Summary
This paper is the second of a series based on a general method to discover and investigate nonlinear evolution equations (NEEs) solvable by the inverse spectral transorm (IST); the results reported are those that obtain by applying this method to the one-dimensional matrix Schrödinger (linear) problem. We give a class of NEEs solvable by IST that is more general than that introduced previously by Wadati and Kamijo, even in the simpler case with only one space variable and constant coefficients; moreover, we introduce classes of NEEs involving more than one space variable and containing coefficients that are not constant. We also introduce and discuss a very general class of Bäcklund transormations (BTs), and derive a number of results (nonlinear superposition principle, multisoliton, ladder, conserved quantities, generalized resolvent formula) implied by them; and we clarify the relationship between BTs and the IST technique. The most remarkable aspect of the results presented in this paper is the discovery that the solitons associated to these NEEs, although possessing all the stability properties that characterize solitons, generally do not move with constant speed (even in the simpler case with one space variable only, that is, discussed in greater detail). This paper is focused on a general presentation of the approach and on the proofs of the results (some of which had been previously reported without proofs); but we also discuss the simpler NEE belonging to this class, and we present some related novel results, including its Lagrangian and the display of some special cases, that constitute interesting novel examples of fairly simple NEEs solvable by IST.
Riassunto
Questo articolo è il secondo di una serie basata su un metodo generale per scoprire e studiare equazioni nonlineari di evoluzione (ENE) risolubili mediante la trasformata spettrale inversa (TSI); i risultati che qui si riportano sono quelli che si ottengono applicando questo metodo al problema unidimensionale (lineare)matriciale di Schrödinger. Si ottiene una classe di ENE risolubili più ampia di quella introdotta precedentemente da Wadati e Kamijo, anche nel caso più semplice con una sola variabile spaziale e coefficienti costanti; si introducono inoltre classi di ENE con più di una variabile spaziale e con coefficienti non costanti. Si introduce e discute una classe assai generale di trasformazioni di Bäcklund (TB), e si ricavano da queste TB numerosi risultati: principio di sovrapposizione non lineare, sequenze di soluzioni multisolitoniche, leggi di conservazione, formula risolvente generalizzata; e si chiarisce la relazione fra TB e TSI. L'aspetto più notevole dei risultati presentati in questo lavoro è la scoperta che i solitoni associati a queste ENE, pur possedendo tutte le proprietà di stabilità che caratterizzano i solitoni, generalmente non si muovono di moto uniforme (anche nel caso più semplice con una sola variabile spaziale, che è discusso nel maggior dettaglio). Questo articolo è prevalentemente dedicato ad una presentazione generale del metodo e alla dimostrazione dei risultati (alcuni dei quali erano stati precedentemente pubblicati senza dimostrazione); si discute però anche la più semplice ENE che appartiene a questa classe, e si presentano per essa alcuni risultati nuovi: l'espressione della funzione Lagrangiana da cui tale equazione può essere dedotta, ed alcuni sottocasi, che costituiscono a loro volta nuovi interessanti esempi di ENE risolubili mediante TSI.
Резюме
Эта статья является второй из серии, посвященной исследованию нелинейных уравнений эволюции, рещаемых с помощью обратного спектрального преобразования. Опубликованные результаты касаются применения этого метода к одномерной матричиой (линейной) проблеме Шредингера. Мы приводим класс нелинейных уравнений эволюции, решаемых с помощью обратного спектрального пребразования, который является более общим, чем класс, введенный ранее Вадати и Камихо, даже в простейшем случае с одной пространственной переменной и постоянными коэффициентами. Кроме того, мы вводим классы нелинейных уравнений эволюции, включающих более одной пространственной переменной и содержащих коэффициенты, которые не являются постоянными. Мы также вводим и обсуждаем общий класс преолразований Бэклунда, с помощью которых получается ряд результатов (нелинейный принцип суперпозиции, многосолитонная ластница, сохраняющиеся величины, формула для обобщенной резольвенты). Мы обсуждаем связь между преобразованиями Бэклунда и техникой обратного спектрального преобразования. Наиболее знаменательный аспект полученных в этой работе результатов состоит в том, что солитоны, связанные с этими нелинеиными уравнениями эволюции, хотя и обладают всеми свойствами устойчивости, присущими солитонам, в общем случае не движутся с постоянной скоростью (даже в более простом случае только с одной пространственной переменной, который обсуждается подробно). Эта работа посбящена общей формулировке подхода и доказательству результатов (некоторые из которых были приведены ранее без доказательств). Также обсуждаются более простые нелинейные уравнения эволюции, принадлежащие этому классу. Мы приводим некоторые новые результаты, включающие Лагранжиан. Рассматриваются частные случаи, которые представляют интересные примеры довольно простых нелинейных уравнений эволюции, решаемых с помощью обратного спектрального преобразования.
