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- T. V. Savina, A . A . Golovin, Stephen H. Davis, Alexander A. Nepomnyashchy, Peter W. Voorhees
- Physical review. E, Statistical, nonlinear, and…
- 2003

Consider faceting of a crystal surface caused by strongly anisotropic surface tension, driven by surface diffusion and accompanied by deposition (etching) due to fluxes normal to the surface. Nonlinear evolution equations describing the faceting of 1+1 and 2+1 crystal surfaces are studied analytically, by means of matched asymptotic expansions for small… (More)

- A . A . Golovin, Alexander A. Nepomnyashchy, Stephen H. Davis, Michael A. Zaks
- Physical review letters
- 2001

In this paper we demonstrate that convective Cahn-Hilliard models, describing phase separation of driven systems (e.g., faceting of growing thermodynamically unstable crystal surfaces), exhibit, with the increase of the driving force, a transition from the usual coarsening regime to a chaotic behavior without coarsening via a pattern-forming state… (More)

- Alexander Oron, Alexander A. Nepomnyashchy
- Physical review. E, Statistical, nonlinear, and…
- 2004

We study the onset of Marangoni instability of the quiescent equilibrium in a binary liquid layer with a nondeformable interface in the presence of the Soret effect. Linear stability analysis shows that both monotonic and oscillatory long-wavelength instabilities are possible depending on the value of the Soret number chi. Sets of long-wavelength nonlinear… (More)

- Michael A. Zaks, Alla Podolny, Alexander A. Nepomnyashchy, Alexander A. Golovin
- SIAM Journal of Applied Mathematics
- 2005

We investigate bifurcations of stationary periodic solutions of a convective Cahn– Hilliard equation, ut+Duux+(u−u+uxx)xx = 0, describing phase separation in driven systems, and study the stability of the main family of these solutions. For the driving parameter D < D0 = √ 2/3, the periodic stationary solutions are unstable. For D > D0, the periodic… (More)

- J. H. Park, Alvin Bayliss, Bernard J. Matkowsky, Alexander A. Nepomnyashchy
- SIAM Journal of Applied Mathematics
- 2006

We study the motion of fronts for an extended version of the nonlinear wave equation, φtt + γφt = 21φ + f (φ)+ h + η1φt with positive 1 in cartesian and polar coordinates and give a local description of the front in terms of its normal velocity, acceleration and curvature. We study analytically and numerically the motion of planar and circular fronts and… (More)

- Alexander A. Nepomnyashchy, Amy Novick-Cohen, Horacio G. Rotstein, Simon Brandon
- SIAM Journal of Applied Mathematics
- 2001

Abstract. We present a phenomenological theory for phase transition dynamics with memory which yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential. Thereafter, we focus on the implications of our theory in the hyperbolic case, and we derive asymptotically an equation of motion in… (More)

- Vladimir A. Volpert, Yana Nec, Alexander A. Nepomnyashchy
- Philosophical transactions. Series A…
- 2013

A review of recent developments in the field of front dynamics in anomalous diffusion-reaction systems is presented. Both fronts between stable phases and those propagating into an unstable phase are considered. A number of models of anomalous diffusion with reaction are discussed, including models with Lévy flights, truncated Lévy flights,… (More)

- E. A. Glasman, Alexander A. Golovin, Alexander A. Nepomnyashchy
- SIAM Journal of Applied Mathematics
- 2004

- Sergey Shklyaev, Alexander A. Nepomnyashchy, Alexander Oron
- Physical review. E, Statistical, nonlinear, and…
- 2011

We consider a long-wave oscillatory Marangoni convection in a layer of a binary liquid in the presence of the Soret effect. A weakly nonlinear analysis is carried out on a hexagonal lattice. It is shown that the derived set of cubic amplitude equations is degenerate. A three-parameter family of asynchronous hexagons (AH), representing a superposition of… (More)