Main results:
- Development of methods for solving one-dimensional inverse problems of
electromagnetic sounding theory.
- Description of the long-time asymptotic behaviour of solutions of
non-linear evolution equations; description of decay of nondecreasing
(step-like) solutions of nonlinear evolution equations into asymptotic
solitons.
- Construction of a new class of asymptotical solutions of
Kadomtsev-Petviashvili equations (curved solitons).
- Development of the homogenization theory for boundary value problems for
domains with fine-grained boundaries, for weekly connected domains,
and for domains with traps.
- Development of the homogenization models for dynamics of fluids with
microstructure.
- Development of the homogenization theory for harmonic fields on
Riemannian manifolds of increasing type.
- Development of models of motion of quantized vortices in the rotating
superfluids and in the Bose-Einstein condensates.
- Construction of the homogenization model of immiscible fluids.
- Existence/nonexistence theorem for a variational problem for
Ginzburg-Landau functional and study of the asymptotic behaviour of
solutions.
- Construction of the asymptotic invariant methods for dynamical systems
describing the bioelectrical activity of brain.
Pricez and awards:
The State Prize of Ukraine (V.A.Marchenko, E.Ya.Khruslov, 1989): for the
series of works "Boundary value problems in domains with fine-grained boundaries".
The Krylov Prize of the National Academy of Sciences of Ukraine (E.Ya.Khruslov,
V.P.Kotlyarov, 1995): for the series of works "The decay of the solutions of
nonlinear evolution equations into asymptotic solitons".
The Lavrentiev Prize of the National Academy of Sciences of Ukraine (V.A.Marchenko,
B.I.Ptashnik, E. Ya.Khruslov, 2007) for the series of works "Analytic and
asymptotic methods of the investigation of non-standard boundary value
problems".
Invited lecture of E.Khruslov at the International Congress of Mathematicians,
Zurich, 1994: “Homogenized models of strongly non-homogeneous media".