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Il Nuovo Cimento B (1971-1996)

, Volume 32, Issue 2, pp 201–242 | Cite as

Nonlinear evolution equations solvable by the inverse spectral transform.—I

  • F. Calogero
  • A. Degasperis
Article

Summary

This paper is the first of a series based on a general method to discover and investigate nonlinear partial differential equations solvable via the inverse spectral transform technique. The results of this paper are those that obtain applying this method to the generalized Zakharov-Shabat linear problem. We give a class of nonlinear evolution equations solvable by the inverse spectral transform, that is more general than that introduced by Ablowitz, Kaup, Newell and Segur because it includes equations involving more than one space variable and containing coefficients that are not constant. We also report a very general class of Bäcklund transformations that includes all such transformations previously considered and clarifies their significance. And we produce, for a somewhat less general class of nonlinear evolution equations (involving only one space variable), a remarkable functional equation that relates the solution at timet to the same solution at timet′. This paper is focussed on a general presentation of the approach and the proof of the main results (some of which had been previously reported without proof). Although the analysis of special equations and special solutions is deferred to subsequent papers of this series, there are here also a few results of this kind, including the explicit display of the exact nonsoliton solution of the sine-Gordon equation corresponding to a double pole of the associated spectral parameter.

Нелинейные уравнения эволюции, решаемые с помощье с помощью обратного спектрального преобразования—I

Резюме

Эта статья является первой стаьей из серии, основанной на общем методе для исслеования иелинейных дифференциальных уравнений в частных производных, рещаемых с помощью техники обратного спектрального преобразования. Результаты, полученные в этой статье, аналогичны результатам, которые получаются при применении зтого метода к обобщенной линейной проблеме захарова-Шабата. Мы приводим класс неинейных уравнений эволюции, решаемых с помошью обратного спектрального преобразования. Этот класс является более общим, чем класс, введенный Абловитцем, Каупом, Невеллом и Сегуром, т.к. он содержит уравнения, включающие более чем одну пространственную переменную и содержащие коэффициенты, которые не являются постоянными. Мы также рассматриваем очень общий класс преобразований Беклунда, который содержит все такие преобразования, которые были рассмотрены ранее. Проводится анализ физического смысла зтих преобразований. Для случая менее общего класса нелинейных уравнений эволюции (включающего только одну пространственную переменную) мы получаем функциональное уравнение, которое связывает рещение в момент времениt с тем же рещением в момент времениt′. №сновное внимание в статье уделяется общему подходу и доказательству основных результатов (некоторые из которых были приведены ранее без доказательств). Хотя анализ специальных уравнений и специальных рещений отложен на последующие статьи этоь серии, в этой работе приводится несколько результатов такого рода, которые включают точное несолитонное рещение уравнения Гордона, соответсующего двойному полюсу ассоциированного спектрального параметра.

Riassunto

Questo lavoro è il primo di una serie dedicata ad un metodo generale per trovare e studiare equazioni non lineari alle derivate parziali risolubili per mezzo della tecnica della trasformata spettrale inversa. In questo articolo si presentano i risultati che si ottengono applicando questo metodo al problema lineare generalizzato di Zakharov-Shabat. Si dà una classe di equazioni di evoluzione nonlineari, solubili con la trasformata, spettrale inversa, che è più generale di quella presentata da Ablowitz, Kaup, Newell e Segur, poiché si includono anche equazioni contenenti coefficienti non costanti e più di una variabile spaziale. Riportiamo inoltre una classe molto generale di trasformazioni di Bäcklund che contiene tutte le trasformazioni già note e ne chiarisce il significato. Infine otteniamo, per una classe più ristretta di equazioni nonlineari di evoluzione (contenenti solo una variabile spaziale), un’interessante equazione funzionale che lega la soluzione al tempot alla stessa soluzione al tempot′. Questo articolo è dedicato ad una presentazione generale del metodo ed alla dimostrazione dei risultati principali (alcuni dei quali sono già stati pubblicati senza dimostrazione). Sebbene l’analisi di equazioni particolari e di soluzioni speciali è rimandata ai lavori successivi di questa serie, alcuni risultati di questo tipo sono già presenti in questo lavoro, tra i quali l’espressione esplicita della soluzione esatta, non di tipo solitone, dell’equazione sine-Gordon, che corrisponde ad un polo doppio dei corrispondenti parametri spettrali.

