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Viktor Tkachenko

Publications and source records attributed to Viktor Tkachenko.

9 recordsLinked to original sources
arXiv2026

An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation

This paper investigates an inverse boundary value problem for a semilinear strongly damped wave equation with Dirichlet boundary conditions in Sobolev spaces of functions bounded in time on $\R$, including periodic and almost periodic functions. In addition to constructing a bounded strong solution, we determine a time-dependent source coefficient via an integral overdetermination condition ensuring well-posedness. After reducing the inverse problem to a direct one, we first establish existence and uniqueness of solutions to an associated problem on finite time intervals. We then extend these solutions to half-lines and construct a bounded strong solution on the whole real line as a limit of such extensions, and subsequently establish its uniqueness. In particular, periodic and almost periodic data yield periodic and almost periodic solutions.

math.AP↗
arXiv2024

Lyapunov function and smooth periodic solutions to quasilinear 1D hyperbolic systems

We apply a Lyapunov function to obtain conditions for the existence and uniqueness of small classical time-periodic solutions to first order quasilinear 1D hyperbolic systems with (nonlinear) nonlocal boundary conditions in a strip. The boundary conditions cover different types of reflections from the boundary as well as integral operators with delays. In the first step we use a Lyapunov approach to derive sufficient conditions for the robust exponential stability of the boundary value problems for a linear(ized) homogeneous problem. Under those conditions and a number of non-resonance conditions, in the second step we prove the existence and uniqueness of smooth time-periodic solutions to the corresponding linear nonhomogeneous problems. In the third step, we prove a perturbation theorem stating that the periodic solutions survive under small perturbations of all coefficients of the hyperbolic system. In the last step, we apply the linear results to construct small and smooth time-periodic solutions to the quasilinear problems.

math.AP↗
arXiv2023

On the bounded smooth solutions to exponentially stable linear nonautonomous hyperbolic systems

We investigate global bounded solutions of higher regularity to boundary value problems for a general linear nonautonomous first order 1D hyperbolic system in a strip. We establish the existence of such solutions under the assumption of exponential stability and certain dissipativity (or non-resonant) conditions. Our results demonstrate the existence and uniqueness of $C^1$- and $C^2$-bounded solutions, in particular, periodic and almost periodic solutions. The number of imposed dissipativity conditions is related to the regularity of solutions. This connection arises due to the nonautonomous nature of the hyperbolic system under consideration, in contrast to autonomous settings. In the general case of global bounded smooth solutions we assume exponential stability in $H^1$. In the case of periodic solutions, the weaker assumption of $L^2$-exponential stability proves to be sufficient.

math.AP↗
arXiv2021

Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems

The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. One of the technical complications we overcome is the "loss of smoothness" property of hyperbolic PDEs.

math.AP↗
arXiv2018

Various damage mechanisms in carbon and silicon materials under femtosecond x-ray irradiation

We review the results of our research on damage mechanisms in materials irradiated with femtosecond free-electron-laser (FEL) pulses. They were obtained using our hybrid approach, XTANT (X-ray-induced Thermal And Nonthermal Transitions). Various damage mechanisms are discussed with respect to the pulse fluence and material properties on examples of diamond, amorphous carbon, C60 crystal, and silicon. We indicate conditions: producing thermal melting of targets as a result of electron-ion energy exchange; nonthermal phase transitions due to modification of the interatomic potential; Coulomb explosion due to accumulated net charge in finite-size systems; spallation or ablation at higher fluences due to detachment of sample fragments; and warm dense matter formation. Transient optical coefficients are compared with experimental data whenever available, proving the validity of our modeling approach. Predicted diffraction patterns can be compared with the results of ongoing or future FEL experiments. Limitations of our model and possible future directions of development are outlined.

cond-mat.mtrl-sci↗
arXiv2016

Almost periodic evolution systems with impulse action at state-dependent moments

We study the existence of almost periodic solutions for semi-linear abstract parabolic evolution equations with impulse action at state-dependent moments. In particular, we present conditions excluding the beating phenomenon in these systems. The main result is illustrated with an example of impulsive diffusive logistic equation.

math.DS↗
arXiv2012

Frequency locking by external forcing in systems with rotational symmetry

We study locking of the modulation frequency of a relative periodic orbit in a general $S^1$-equivariant system of ordinary differential equations under an external forcing of modulated wave type. Our main result describes the shape of the locking region in the three-dimensional space of the forcing parameters: intensity, wave frequency, and modulation frequency. The difference of the wave frequencies of the relative periodic orbit and the forcing is assumed to be large and differences of modulation frequencies to be small. The intensity of the forcing is small in the generic case and can be large in the degenerate case, when the first order averaging vanishes. Applications are external electrical and/or optical forcing of selfpulsating states of lasers.

math.DS↗
arXiv2010

Frequency locking of modulated waves

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.

math.DS↗