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References
F. Calogero andA. Degasperis:Nuovo Cimento,32 B, 201 (1976) (hereafter referred to as I).
Here, and always in the following, a subscripted variable indicates partial differentiation.
V. E. Zakharov andA. B. Shabat:Funk. Anal. Appl.,8, 43 (1974).
See the review paper byV. E. Zakharov in the forthcoming Springer monograph edited byR. K. Bullough.
F. Calogero:Lett. Nuovo Cimento,14, 537 (1975).
F. Calogero andA. Degasperis:Lett. Nuovo Cimento,16, 181 (1976).
M. Wadati andT. Kamijo:Prog. Theor. Phys.,52, 397 (1974).
F. Calogero andA. Degasperis:Lett. Nuovo Cimento,15, 65 (1976) (hereafter referred to as L).
It should be emphasized that this happens even though the equations are translation invariant. This phenomenon is therefore essentially different from that discovered, in a special case, byH. H. Cheng andC. S. Liu:Solitons in nonuniform media, Maryland preprint; and byR. Hirota andJ. Satsuma:N-soliton solution of the KdVequation with loss and nonuniformity terms, preprint, quoted byM. Wadati andM. Ablowitz, that can be indeed eliminated by appropriate changes of (dependent and independent) variables (private communication byM. Wadati andM. Ablowitz).
F. Calogero andA. Degasperis:Lett. Nuovo Cimento,16, 425, 434 (1976) (hereafter referred to as B1 and B2).
Z. S. Agranovich andV. A. Marchenko:The Inverse Problem of Scattering Theory (translated from the Russian byB. D. Seckler) (New York, N. Y., 1963); note, however, that this book concentrates on the problem within the interval 0≤x<∞ (that obtains from the 3-dimensional Schrödinger equation with spherically symmetrical potentials), rather than the one-dimensional Schrödinger problem. For a treatment of the problem within the interval—∞<x<∞, see ref. (7).M. Wadati andT. Kamijo:Prog. Theor. Phys.,52, 397 (1974).
Throughout this paper we assume, for the sake of simplicity, the discrete eigenvalues to be nondegenerate. Results for the (exceptional) case of degenerate eigen-values can be easily recovered by a suitable limiting process (coalescence of two, or more, nondegenerate eigenvalues).
IfQ(x) vanishes asymptotically exponentially, as it is suggested by (2.2.3), the formula (2.2.9) may, but need not, hold. For a detailed treatment of this phenomenon in the simpler context of the more usual single-channel potential scattering theory, seeF. Calogero andJ. R. Cox:Nuovo Cimento,55 A, 786 (1968); the situation in the present context is analogous.
For a detailed treatment (in the single-channel case) see the paper byF. Calogero;Nuovo Cimento,31 B, 229 (1976).
It should be emphasized that in this formula, as well as in all the similar ones given below, the operators never act on the wave functions, even though one of these is always written on the r.h.s (and could not be written on the l.h.s., since one is generally dealing with noncommuting matrices).
For analogous treatments see I and ref. (16).
There is an obvious misprint in the definition ofG given by eq. (3) of L.
Here we assume thatQ is Hermitian (it is this condition that constrains the discrete eigenvaluesk (j)to occur on the positive imaginary axis); but the results given below are valid more generally.
Note that, even when it enters as the argument of a function being integrated overt′, p is evaluated at timet, nott′.
And it does depend on its first argument; otherwise they-dependence is only apparent and can be gotten rid of by an appropriate redefinition of the time variable.
It is actually easy to obtain in this manner then-soliton solution.
Note that in this case the projectorsE karet-independent, as well as the quantitiese k.
This term was actually introduced by us in the study of a special NEE, that need not have the property of eq. (4.5.14); see subsect.4·7.