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References

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Presumably another useful reference, that we have however not yet been able to consult, isNonlinear Wave Motion, edited byA. C. Newell,lectures in Applied Math.,15 (Providence, R. I., 1974). Results for Bäcklund transformations have been obtained and discussed, for some special equations (KdV, modified KdV, nonlinear Schrödinger, sine-Gordon), by AKNS and by many others; see, for instance,M. Wadati, H. Sanuki andK. Konno:Prog. Theor. Phys.,53 419 (1975), the papers of ref. (4) We list again only a few contributions, particularly significant in the context of this paper:H. D. Wahlquist andF. B. Estabrook:a)Phys. Rev. Lett.,31, 1386 (1973);b)Journ. Math. Phys.,16, 1 (1975);c)D. W. McLaughlin andA. C. Scott:Journ. Math. Phys.,14, 1817 (1973);d)G. L. Lamb jr.:Journ. Math. Phys.,15, 2157 (1974);e)F. Calogero:Lett. Nuovo Cimento,14, 537 (1975); see also the papers of ref. (3) Out of the extensive literature on this topic we list here only the most significant contributions, selected on the basis of their review nature, their landmark character or their technical closeness to the approach of this paper:a)A. C. Scott, F. Y. F. Chu andD. W. McLaughlin:Proc. IEEE,61, 1443 (1973);b)G. B. Whitham:Linear and Nonlinear Waves (New York, N. Y., 1974);c)J. Moser, Editor:Dynamical Systems, Theory and Applications (Berlin, 1974) (see in particular the papers byM. Kruskal and byH. Flaschka andA. C. Newell);d)P. D. Lax:Comm. Pure Appl. Math.,21, 467 (1968);e)V. E. Zakharov andL. D. Faddeev:Func. Anal. Appl.,5, 280 (1971);f)V. E Zakharov andA. B. Shabat:Sov. Phys. JETP,34, 62 (1972);g)M. J. Ablowitz, D. J. Kaup, A. C. Newell andH. Segur:Stud. Appl. Math.,53, 249 (1974), hereafter referred to as AKNS.;h)F. Calogero:Lett. Nuovo Cimento,14, 443 (1965);i)T. Kotera andK. Sawada:Journ. Phys. Soc. Japan,39, 501 (1975). 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Math.,53, 249 (1974), hereafter referred to as AKNS;h)F. Calogero:Lett. Nuovo Cimento,14, 443 (1965);i)T. Kotera andK. Sawada:Journ. Phys. Soc. Japan,39, 501 (1975). Presumably another useful reference, that we have however not yet been able to consult, isNonlinear Wave Motion, edited byA. C. Newell,Lectures in Applied Math.,15 (Providence, R. I., 1974). After this paper was partially drafted (and the two papers of ref. (16)F. Calogero andA. Degasperis:a) Phys. Rev. Lett (submitted to);b) Lett. Nuovo Cimento,15, 65 (1976) had been submitted for publication) we received a preprint byH. Flaschka andD. W. McLaughlin (Some comments on Bäcklund, transformations, canonical transformations and the inverse scattering method, to be published) that takes a point of view similar to that of this paper, and reports some results that coincide with special cases of those given here.Note added in proofs.—The fact that the same Bäcklund transformation applies to all the equations of the AKNS class had been previously noted byH. H. Chen:Phys. Rev. Lett.,33, 925 (1974) (but the only consideredt he simple Bäcklund transformations that are included in the class of eq. (4.2.13a) below, since the more general Bäcklund transformations introduced here, eq. (4.2.1), were not known, nor their spectral significance, eq. (4.2.2), understood).MathSciNetADSCrossRefGoogle Scholar
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Copyright information

© Società Italiana di Fisica 1976

Authors and Affiliations

  • F. Calogero
    • 1
    • 2
  • A. Degasperis
    • 1
    • 2
    • 3
  1. 1.Istituto di Fisica dell’UniversitàRomaItalia
  2. 2.Sezione di RomaIstituto Nazionale di Fisica NucleareRomaItalia
  3. 3.Istituto di Fisica dell’UniversitàLecce

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