Antonio Degasperis

A new integrable equation with peakon solutions

We consider a new partial differential equation recently obtained by Degasperis and Procesi using the method of asymptotic integrability; this equation has a form similar to the Camassa–Holm shallow water wave equation. We prove the exact integrability of the new equation by constructing its Lax pair and explain its relation to a negative flow in the Kaup–Kupershmidt hierarchy via a reciprocal transformation. The infinite sequence of conserved quantities is derived together with a proposed bi-Hamiltonian structure. The equation admits exact solutions as a superposition of multipeakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa–Holm peakons.

Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves.

We construct and discuss a semi-rational, multi-parametric vector solution of coupled nonlinear Schrödinger equations (Manakov system). This family of solutions includes known vector Peregrine solutions, bright-dark-rogue solutions, and novel vector unusual freak waves. The vector freak (or rogue) waves could be of great interest in a variety of complex systems, from optics to Bose-Einstein condensates and finance.

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....

Vector Rogue Waves and Baseband Modulation Instability in the Defocusing Regime

We report and discuss analytical solutions of the vector nonlinear Schrödinger equation that describe rogue waves in the defocusing regime. This family of solutions includes bright-dark and dark-dark rogue waves. The link between modulational instability (MI) and rogue waves is displayed by showing that only a peculiar kind of MI, namely baseband MI, can sustain rogue-wave formation. The existence of vector rogue waves in the defocusing regime is expected to be a crucial progress in explaining extreme waves in a variety of physical scenarios described by multicomponent systems, from oceanography to optics and plasma physics.

Integrable and non-integrable equations with peakons

We consider a one-parameter family of non-evolutionary partial differential equations which includes the integrable Camassa-Holm equation and a new integrable equation first isolated by Degasperis and Procesi. A Lagrangian and Hamiltonian formulation is presented for the whole family of equations, and we discuss how this fits into a bi-Hamiltonian framework in the integrable cases. The Hamiltonian dynamics of peakons and some other special finite-dimensional reductions are also described.

Rational solitons of wave resonant-interaction models

Integrable models of resonant interaction of two or more waves in 1+1 dimensions are known to be of applicative interest in several areas. Here we consider a system of three coupled wave equations which includes as special cases the vector nonlinear Schrödinger equations and the equations describing the resonant interaction of three waves. The Darboux-Dressing construction of soliton solutions is applied under the condition that the solutions have rational, or mixed rational-exponential, dependence on coordinates. Our algebraic construction relies on the use of nilpotent matrices and their Jordan form. We systematically search for all bounded rational (mixed rational-exponential) solutions and find a broad family of such solutions of the three wave resonant interaction equations.

Stable control of pulse speed in parametric three-wave solitons.

We analyze the control of the propagation speed of three wave packets interacting in a medium with quadratic nonlinearity and dispersion. We find analytical expressions for mutually trapped pulses with a common velocity in the form of a three-parameter family of solutions of the three-wave resonant interaction. The stability of these novel parametric solitons is simply related to the value of their common group velocity.

Three-dimensional optical beam propagation and solitons in photorefractive crystals

The model equations for beam propagation in photorefractive material are simplified under appropriate conditions. The possibility of obtaining bright and dark screening soliton solutions in 2+12+1 dimensions is investigated, and, whenever possible, their amplitude–size relation is displayed.

Multiple-scale perturbation beyond the nonlinear Schroedinger equation. I

Abstract We consider the effect of weak nonlinearity on the propagation of one-dimensional strongly dispersive waves. In the standard quasi-monochromatic approximation, it is well known that the first-order amplitude modulation satisfies the nonlinear Schroedinger equation in “slow variables”. With the assumption that this amplitude modulation has no discrete spectrum component (i.e. no soliton), we explore the higher-order effects, and provide a well-defined procedure to compute all coefficients of the perturbative expansion as relatively bounded functions (no secularity effects) of slow variables. The main effect of this procedure is that the leading amplitude modulation satisfies, at higher orders, the nonlinear Schroedinger hierarchy of evolution equations in slow variables. Key ingredient is the integrability of the nonlinear Schroedinger equation and its commuting symmetries.