Search results for "Integrable system"
showing 10 items of 354 documents
On Spaces of Bochner and Pettis Integrable Functions and Their Set-Valued Counterparts
2011
The aim of this paper is to give a brief summary of the Pettis and Bochner integrals, how they are related, how they are generalized to the set-valued setting and the canonical Banach spaces of bounded maps between Banach spaces that they generate. The main tool that we use to relate the Banach space-valued case to the set-valued case, is the R ̊adstr ̈om embedding theorem.
The bi-Hamiltonian theory of the Harry Dym equation
2002
We describe how the Harry Dym equation fits into the the bi-Hamiltonian formalism for the Korteweg-de Vries equation and other soliton equations. This is achieved using a certain Poisson pencil constructed from two compatible Poisson structures. We obtain an analogue of the Kadomtsev-Petviashivili hierarchy whose reduction leads to the Harry Dym hierarchy. We call such a system the HD-KP hierarchy. We then construct an infinite system of ordinary differential equations (in infinitely many variables) that is equivalent to the HD-KP hierarchy. Its role is analogous to the role of the Central System in the Kadomtsev-Petviashivili hierarchy.
Temporal Soliton “Molecules” in Mode-Locked Lasers: Collisions, Pulsations, and Vibrations
2008
A few years after the discovery of the stable dissipative soliton pairs in passively mode-locked lasers, a large variety of multi-soliton complexes were studied in both experiments and numerical simulations, revealing interesting new behaviors. This chapter focuses on the following three subjects: collisions between dissipative solitons, pulsations of dissipative solitons, and vibrations of soliton pairs. Different outcomes of collisions between a soliton pair and a soliton singlet are discussed, showing possible experimental control in the formation or dissociation of ‘soliton molecules’. Long-period pulsations of single and multiple dissipative solitons are presented as limit cycles and o…
Spectral incoherent solitons
2009
Solitons have been usually considered as inherently coherent localized structures and the discovery of incoherent optical solitons has represented a significant progress [1]. As occurs for standard coherent solitons, incoherent solitons are characterized by a confinement of the field in the spatial or in the temporal domain. We introduce here a novel type of incoherent solitons that are neither spatial nor temporal, i.e., the incoherent field does not exhibit any confinement in the spatiotemporal domain; however, the uncorrelated frequency components that constitute the incoherent field exhibit a localized soliton behavior in the frequency domain [2].
Generation of High-Repetition-Rate Dark Soliton Trains and Frequency Conversion in Optical Fibers
1998
Induced modurational polarization instability in birefringent fibers leads to trains of dark soliton-like pulses. Optimal large-signal cw and soliton frequency conversion is also analysed.
Hydrodynamics of periodic breathers
2014
We report the first experimental observation of periodic breathers in water waves. One of them is Kuznetsov–Ma soliton and another one is Akhmediev breather. Each of them is a localized solution of the nonlinear Schrödinger equation (NLS) on a constant background. The difference is in localization which is either in time or in space. The experiments conducted in a water wave flume show results that are in good agreement with the NLS theory. Basic features of the breathers that include the maximal amplitudes and spectra are consistent with the theoretical predictions.
Collision of Akhmediev Breathers in Nonlinear Fiber Optics
2013
We report here a novel fiber-based test bed using tailored spectral shaping of an optical-frequency comb to excite the formation of two Akhmediev breathers that collide during propagation. We have found specific initial conditions by controlling the phase and velocity differences between breathers that lead, with certainty, to their efficient collision and the appearance of a giant-amplitude wave. Temporal and spectral characteristics of the collision dynamics are in agreement with the corresponding analytical solution. We anticipate that experimental evidence of breather-collision dynamics is of fundamental importance in the understanding of extreme ocean waves and in other disciplines dri…
Stability and Chaos
2010
In this chapter we study a larger class of dynamical systems that include but go beyond Hamiltonian systems. We are interested, on the one hand, in dissipative systems, i.e. systems that lose energy through frictional forces or into which energy is fed from exterior sources, and, on the other hand, in discrete, or discretized, systems such as those generated by studying flows by means of the Poincare mapping. The occurence of dissipation implies that the system is coupled to other, external systems, in a controllable manner. The strength of such couplings appears in the set of solutions, usually in the form of parameters. If these parameters are varied it may happen that the flow undergoes …
Inelastic scattering and interactions of three-wave parametric solitons.
2006
We study the interactions of velocity-locked three-wave parametric solitons in a medium with quadratic nonlinearity and dispersion. We reveal that the inelastic scattering between three-wave solitons and linear waves may be described in terms of analytical solutions with dynamically varying group velocity, or boomerons. Moreover, we demonstrate the elastic nature of three-wave soliton-soliton collisions and interactions.
Dark-and-bright rogue waves in long wave-short wave resonance
2014
Nonlinear Photonics, Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, in Proceedings Advanced Photonics, Part of Advanced Photonics, Barcelona, Spain, 28-31 July 2014