The Fourier method for the linearized Davey-Stewartson I equation

The linearized Davey-Stewartson equation with varing coefficients is solved by Fourier method. The approach uses the inverse scattering transform for the Davey-Stewartson equation.Comment: 4 pages, LaTe

Emission of autoresonant trajectories and thresholds of resonant pumping

We study an autoresonant asymptotic behaviour for nonlinear oscillators under slowly changing frequency and amplitude of external driver. As a result we obtain formulas for threshold values of amplitude and frequency of the driver when autoresonant behaviour for the nonlinear oscillator is observed. Also we study a capture into resonance and emission out of the resonance for trajectories of the oscillator. A measure of autoresonant asymptotic behaviours for nonlinear oscillator is obtained.Comment: 15 pages, 6 figure

Asymptotics of soliton solution for the perturbed Davey-Stewartson-1 equations

The dromion of the Davey-Stewartson-1 equation is studied under perturbation on the large time.Comment: Mistake and misprints are corrected. Latex, 14 page

Asymptotic approach for the rigid condition of appearance of the oscillations in the solution of the Painleve-2 equation

The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point $t_*$ and fast oscillating behavior after the point $t_*$. In the transition layer the behavior of the asymptotic solution is more complicated. The leading term of the asymptotics satisfies the Painleve-1 equation and some elliptic equation with constant coefficients, where the solution of the Painleve-1 equation has poles. The uniform smooth asymptotics are constructed in the interval, containing the critical point $t_*$.Comment: Latex, 18 page

The slowly passage through the resonances and wave packets with the different carriers

Solution of the nonlinear Klein-Gordon equation perturbed by small external force is investigated. The perturbation is represented by finite collections of harmonics. The frequencies of the perturbation vary slowly and pass through the resonant values consecutively. The resonances lead to the sequence of the wave packets with the different fast oscillated carriers. Full asymptotic description of this process is presented.Comment: 24 pages, LaTe

Scattering of solitons on resonance

We investigate a propagation of solitons for nonlinear Schrodinger equation under small driving force. The driving force passes the resonance. The process of scattering on the resonance leads to changing of number of solitons. After the resonance the number of solitons depends on the amplitude of the driving force.Comment: LaTeX, 10 page

The solution of the Painleve equations as special functions of catastrophes, defined by a rejection in these equations of terms with derivative

The relation between the Painleve equations and the algebraic equations with the catastrophe theory point of view are considered. The asymptotic solutions with respect to the small parameter of the Painleve equations different types are discussed. The qualitative analysis of the relation between algebraic and fast oscillating solutions is done for Painleve-2 as an example.Comment: Latex, 15 page

The capture into parametric autoresonance

In this work we show that the capture into parametric resonance may be explained as the pitchfork bifurcation in the primary parametric resonance equation. We prove that the solution close to the moment of the capture is described by the Painleve-2 equation. We obtain the connection formulas for the asymptotic solution of the primary parametric resonance equation before and after the capture using the matching of the asymptotic expansions.Comment: LaTeX, 21 pages, 3 figure

Asymptotic description of nonlinear resonance

We study a hard regime of stimulation of two-frequency oscillations in the main resonance equation with a fast oscillating external force: \ve i \psi' + |\psi|^2\psi = \exp\big(it^2/ (2\ve)\big), 0<\ve\ll1. This phenomenon is caused by resonance between an eigenmode and the external force. The asymptotic solution before, inside and after the resonance layer is studied in detail and matched.Comment: LaTeX, 48 pages, two figure

Asymptotics of perturbed soliton for Davey--Stewartson II equation

It is shown that, under a small perturbation of lump (soliton) for Davey--Stewartson (DS-II) equation, the scattering data gain the nonsoliton structure. As a result, the solution has the form of Fourier type integral. Asymptotic analysis shows that, in spite of dispertion, the principal term of the asymptotic expansion for the solution has the solitary wave form up to large time.Comment: In this replaced version the formula for the perturbed parameter of the soliton is corrected. Amstex, 13 page