CNRS-Luminy CPT-85/P.1170.
The development of the numerical method is essentially due toA. Pascolini.
A. Goldberg, H. M. Schey andJ. L. Schwartz:Am. J. Phys.,35, 177 (1967).
M. Delfour, M. Fortin andG. Payre:J. Comput. Phys.,44, 277 (1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bon, P., Galaverni, M., Minelli, T.A. et al. Thermodynamical description of the solutions of the nonlinear Schrödinger equation under slow perturbations. Nuov Cim A 97, 385–418 (1987). https://doi.org/10.1007/BF02734465
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02734465