Transmission of solitary pulse in dissipative nonlinear transmission lines

A class of dissipative complex Ginzburg–Landau (DCGL) equations that govern the wave propagation in dissipative nonlinear transmission lines is solved exactly by means of the Hirota bilinear method. Two-soliton solutions of the DCGL equations, from which the one-soliton solutions are deduced, are obtained in analytical form. The modified Hirota method imposes some restrictions on the coefficients equations. Namely, the second-order dispersion must be real. The physical requirement of the solutions imposes complementary conditions on the combination of the dispersion and nonlinear gain/loss terms of the equation, as well as on the coefficient of the Kerr nonlinearity. The analytical solutions for one-solitary pulses are tested in direct simulations.     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Periodic perturbation of genetic oscillations

Kinetics of gene expression may be oscillatory due to the feedbacks between the mRNA, non-coding RNA and protein synthesis. In complex genetic networks, the kinetic oscillations generated in one subnetwork may influence oscillations in another subnetwork. To clarify what may happen in such situations, we have performed a mean-field analysis and Monte Carlo simulations of periodic perturbation of the oscillatory kinetics of the simplest genetic network including a gene with negative regulation of the mRNA production by protein, obtained via mRNA translation and two steps of conversion. Our analysis shows universal and specific features of the kinetics under consideration. Our simulations indicate that due to fluctuations only some of these features can really be observed. Specifically, the main frequencies obtained by the Fourier expansion of the mean-field and Monte Carlo kinetics are found to be often similar except that the Monte Carlo distribution of frequencies near the main frequencies is somewhat wider.     

Invariant tori in nonlinear oscillations

The boundedness of all the solutions for semilinear Duffing equationx″ + ω² x + φ(x) =p(t), ω ∈ ℝ⁺ℕ is proved, wherep (t) is a smooth 2π-periodic function and the perturbation ⌽(x) is bounded.     

Analytical studies of soliton pulses along two-dimensional coupled nonlinear transmission lines

A nonlinear network with many coupled nonlinear LC dispersive transmission lines is considered, each line of the network containing a finite number of cells. In the semi-discrete limit, we apply the reductive perturbation method and show that the wave propagation along the network is governed by a two-dimensional nonlinear partial differential equation (2-D NPDE) of Schrödinger type. Two regimes of wave propagation, the high-frequency and the low-frequency are detected. By the means of exact soliton solution of the 2-D NPDE, we investigate analytically the soliton pulse propagation in the network. Our results show that the network under consideration supports the propagation of kink and dark solitons.     

On forced nonlinear oscillations

The existence of a 1-periodic solution of the generalized Liénard equation x″ + g(x)x′ + f(t, x) = e(t), where g(x) is continuous, e(t) is continuous, periodic of period 1 and with mean value 0 and f is continuous, periodic of period 1 in t, is proved under one of the following conditions: (i) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$ and

$\text{lim sup}\text{|x|?+∞}\text{|f(t,x)|}\text{| x |}\text{<}\text{4π}{}^2\text{2π + 1}$

(ii) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$. Earlier results of A. C. Lazer, J. Mawhin and R. Reissig are obtained as particular cases.     

Analytical study for the ability of nonlinear transmission lines to generate solitons

The ability of the nonlinear transmission lines (NLTL) has been studied analytically, in this paper to generate solitons and to cause waveform spreading. This can be achieved by balancing nonlinearity and dispersion. A new technique of improved tanh method (ITM) and improved sech methods (ISM) is applied to the nonlinear partial differential equation that describes the NLTL. It is found that the parameters of the transmission line play an important role in controlling the shape of the soliton.     

Exact traveling discrete kink-soliton solutions for the discrete nonlinear electrical transmission lines

We investigate exact soliton solutions for the discrete nonlinear electrical transmission line by performing the simplest equation method from a trivial seed solution. Starting from the nonlinear propagation of signals in electrical transmission lines, we derive exact traveling kink and antikink solitary wave solutions. It is shown that under a safe range of parameter, the shape of kink soliton can be controlled well by adjusting the parameter of the line. The analytical solutions for the kink and antikink solitary waves are tested in direct simulations.     

Periodic solutions of nonlinear wave equations with damping

A known existence theorem for doubly periodic solutions of nonlinear wave equations with linear damping is being proved in a direct manner by an approach which has been developed by the authors in [5, 6, 7] for hyperbolic problems, when the kernel of the underlying linear operator is infinite dimensional.     

Linearized oscillations in nonlinear delay difference equations

Consider the nonlinear delay difference equation We establish a linearized oscillation result of this equation, which is the extension of the result in the paper [1]. Supported by the National Natural Science Foundation of China     

Formation of nonlinear magnetization oscillations by spin waves transmission through the boundary of two uniaxial ferromagnets

In this paper, the possibility of nonlinear spin wave formation at transmission of a bulk spin wave through the boundary of two ferromagnets with uniaxial magnetic anisotropy in the external magnetic field was shown at the certain conditions. The bulk spin waves with different values of precession angles of magnetizations can propagate in the second ferromagnet so both linear and nonlinear states can be excited. Also, expressions for coefficients of reflection and transmission of spin waves were obtained depending on value of a precession angle of magnetization in the second ferromagnet.