2,030 research outputs found
On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients
For a mixed (advanced--delay) differential equation with variable delays and
coefficients
where explicit
nonoscillation conditions are obtained.Comment: 17 pages; 2 figures; to appear in Computers & Mathematics with
Application
Chaotic Phenomenon in Nonlinear Gyrotropic Medium
Nonlinear gyrotropic medium is a medium, whose natural optical activity
depends on the intensity of the incident light wave. The Kuhn's model is used
to study nonlinear gyrotropic medium with great success. The Kuhn's model
presents itself a model of nonlinear coupled oscillators. This article is
devoted to the study of the Kuhn's nonlinear model. In the first paragraph of
the paper we study classical dynamics in case of weak as well as strong
nonlinearity. In case of week nonlinearity we have obtained the analytical
solutions, which are in good agreement with the numerical solutions. In case of
strong nonlinearity we have determined the values of those parameters for which
chaos is formed in the system under study. The second paragraph of the paper
refers to the question of the Kuhn's model integrability. It is shown, that at
the certain values of the interaction potential this model is exactly
integrable and under certain conditions it is reduced to so-called universal
Hamiltonian. The third paragraph of the paper is devoted to quantum-mechanical
consideration. It shows the possibility of stochastic absorption of external
field energy by nonlinear gyrotropic medium. The last forth paragraph of the
paper is devoted to generalization of the Kuhn's model for infinite chain of
interacting oscillators
Oscillation criteria for nonlinear delay differential equations of second order
We prove oscillation theorems for the nonlinear delay differential equation
where and locally integrable on is a continuous function satisfiying and lim The results obtained essentially improve the known results in the literature and can be applied to linear and half-linear delay type differential equations
Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
We consider two physically and mathematically distinct regularization
mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the
combination of diffusion and dispersion are known to give rise to monotonic and
oscillatory traveling waves that approximate shock waves. The zero-diffusion
limits of these traveling waves are dynamically expanding dispersive shock
waves (DSWs). A richer set of wave solutions can be found when the flux is
non-convex. This review compares the structure of solutions of Riemann problems
for a conservation law with non-convex, cubic flux regularized by two different
mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation;
and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation.
In the first case, the possible dynamics involve two qualitatively different
types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the
second case, in addition to RWs, there are traveling wave solutions
approximating both classical (Lax) and non-classical (undercompressive) shock
waves. Despite the singular nature of the zero-diffusion limit and rather
differing analytical approaches employed in the descriptions of dispersive and
diffusive-dispersive regularization, the resulting comparison of the two cases
reveals a number of striking parallels. In contrast to the case of convex flux,
the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is
identified as an undercompressive DSW. Other prominent features, such as
shock-rarefactions, also find their purely dispersive counterparts involving
special contact DSWs, which exhibit features analogous to contact
discontinuities. This review describes an important link between two major
areas of applied mathematics, hyperbolic conservation laws and nonlinear
dispersive waves.Comment: Revision from v2; 57 pages, 19 figure
The asymptotic nature of a class of second order nonlinear system
In this paper, we obtain some results on the nonoscillatory behaviour of the system (1), which contains as particular cases, some well known systems. By negation, oscillation criteria are derived for these systems. In the last section we present some examples and remarks, and various well known oscillation criteria are obtained
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