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.
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.
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.
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References
Throughout this paper whenever we mention NLPDE’s we include also the possibility that these be integro-differential equations (that may, or may not, reduce to pure partial differential equations, possibly by an appropriate redefinition of the dependent variable).
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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). 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).
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Throughout this paper we occasionally differ, to streamline our presentation, from the notation used previously.
The literature on the sine-Gordon equation and its applications is large (see,e.g., ref. (3a) ; its complete solution was first given byM. J. Ablowitz, D. J. Kaup, A. C. Newell andH. Segur:Phys. Rev. Lett.,30, 1262 (1973); and byL. D. Faddeev andL. A. Takhtajan: Commun. JINR Dubna, E2-7998 (1974). See alsoD. J. Kaup:Stud. Appl. Math.,54, 165 (1975).
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F. Calogero:Nuovo Cimento,29 B, 509 (1975). See also the paper by the same author in the forthcoming Festschrift in honor ofV. Bargmann, edited byB. Simon andA. S. Wightman.
F. Calogero andA. Degasperis:a) Phys. Rev. Lett (submitted to);b) Lett. Nuovo Cimento,15, 65 (1976).
Indeed, even in the case with one space variable only, only a more limited class of NLPDE’s than the one reported here (and in ref. (16b)Lett. Nuovo Cimento,15, 65 (1976)) had been identified as solvable by the IST associated with the multichannel Schrödinger equation (7).
The difference between the integral operatorsI_ andI, eqs. (1.4) and (1.9), compensates exactly the sign differences between the definitions ofL_ andL, eqs. (1.3) and (1.8); see the appendix.
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. and some of 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). 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.
A preprint byD. J. Kaup:the closure of the squared Zakharov-Shabat eigenstates (to appear inJourn. Math. Anal. Appl.) also takes a somewhat similar point of view to that of this paper.
This point is ignored in-ref. (7),, where the interested reader may find the explicit connection between the differential equations of the two problems.
The significance of such a restriction is well understood in the context of the usual Schrödinger scattering problem; see, for instance,F. Calogero andJ. R. Cox:Nuovo Cimento,55 A, 786 (1968). In the present context the limitation is not a serious one, but it deserves a separate discussion, in view of its relevance for soliton solutions (that are, however, already included in the present treatment; see below).
For an outline of the difficulties that might originate from this formal step we refer to previous works, such as-ref. (3g,3h), ; postponing a more detailed discussion to subsequent papers of this series.
A condition, that we have not, for simplicity, mentioned previously (16),F. Calogero andA. Degasperis:a)Phys. Rev. Lett (submitted to);b) Lett. Nuovo Cimento,15, 65 (1976), but that is clearly implied by (4.1.10), is that γ andv have the same singularity structure in the finite part of the complexz-plane. Let us however also mention at this point that the requirement that these be entire (or ratios of entire) functions is sufficient, but not necessary, for the validity of all these results, that might indeed also hold for nonentire functions provided a suitable definition is given of the operator that obtains after replacing the argument of such a function by an operator.
The basic idea that is used to extract the conserved quantities is a fairly old one in potential scattering theory, that may be traced back to papers byN. Levinson, R. G. Newton andL. D. Faddeev; see, for instance,F. Calogero andA. Degasperis:Journ. Math. Phys.,9, 90 (1968).
The relation (4.2.3) is a generalized version of the formulae often referred to in the literature as «one half» of a Bäcklund transformation: see 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). and, more specifically, those of ref. (4,19,20) 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). 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). 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). and some of 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). 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).
Note that, since in this case bothα (±) andα (±)′ vanish, we are in fact extrapolating our results to a case in which the discrete spectrum cannot be obtained by analytic continuation from the alphas. A discussion of this point is deferred to a subsequent paper, as well as a more detailed analysis of this «soliton» solution when a nontrivial γ-dependence is present (in which case in general it does not behave like a soliton at all).
Write explicitly the linear Bäcklund transformation for the fields (using eqs. (4.2.3) and (4.2.13a)), introduce the function\(w(x) - \int\limits_x^\infty {dx' q^2 (x')} \) (andw′(x), similarly related toq′(x)), solve forw′−w in terms ofq′+q (choosing appropriately the sign in the solution of the second-degree equation), differentiate, simplify, and finally integrate using the asymptotic boundary conditionsQ(+∞)=Q′(+∞)=q(+∞)=q′(+∞)=0.
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Calogero, F., Degasperis, A. Nonlinear evolution equations solvable by the inverse spectral transform.—I. Nuov Cim B 32, 201–242 (1976). https://doi.org/10.1007/BF02727634
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DOI: https://doi.org/10.1007/BF02727634