The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to 1 + 2 dimensions. Perturbation theory is the ...

The dynamics of a soliton propagating in a single mode optical fiber with Raman coupling to thermal phonons is analyzed. Using both soliton perturbation theory, and exact numerical techniques, we predict that intrinsic thermal quantum noise effects due to the phonon reservoirs is a dominant source of jitter and other perturbations for short pulses (∼ 1ps). The size of the Raman timing jitter is...

The three-soliton solutions to the sine-Gordon equation describing the collision of a kink with a breather or with a kink–antikink pair are given and several separatrix three-soliton solutions are extracted from these solutions. The influence of a small perturbation on the threesoliton collisions is studied numerically. As the perturbed system the Frenkel–Kontorova model with a small degree of ...

Optical-fiber transmission of pulses can be modeled with the complex Ginzburg–Landau equation. We find novel stable soliton pairs and trains, which are relevant in this case, and analyze them. We suggest that the distance between the pulses and the phase difference between them is defined by energy and momentum balance equations, rather than by equations of standard perturbation theory. We pres...

Complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory. In this article, we explicitly construct such eigenfunctions for a large class of integrable equations including the KdV, NLS and mKdV hierarchies. We establish the striking result that the linearization operators of all equations in the...

We study the dynamics of cavity solitons under the influence of spatial inhomogeneities and derive generalized equations of motions. For perturbations large compared to the soliton size we find the modulus of the soliton velocity to be proportional to the gradient of the respective perturbation and that the proportionality coefficient changes sign when the soliton peak power drives the cavity b...

For the sine-Gordon and nonlinear Schr~Sdinger equations with conservative perturbations three-particle effects in collision of a fast soliton with two more slow ones are considered in the first order of the perturbation theory. The momentum and energy exchanged between the fast soliton and slow ones are calculated. For the latter system the three-particle contribution to the perturbation-induc...

A variational method is used to obtain the growth rate of a transverse long-wavelength perturbation applied to the soliton solution of a nonlinear Schrödinger equation with a three-half order potential. We demonstrate numerically that this unstable perturbed soliton will eventually transform into a cylindrical soliton. Keywords—soliton, instability, variational method, spectral method

We study nonlinear dynamics of inhomogeneous DNA double helical chain under dynamic plane-base rotator model by considering angular rotation of bases in a plane normal to the helical axis. The DNA dynamics in this case is found to be governed by a perturbed sine-Gordon equation while taking into account the interstrand hydrogen bonding energy between bases and the intrastrand inhomogeneous stac...

We analyze the exact two-soliton solution to the unperturbed nonlinear Schrödinger equation and predict that in a weakly perturbed system (i) soliton collisions can be strongly inelastic, (ii) inelastic collisions are of almost nonradiating type, (iii) results of a collision are extremely sensitive to the relative phase of solitons, and (iv) the effect is independent on the particular type of p...