Francesco Calogero

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

SummaryThis 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.RiassuntoQuesto 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.РезюмеЭта 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.

Spectral Transform and Solitons: How to Solve and Investigate Nonlinear Evolution Equations

The soliton was discovered (and named) in 1965 by Zabusky and Kruskal,(1) who were experimenting with the numerical solution by computer of the Korteweg-de Vries (KdV) equation. This nonlinear partial differential equation had been introduced at the end of the last century to describe wave motion in shallow canals.(2) Zabusky and Kruskal studied the equation because of its relevance to plasma physics, as well as to the Fermi-Pasta-Ulam puzzle(3) (for a fascinating account of the motivations that led to the “birth of the soliton,” see Kruskal.(4)) The first scientific description of the soliton as a natural phenomenon, however, goes back to the first half of the nineteenth century, and was reported by J. Scott-Russell in the following prose:(5) I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon....

Motion of Poles and Zeros of Special Solutions of Nonlinear and Linear Partial Differential Equations and Related Solvable Many Body Problems

SummaryThe motion of the poles and zeros of special solutions of certain nonlinear and linear partial differential equations is shown to be interpretable in terms of equivalent many-body problems. Several solvable many-body models are thus introduced and discussed. The treatment is limited to problems involving a finite number of particles moving in one space dimension.RiassuntoSi mostra come il moto degli zeri e dei poli di soluzioni particolari di alcune equazioni alle derivate parziali lineari e non lineari possa essere interpretato come un problema a molti corpi. Si introducono in tal modo numerosi esempi di problemi a molti corpi risohibili. L’analisi è limitata a modelli con un numero finito di particelle che si muovono in una dimensione.РЕжУМЕпОкАжыВАЕтсь, ЧтО ДВИ жЕНИЕ пОлУсОВ И НУлЕИ ЧАстНых РЕшЕНИИ НЕкО тОРых НЕлИНЕИНых И лИ НЕИНых ДИФФЕРЕНцИАл ьНых УРАВНЕНИИ В ЧАст Ных пРОИжВОДНых М ДИФФЕРЕНцИАльНых УР АВНЕНИИ В ЧАстНых пРО ИжВОДНых МОжЕт Быть И НтЕРпРЕтИРОВАНО В тЕ РМИНАх ЁкВИВАлЕНтНы х пРОБлЕМ МНОгИх тЕл. О БсУжДАУтсь НЕкОтОРы Е РЕшА МОжЕт Быть ИНтЕРпРЕт ИРОВАНО В тЕРМИНАх Ёк ВИВАлЕНтНых пРОБлЕМ МНОгИх тЕл. ОБсУжДАУт сь НЕкОтОРыЕ РЕшАЕМы Е МОДЕлИ МНОгИх тЕл. РА ссМОтРЕНИЕ ОгРАНИЧИ ВАЕтсь пРОБлЕМАМИ, Вк лУЧАУЩИМИ кОНЕЧНОЕ Ч ИслО ЧАстИц, ДВИжУ пРОБлЕМ МНОгИх тЕл. ОБ сУжДАУтсь НЕкОтОРыЕ РЕшАЕМыЕ МОДЕлИ МНОг Их тЕл. РАссМОтРЕНИЕ О гРАНИЧИВАЕтсь пРОБл ЕМАМИ, ВклУЧАУЩИМИ кО НЕЧНОЕ ЧИслО ЧАстИц, Д ВИжУЩИхсь МОДЕлИ МНОгИх тЕл. РАс сМОтРЕНИЕ ОгРАНИЧИВ АЕтсь пРОБлЕМАМИ, Вкл УЧАУЩИМИ кОНЕЧНОЕ ЧИ слО ЧАстИц, ДВИжУЩИхс ь пРОБлЕМАМИ, ВклУЧАУЩ ИМИ кОНЕЧНОЕ ЧИслО ЧА стИц, ДВИжУЩИхсь ДВИжУЩИхсь В ОДНОМЕстНОМ пДОстР АНстВЕ.

The neatest many-body problem amenable to exact treatments (a “goldfish”?)

Various formulations, findings and conjectures are reviewed, which relate to a many-body problem that is arguably the neatest nontrivial such model amenable to exact treatments.

A class of integrable Hamiltonian systems whose solutions are (perhaps) all completely periodic

We show that the dynamical system characterized by the (complex) equations of motion qj+iΩqj=∑k=1,k≠jnqjqkf(qj−qk), j=1,…,n, with f(x)=−λ℘′(λx)/[℘(λx)−℘(λμ)], is Hamiltonian and integrable, and we conjecture that all its solutions qj(t), j=1,…,n are completely periodic, with a period that is a finite integral multiple of T=2π/Ω. Here n is an arbitrary positive integer, Ω is an arbitrary (nonvanishing) real constant, ℘(y)≡℘(y|ω,ω′) is the Weierstrass function (with arbitrary semiperiods ω,ω′), and λ,μ are two arbitrary constants; special cases are f(x)=2λ coth(λx)/[1+r2 sinh2(λx)], f(x)=2λ coth(λx), f(x)=2λ/sinh(λx), f(x)=2/[x(1+λ2x2)], and of course f(x)=2/x. These findings, as well as the conjecture (which is shown to be true in some of these special cases), are based on the possibility to recast these equations of motion in the modified Lax form L+iΩL=[L,M] with L and M appropriate (n×n)-matrix functions of the n dynamical variables qj and of their time-derivatives qj.

The transition from regular to irregular motions, explained as travel on Riemann surfaces

We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a three-body problem in the (complex) plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology—illustrating the onset in a deterministic context of irregular motions—is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. Here only some of our main findings are reported, without detailing their proofs: a more complete presentation will be published elsewhere.

Lax pairs galore

A formula that yields an (apparently—but only apparently—nontrivial) Lax pair for any nonlinear evolution PDE in 1+1 dimensions possessing a local conservation law is presented. Several examples are exhibited.

The Burgers equation on the semi-infinite and finite intervals

The initial/boundary value problem on the semiline and on a finite interval, for the Burgers equations ut=uxx+2uxu, is solved, i.e. reduced, by quadratures, to a linear integral equation of Volterra type in one independent variable, which can itself be solved by quadratures if the boundary data are time independent.

An integrable Hamiltonian system

Abstract For any n , the dynamical system characterized by the Hamiltonian H = ∑ j , k = 1 n p j p k { λ + μ cos[ v ( q j − q k )]} is completely integrable: n constants of motion in involution are explicitly given, its initial-value problem is solved in completely explicit form.

Cosmic origin of quantization

Abstract The origin of quantization is attributed - via the mechanism of “stochastic quantization” - to the universal interaction of every particle with the background gravitational force due to all other particles of the Universe. A formula for Plancks action constant h , obtained on the basis of this idea, yields the correct order of magnitude for h when implemented with current cosmological data.