arXiv subjects

V. S. Dryuma

Publications and source records attributed to V. S. Dryuma.

4 recordsLinked to original sources

arXiv2007

Dunajski generalization of the second heavenly equation: dressing method and the hierarchy

Dunajski generalization of the second heavenly equation is studied. A dressing scheme applicable to Dunajski equation is developed, an example of constructing solutions in terms of implicit functions is considered. Dunajski equation hierarchy is described, its Lax-Sato form is presented. Dunajsky equation hierarchy is characterized by conservation of three-dimensional volume form, in which a spectral variable is taken into account.

nlin.SI↗

arXiv2006

On the dressing method for Dunajski anti-self-duality equation

A dressing scheme applicable to Dunajski equation is developed. Simple example of constructing solutions in terms of implicit functions is considered. Dunajski equation hierarchy is described, its Lax-Sato form is presented. Dunajsky equation hierarchy is characterised by conservation of three-dimensional volume form, in which a spectral variable is taken into account. Some reductions of Dunajsky equation hierarchy, including waterbag-type reduction, are studied.

nlin.SI↗

arXiv1997

Investigation of dynamical systems using tools of the theory of invariants and projective geometry

The investigation of nonlinear dynamical systems of the type $\dot{x}=P(x,y,z),\dot{y}=Q(x,y,z),\dot{z}=R(x,y,z)$ by means of reduction to some ordinary differential equations of the second order in the form $y''+a_1(x,y)y'^3+3a_2(x,y)y'^2+3a_3(x,y)y'+a_4(x,y)=0$ is done. The main backbone of this investigation was provided by the theory of invariants developed by S. Lie, R. Liouville and A. Tresse at the end of the 19th century and the projective geometry of E. Cartan. In our work two, in some sense supplementary, systems are considered: the Lorenz system $\dot{x}=σ(y-x), \dot{y}=rx-y-zx,\dot{z}=xy-bz $ and the Rößler system $\dot{x}=-y-z,\dot{y}=x+ay,\dot{z}=b+xz-cz.$. The invarinats for the ordinary differential equations, which correspond to the systems mentioned abouve, are evaluated. The connection of values of the invariants with characteristics of dynamical systems is established.

solv-int↗

arXiv1997

On equation of geodesic deviation and its solutions

Equations of geodesic deviation for the 3-dimensional and 4-dimensional Riemann spaces are discussed. Availability of wide classes of exact solutions of such equations, due to recent results for the matrix Schrödinger equation, is demonstrated. Particular classes of exact solutions for the geodesic deviation equation as well as for the Raychaudhuri and generalized Raychaudhuri equation are presented. Solutions of geodesic deviation equation for the Schwarzshild and Kasner metrics are found.

solv-int↗