Lecture Notes in Physics

Transcription

1 Lecture Notes in Physics Edited by J. Ehlers, Miinchen, K. Hepp, Zijrich R. Kippenhahn, Miinchen, H. A. Weidenmiiller, Heidelberg and J. Zittartz, Kijln Managing Editor: W. Beiglbijck, Heidelberg 120 Nonlinear Evolution Equations and Dynamical Systems Proceedings of the Meeting Held at the University of Lecce June 20-23,1979 Edited by M. Boiti, F. Pempinelli, and G. Soliani Springer-Verlag Berlin Heidelberg New York 1980

2 Editors M. Boiti F. Pempinelli G. Soliani UniversitA degli Studi di Lecce lstituto di Fisica Lecce Via Arnesano Italy ISBN Springer-Verlag Berlin Heidelberg New York ISBN o Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 5 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 0 by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/

3 C O N T E N T S I. F. Calogero NONLINEAR EVOLUTION EQUATIONS SOLVABLE BY THE SPECTRAL TRANSFORM: SOME RECENT RESULTS B. Kostant QUANTIZATION OF COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS A. Degasperis REDUCTION TECHNIQUE FOR MATRIX NONLINEAR EVOLUTION EQUATIONS M. Boiti F. Pempinelli SIMILARITY SOLUTIONS OF THE KORTEWEG- DE VRIES EQUATION, B~CKLUND TRANSFORMATIONS AND PAINLEVE TRANSCENDENTS A lacob S. Sternberg COADJOINT STRUCTURES, SOLITONS, AND INTEGRABILITY P.C. Sabatier AROUND THE CLASSICAL STRING PROBLEM D.V. Chudnovsky THE GENERALIZED RIEMANN-HILBERT PROBLEM AND THE SPECTRAL INTERPRETATION G.V. Chudnovsky THE INVERSE SCATTERING PROBLEM AND APPLICATIONS TO ARITHMETICS, APPROXIMATION THEORY AND TRANSCENDENTAL NUMBERS D. ter Haar THE SOLITON THEORY OF STRONG LANGMUIR TURBULENCE F.A.E. Pirani GEOMETRY OF B~CKLUND TRANSFORMATIONS... 2]2 1. J. Ginibre G. Velo EXISTENCE OF SOLUTIONS AND SCATTERING THEORY FOR THE NONLINEAR SCHRODINGER EQUATION F. Magri A GEOMETRICAL APPROACH TO THE NONLINEAR SOLVABLE EQUATIONS

4 13. M. Leo R.A. Leo G. Soliani PROPERTIES OF A SPECIAL FUNCTION RELATED TO SELF-SIMILAR SOLUTIONS OF CERTAIN NON- LINEAR WAVE EQUATIONS J. Burbea VORTEX MOTIONS AND CONFORMAL MAPPINGS B. Savo ANALYTICAL SOLUTIONS OF THE SINE-GORDON EQUATION AND THEIR APPLICATION TO JOSEPHSON TUNNEL JUNCTIONS R. Sasaki R.K. Bullough GEOMETRY OF THE AKNS-ZS INVERSE SCATTERING SCHEME S.N. Bagchi FUNDAMENTAL EQUATIONS OF CLASSICAL AND OF QUANTUM PHYSICS AS SPECIAL CASES OF A NON- LINEAR EQUATION M.J. McGuinness NOETHER'S THEOREM AND INFINITIES OF POLYNOMIAL CONSERVED DENSITIES

5 FOREWORD This volume contains the Proceedings of the Meeting on "Nonlinear Evolution Equations and Dynamical Systems" held at the University of Lecce (Italy) from June 2Oth-23rd, The main emphasis of the meeting was on recent advances and new trends in nonlinear evolution equations, their application to physics and the mathematical development which has followed from their study. These research areas have developed rapidly over the last decade from the pioneering period, when solitons were discovered, to a much more professional and specilized stage. The collection of papers in this volume is a representative selection of work carried out recently. These papers are an expanded version of must of the talks given at the meeting and include many additional details and results not presented then. We are particularly pleased to thank the authors of these papers for their hard work and cooperation in preparing the manuscripts and, in general, to all partecipants whose confidence in the meeting made its success possible. The International Scientific Committee of the meeting was composed of Proff. R.K.Boullough, F.Calogero, G.F.Dell'Antonio, F.Pirani and T. Regge. We would like to thank each of them for their collaboration. We are especially grateful to Prof.Calogero for his helpful advice in the early stages of the organization of the meeting. Financial support from the University of Lecce, from the Italian National Research Council (CNR), from the Banca Piccolo Credito Salentino and from the Banca Fratelli Vallone made the meeting possible. Contributions from the Banca Agricola Popolare di Matino e Lecce, Banca Popolare di Parabita, Banca Venturi, Banca Commerciale Italiana (Lecce) and Banca dei Paschi di Siena (Lecce) were equally important in meeting

6 VI our most immediate financial obbligations. The social events were highlighted by a visit to Otranto and the coast of Salento organised by the Ente Provinciale del Turismo of Lecce (the ufficial tourist organization). We would like to thank all the Organizations and Institutions men- tioned above for having contributed to the Meeting's success. Finally we want to express here our thank and appreciation to Proff. M.Leo, R.A.Leo, A.Luches and P.Rotelli of the Local Organizing Committee for their aid in overcoming all kinds of serious difficulties, to the secretaries Mrs. G Mazzotta and Mrs. A.Rossi for their invaluable help in the preparation and organization of the meeting and to Mr. C. Laddomada for his excellent typing of the photo-ready copy of the manuscripts. Lecce, January 1980 M. Boiti F. Pempinelli G. Soliani NOTE. In order to speed up the pubblication of the Proceedings most of the proofs have not been revised by the authors.