A system of two Korteweg-de Vries equations coupled by small linear and nonlinear terms, which is a model of a resonant interaction between two internal gravity-wave modes in a shallow stratified liquid, is considered. The present paper is a continuation of a preceding one, where only the linear coupling was dealt with. Various dynamical processes involving one-mode solitons are investigated. I...

We develop a perturbation theory that describes bound states of solitons localized in a confined area. External forces and influence of inhomogeneities are taken into account as perturbations to exact solutions of the sine-Gordon equation. We have investigated two special cases of fluxon trapped by a microresistor and decay of a breather under dissipation. Also, we have carried out numerical si...

We describe spectral or dynamical methods that can be used to determine the stability and noise performance of modelocked lasers. We first reviewmethods that have been used to date to theoretically and computationally study passively modelocked lasers, contrasting evolutionary and dynamical approaches and their application to full, averaged, and reduced models. We then develop the spectral meth...

Small perturbation of the Liouville equation under smooth initial data is considered. Asymptotic solution which is available for a long time interval is constructed by the two scale method. The Cauchy problem for the Liouville equation with a small perturbation ∂ 2 t u − ∂ 2 x u + 8 exp u = εF[u], 0 < ε ≪ 1, (0.1) u| t=0 = ψ 0 (x), ∂ t u| t=0 = ψ 1 (x), x ∈ R (0.2) is considered. The problem is...

The interaction of a fluxon with a finite-size inhomogeneity in a long Josephson junction is studied within the framework of the perturbation theory for solitons. The threshold values of the bias current density either for capture of the fluxon or for its escape on the inhomogeneity are calculated. Induced by the resonant fluxon emission hystereses on the first zero-field step of the Josephson ...

We report on the possibility of unidirectional motion of a topological soliton in a dissipative AC-driven sine-Gordon equation. This motion is due to breaking of certain space-time symmetries. It can be achieved by the proper choice of the potential or the AC force. We formulate the necessary symmetry conditions for the AC force and confirm them with the first order perturbation theory and nume...

The system of coupled nonlinear evolution equations describing gap solitons is shown to be approximately equivalent in the neighborhood of the soliton solution to the massive Thirring model. The equivalence is then used for construction of the perturbation theory for gap solitons based on the RiemannHilbert problem. The value of deviation of the initial condition from the gap soliton shape serv...

Abstract We consider a randomly perturbed Korteweg-de Vries equation. The perturbation is a random potential depending both on space and time, with a white noise behavior in time, and a regular, but stationary behavior in space. We investigate the dynamics of the soliton of the KdV equation in the presence of this random perturbation, assuming that the amplitude of the perturbation is small. We...

It is shown that, if the driving force and damping are sufficiently small, a small amplitude phase-locked soliton is a fixed point attractor of the Landau-Lifshitz equation of motion which describes a uniaxial one-dimensional ferromagnet. Recent studies have shown that a soliton can be a low dimensional attractor of the driven damped nonlinear Schrijdinger equation [I]. In particular, if an osc...

An efficient formalism is elaborated to analytically describe dynamics of the Ablowitz-Ladik soliton in the presence of perturbations. This formalism is based on using the Riemann-Hilbert problem and provides the means of calculating evolution of the discrete soliton parameters, as well as shape distortion and perturbation-induced radiation effects. As an example, soliton characteristics are ca...