Alim Sukhtayev

Vakhitov–Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields

We study the linear stability of localized modes in self-interacting spinor fields, analyzing the spectrum of the operator corresponding to linearization at solitary waves. Following the generalization of the Vakhitov--Kolokolov approach, we show that the bifurcation of real eigenvalues from the origin is completely characterized by the Vakhitov--Kolokolov condition

The Morse and Maslov indices for Schrödinger operators

dQ/d\omega=0

Instability of pulses in gradient reaction–diffusion systems: a symplectic approach

and by the vanishing of the energy functional. We give the numerical data on the linear stability in the generalized Gross--Neveu model and the generalized massive Thirring model in the charge-subcritical, critical, and supercritical cases, showing the agreement with the Vakhitov--Kolokolov and the energy vanishing conditions.

Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model

We study the spectrum of Schrödinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger operators with periodic potentials. For quasi-convex domains in Rn, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.

The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited

In a scalar reaction–diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state’s critical points. In particular, this implies that pulse solutions are unstable. We extend this picture to pulses in reaction–diffusion systems with gradient nonlinearity. In particular, we associate a Maslov index to any asymptotically constant state, generalizing existing definitions of the Maslov index for homoclinic orbits. It is shown that this index equals the number of unstable eigenvalues for the linearized evolution equation. Finally, we use a symmetry argument to show that any pulse solution must have non-zero Maslov index, and hence be unstable. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials

Applying the Lyapunov–Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift–Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction–diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg–Landau amplitude equations, rigorously validating the standard weakly unstable approximation and the Eckhaus criterion.