Eqs. (5) of B2 obtain from this formula in the special caseN=2 (and by using the notation introduced in footnote (4) See the review paper byV. E. Zakharov in the forthcoming Springer monograph edited byR. K. Bullough of B1).
One of us (F.C.) would like to acknowledge useful discussions about this result with the partecipants to the «soliton» meeting at theCentre de Physique Theorique of the CNRSin Marseille, July 5–9, 1976, and in particular withR. Stora.
Here and below the indices refer to the Cartesian components of the 3-vectorsa, b andV; of course the theorem can be trivially extended by performing a common rotation on the vectorsa, b andV. Note that the nonreality ofa, b andV implied by (4.7.14) and (4.7.15) yields anon-Hermitian potentialQ in the associated 2×2 matrix Schrödinger problem.
This corrects eq. (17) of L; the following equation of L should also be similarly corrected.
For instance, in the caseN=2 and for the simpler BT witha μ andb μ constant, it can be easily shown that (5.2.1), together with the Hermiticity requirement just mentioned, impliesa n=b n=0,n=1, 2, 3.
Actually eq. (4.2.4) requires thatf(z) andg(z) be real for negativez only; butf(z) andg(z) must also be analytic inz for eq. (5.2.3) to make sense.
It should be emphasized that, while this feature of the BT (5.2.3) can be immediately seen from the formula (5.2.4), it is less than evident from the BT formula itself, eq. (5.2.3).
Typically, this forbids ϕ(k) (that is generally a meromorphic function ofk) from having poles of higher order than the first, or from having (even simple) poles coinciding with poles ofR(k) or from having poles in disallowed regions of the complexk-plane (such as the region Rek≠0, Imk>0, if the potentials are required to be Hermitian and to vanish asymptotically faster than exponentially).
One difficulty is apparent from eq. (5.2.7), wherep is by assumption independent ofy andt, while we know from the results of subsect.44 that the poles of the reflection coefficient on the positive imaginary axis, corresponding to the discrete eigenvalues of the Schrödinger problem, generally do not remain constant in time if they-variable is present. Note, however, that there is no contradiction with the constancy ofp in (5.2.7), since also (4.4.2) yields at- andy-independentp if, at the initial time,p isy-independent. But of course this raises the question of they-dependence ofQ′(x,y,t); indeed, generally, even ifQ(x,y,t) is localized iny (i.e. vanishing when |y| diverges), theQ′(x,y,t) related toQ(x,y,t) by (5.2.8) cannot also be localized, although it does generally satisfy (4.2.4), ifQ(x,y,t) does (and if the «constants of integration» that obtain by integrating (5.2.8) depend appropriately ony andt).
See eqs. (1) of B2, and differentiate the second with respect tox.
To derive this equation rewrite (5.2.8) by using (3.2.12) and (5.2.9) (and its analog forQ′ andW′), then integrate overx by using the boundary condition (5.2.10) (and its analog forW′) and then combine this equation with the analogous one obtained by inverting the BT,i.e. by replacingp with −p,W withW′ andW′ withW.
In the special caseN=2, this becomes eqs. (9) of B2 (for the connection between the notations used there and here, see footnote (4) See the review paper byV. E. Zakharov in the forthcoming Springer monograph edited byR. K. Bullough of B1).
To solve (5.2.11), it is convenient to perform the substitutionW′=Y −1.
Admittedly, we are somewhat cavalier in drawing this implication, that, to be completely justified, would require an analysis of the effect of BTs on the discrete-spectrum parameters, that we propose, instead to give in a separate paper. But the «structural» nature of the results derived here (see in particular eq. (5.3.7) below) justifies their general validity, that is also supported by the verification mentioned at the end of this section.
Take the trace of (5.4.1), thereby eliminating all commutators on the r.h.s., and then integrate both sides overx.
To complete the proof of this result one should also look at the evolution of the discrete-spectrum parameters; but this is trivial. Note that the formula defininga μ andb μ holds for all (complex) values ofz, since it relates analytic, indeed entire, functions ofz.
These, and the following equations of this section, supersede eqs. (22)–(28) of L.
For analogous formulae for other classes of NEEs see ref.(6).
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Calogero, F., Degasperis, A. Nonlinear evolution equations solvable by the inverse spectral transform.— II. Nuovo Cim B 39, 1–54 (1977). https://doi.org/10.1007/BF02738174
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DOI: https://doi.org/10.1007/BF02738174