M. Bruschi

Properties of the zeros of the classical polynomials and of the Bbessel functions

SummaryMany algebraic equations satisfied by the zeros of the classical polynomials and of the Bessel functions are reported. Some of them are collected from recent papers; several of them are new; most of them display remarkable diophantine features. Certain matrices constructed with arbitrary numbers rather than the zeros of special functions, but displaying analogous diophantine properties, are also exhibited.RiassuntoSi riportano molte equazioni algebriche soddisfatte dagli zeri dei polinomi classici e delle funzioni di Bessel. Alcune sono riprese da lavori recenti; numerose sono nuove; la maggioranza esibisce notevoli proprietà di carattere diofantino. Si esibiscono inoltre alcune matrici, costruite con numeri arbitrari anziché con gli zeri di funzioni speciali, che anche loro posseggono però delle notevoli proprietà diofantine.РезюмеПриводится большое число алгебраических уравнений, которым удовлетворяют нули классических полиномов и бесселевых функций. Некоторые из уравнений взяты из недавних работ, некоторые из них новые. Большинство из них обнаруживают интересные особенности. Указываются некоторые матрицы, сконструированные с помощью произвольного числа нулей специальных функций.

Continuous and discrete matrix Burgers’ hierarchies

SummaryWe derive two hierarchies of matrix nonlinear evolution equations which reduce to the Burgers’ hierarchy in the scalar case and can be linearized by a matrix analogue of the Hopf-Cole transformation: for these hierarchies we display the associated class of Bäcklund transformations and show some special kinds of explicit solutions. More-over, by exploiting a discrete version of the Hopf-Cole transformation, we are also able to construct two hierarchies of linearizable nonlinear difference evolution equations and to derive for them Bäcklund trans-formations and explicit solutions.RiassuntoIn questo lavoro si derivano due gerarchie di equazioni di evoluzione nonlineari matriciali che possono essere linearizzate mediante un analogo matriciale della trasformazione di Hopf-Cole e si riducono nel caso scalare alla già nota gerarchia di Burgers. Per queste due gerarchie, come pure per le loro versioni discrete (anch’esse linearizzabili) si ottengono le trasformazioni di Bäcklund e si mostrano alcuni tipi significativi di soluzioni esplicite.

Extension of the Zakharov-Shabat generalized inverse method to solve differential-difference and difference-difference equations

SummaryWe extend the generalized inverse method recently introduced by Zakharovet al. in order to solve differential-difference and difference-difference equations and give the explicit form of the nondegenerateN-soliton solution. As an example we consider a discrete version of the sine-Gordon equation.RiassuntoIl « metodo inverso generalizzato » recentemente introdotto da Zakharovet al. è esteso ed applicato per risolvere equazioni non lineari alle differenze finite. Per tali equazioni viene data la forma esplicita della soluzione aN solitoni non degenere. Come esempio si considera un analogo discreto dell’equazione di sine-Gordon.РеэюмеМы расщиряем обобшенный универсальный метод, недавно предложенный Захаровым и др., для рещения дифференциально-раэ ностных и раэностно-раэностных уравнений. Для зтих уравнений приводится явная форма рещения сN солитонами. Как пример, рассматривается дискретный аналог уравнения синус-Гордона.

Soliton surfaces. VII: Relativistic string in external field: general integral and particular solutions

SummaryThe general integral of the equations of motion of a relativistic string in a constant external field is given. TheN-soliton excitations are constructed and a particular solution (one-soliton excitation) is discussed. The one soliton excitation gives rise to a motion analogous to the helicoidal motion of an electron in a constant magnetic field.

Discrete version of the nonlinear Schrödinger equation with linearlyx-dependent coefficients

SummaryIn this paper we discuss the differential-difference analog of the nonlinear Schrödinger equation with variable coefficients. The behaviour of the single-soliton solution is thoroughly investigated and the continuum limit is tersely derived.RiassuntoIn questo articolo si discute l’equazione differenziale alle differenze finite che è l’analogo discreto dell’equazione non lineare di Schrödinger con coefficienti variabili. Si studia in dettaglio l’andamento della soluzione a un solitone e si deriva brevemente il limite continuo.РезюмеВ этой статье мы обсуждаем дифференциально-разностный аналог уравнения Шрединтера с переменными коэффициентами. Подробно исследуется поведение одно-солитонного решения. Рассматривается предельный непрерывный случай.

Nonlinear partial differential equations and bäcklund transformations related to the 4-dimensional self-dual Yang-Mills equations

SummaryWe write down without proof a class of integrable nonlinear partial differential equations containing as one of its members the 4-dimensional Yang-Mills equation together with its associated Darboux and Bäcklund transformations.

The quantum Toda lattice

Abstract We investigate the quantum version of the nonperiodic Toda lattice, deriving an explicit formula for the wave function in the momentum and coordinate space in the case of two- and three-body systems.

Discrete version of the modified Korteweg-De Vries equation withx-dependent coefficients

SummaryIn this paper we investigate the differential-difference version of the modified Korteweg-de Vries equation withx-dependent coefficients, displaying the characteristic behaviour of the single-soliton solution. The continuum limit of the equation and of the soliton solution is thoroughly discussed.RiassuntoIn questo articolo si studia l’equazione alle differenze finite che è lanalogo discreto dell’equazione di Korteweg-de Vries modificata con coefficienti dipendenti dallax. Si mostra in particolare l’andamento caratteristico della soluzione a un solitone. Infine si discute in dettaglio il limite continuo dell’equazione e della corrispondente soluzione solitonica.

The discrete chiral-field hierarchy

SummaryWe construct the whole class of nonlinear matrix differential difference equations which contains discrete versions of the sine-Gordon and chiral-field equations. For any member of this class we show how to construct the Lax operators and the Bäcklund transformations.

Evolution equations associated to the triangular-matrix Schrödinger problem solvable by the inverse spectral transform

SummaryIn this paper we consider the evolution equations which are associated to the Schrödinger matrix problem for triangular matrices, a very simple subclass of the non-Hermitian matrices. These evolution equations, though linear in the simple case considered in the present paper, are not all solvable by means of standard techniques, thus making it worthwhile exploiting the applicability of the inverse spectral transform to obtain particular solutions. The 2 × 2 case is investigated thoroughly.RiassuntoIn questo articolo si studiano le equazioni di evoluzione associate all’equazione matriciale di Schrödinger nel easo di matrici triangolari, una classe assai semplice di matrici non hermitiane. Tali equazioni di evoluzione, anche se lineari nel caso qui considerato, non sono tutte risolubili con tecniche standard: di qui l’interesse di utilizzare la tecnica della trasformata spettrale inversa per trovare delle soluzioni particolari. Si esamina in dettaglio il caso di matrici triangolari 2 × 2.РеэюмеВ зтой работе мы рассматриваем уравнения зволюции, свяэанные с проблемой Щредингера для треугольных матриц, очень простого подкласса не-эрмитовых матриц. Эти уравнения зволюции, хотя они являются линейными в рассмотренном в работе случае, не могут быть рещены с помощью стандартной техники. В свяэи с зтим испольэуется обратное спектральное преобраэование для получения частных рещений. Исследуется случай 2 × 2.