arXiv subjects

Aleksandr O. Smirnov

Publications and source records attributed to Aleksandr O. Smirnov.

4 recordsLinked to original sources

arXiv2022

The vector form of Kundu-Eckhaus equation and its simplest solutions

In our work a hierarchy of integrable vector nonlinear differential equations depending on the functional parameter $r$ is constructed using a monodromy matrix. The first equation of this hierarchy for $r=α(\mathbf{p}^t\mathbf{q})$ is vector analogue of the Kundu-Eckhaus equation. When $α=0$, the equations of this hierarchy turn into equations of the Manakov system hierarchy. New elliptic solutions to vector analogue of the Kundu-Eckhaus and Manakov system are presented. In conclusion, it is shown that there exist linear transformations of solutions to vector integrable nonlinear equations into other solutions to the same equations.

nlin.SI↗

arXiv2021

Fundamental Analytic Solutions for the Kulish-Sklyanin Model with Constant Boundary Conditions

In the present paper we analyze the construction of fundamental analytic solutions (FAS) for the generalized Kulish-Sklyanin models (KSM) for vanishing (VBC) and constant boundary conditions (CBC). Using FAS one can reduce the direct and inverse scattering problems for the Lax operator to a Riemann-Hilbert problem (RHP). For VBC we find two FAS $χ^+(x,t,λ) $ and $χ^-(x,t,λ) $ analytic in the upper/lower $\mathbb{C}_\pm$ complex $λ$-plane. The RHP consists in: given the sewing function $G(x,t,λ)$ to constructing both $χ^\pm(x,t,λ) $ in their regions of analyticity. For CBC the problem becomes more complicated, because now the RHP must be formulated on a Riemannian surface of genus 1.

nlin.SI↗

arXiv2017

Spectral curves for the rogue waves

Here we find the spectral curves, corresponding to the known rational or quasi-rational solutions of AKNS hierarchy equations, ultimately connected with the modeling of the rogue waves events in the optical waveguides and in hydrodynamics. We also determine spectral curves for the multi-phase trigonometric, hyperbolic and elliptic solutions for the same hierarchy. It seams that the nature of the related spectral curves was not sufficiently discussed in existing literature.

math-ph↗

arXiv2015

Three-Phase Freak Waves

In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schrödinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic "freak waves". We also study the dependency of the solution parameters on the spectral curves.

math-ph↗