arXiv subjects
Publications and source records attributed to O. M. Kiselev.
We study a hard regime of stimulation of two-frequency oscillations in the main resonance equation with a fast oscillating external force: $ \ve i ψ' + |ψ|^2ψ= \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.
An asymptotic behaviour of solution of Kadomtsev-Petviashvili-2 equation is obtained as $t\to\infty$ uniformly with respect to spatial variables.
The dromion of the Davey-Stewartson-1 equation is studied under perturbation on the large time.
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 $t_*$.
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.
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.
We show that lumps (solitons) of the Davey--Stewartson II equation fail under small perturbations of initial data.
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.