Gro Hovhannisyan

On Oscillations of Solutions of Third-Order Dynamic Equation

We are proving the new oscillation theorems for the solutions of third-order linear nonautonomous differential equation with complex coefficients. In the case of real coefficients we derive the oscillation criterion that is invariant with respect to the adjoint transformation. Our main tool is a new version of Levinsons asymptotic theorem.

Asymptotic stability for dynamic equations on time scales

We examine the conditions of asymptotic stability of second-order linear dynamic equations on time scales. To establish asymptotic stability we prove the stability estimates by using integral representations of the solutions via asymptotic solutions, error estimates, and calculus on time scales.

Ablowitz-Ladik hierarchy of integrable equations on a time-space scale

We derive the Todas lattice, the Hirotas network, and the nonlinear Schrodinger dynamic equations on a time-space scale by extension on a time-space scale the Ablowitz-Ladik hierarchy of integrable dynamic systems.

On nonautonomous Dirac equation

We construct the fundamental solution of time dependent linear ordinary Dirac system in terms of unknown phase functions. This construction gives approximate representation of solutions which is useful for the study of asymptotic behavior. Introducing analog of Rayleigh quotient for differential equations we generalize Hartman–Wintner asymptotic integration theorems with the error estimates for applications to the Dirac system. We also introduce the adiabatic invariants for the Dirac system, which are similar to the adiabatic invariant of Lorentz’s pendulum. Using a small parameter method it is shown that the change in the adiabatic invariants approaches zero with the power speed as a small parameter approaches zero. As another application we calculate the transition probabilities for the Dirac system. We show that for the special choice of electromagnetic field, the only transition of an electron to the positron with the opposite spin orientation is possible.

Wiman’s formula for a second order dynamic equation

We derive Wiman’s asymptotic formula for the number of generalized zeros of (nontrivial) solutions of a second order dynamic equation on a time scale. The proof is based on the asymptotic representation of solutions via exponential functions on a time scale. By using the Jeffreys et al. approximation we prove Wiman’s formula for a dynamic equation on a time scale. Further we show that using the Hartman-Wintner approximation one can derive another version of Wiman’s formula. We also prove some new oscillation theorems and discuss the results by means of several examples.MSC:34E20, 34N05.

Schwarzian derivative and Ermakov equation on a time scale

We introduce the Schwarzian, simplified Schwarzian, and Schwarzian Korteweg-de Vries equations on a time scale that are invariant under the fractional linear transformations. As an application, we derive their solutions and establish the invariant disconjugacy condition for second order dynamic equations on a time scale. Furthermore, we consider the Ermakov dynamic equation and the Ermakov-Lewis adiabatic invariant on a time scale. We discuss specific examples of discrete, quantum, and continuous time scales to compare our equations with the well-known ones. We also derive the linearization of the Ermakov equation and the corresponding Painleve equation on a time scale.We introduce the Schwarzian, simplified Schwarzian, and Schwarzian Korteweg-de Vries equations on a time scale that are invariant under the fractional linear transformations. As an application, we derive their solutions and establish the invariant disconjugacy condition for second order dynamic equations on a time scale. Furthermore, we consider the Ermakov dynamic equation and the Ermakov-Lewis adiabatic invariant on a time scale. We discuss specific examples of discrete, quantum, and continuous time scales to compare our equations with the well-known ones. We also derive the linearization of the Ermakov equation and the corresponding Painleve equation on a time scale.

Asymptotic behavior of a planar dynamic system

We investigate the asymptotic solutions of the planar dynamic systems and the second order equations on a time scale by using a new version of Levinson’s asymptotic theorem. In this version the error estimate is given in terms of the characteristic (Riccati) functions which are constructed from the phase functions of an asymptotic solution. It means that the improvement of the approximation depends essentially on the asymptotic behavior of the Riccati functions. We describe many different approximations using the flexibility of this approach. As an application we derive the analogue of D’Alembert’s formula for the one dimensional wave equation in a discrete time.

On Dirac equation on a time scale

We consider the non-autonomous linear Dirac equation on a time scale containing important discrete, continuous, and quantum time scales. A representation of the solutions is established via an approximate solutions in terms of unknown phase functions with the error estimates. JWKB and other asymptotic representations are discussed. The adiabatic invariants of the Dirac equation are described by using a small parameter method. We also calculate the transition probabilities for the Dirac equation. Using the asymptotic solutions we show that the electron-positron transition probability during a long period of time is about 1/3. Since this probability is high, there is a simple explanation of the stability of the revolution of an electron about the proton only by the electromagnetic field. Indeed when the electron is far from the proton, it is attracted by the electromagnetic field of the proton. When the electron approaches closer to the proton, it turns to the positron which is repelling from the proton by ...

WKB Estimates for 2 × 2 Linear Dynamic Systems on Time Scales

AbstractsWe establish WKB estimates for linear dynamic systems with a small parameter on a time scale unifying continuous and discrete WKB method. We introduce an adiabatic invariant for dynamic system on a time scale, which is a generalization of adiabatic invariant of Lorentz_s pendulum. As an application we prove that the change of adiabatic invariant is vanishing as approaches zero. This result was known before only for a continuous time scale. We show that it is true for the discrete scale only for the appropriate choice of graininess depending on a parameter . The proof is based on the truncation of WKB series and WKB estimates.We establish WKB estimates for Open image in new window linear dynamic systems with a small parameter Open image in new window on a time scale unifying continuous and discrete WKB method. We introduce an adiabatic invariant for Open image in new window dynamic system on a time scale, which is a generalization of adiabatic invariant of Lorentz_s pendulum. As an application we prove that the change of adiabatic invariant is vanishing as Open image in new window approaches zero. This result was known before only for a continuous time scale. We show that it is true for the discrete scale only for the appropriate choice of graininess depending on a parameter Open image in new window. The proof is based on the truncation of WKB series and WKB estimates.

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinsons asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.