A type of bounded traveling wave solutions for the Fornberg-Whitham equation

In this paper, by using bifurcation method, we successfully find the Fornberg-Whitham equation has a type of traveling wave solutions called kink-like wave solutions and antikinklike wave solutions. They are defined on some semifinal bounded domains and possess properties of kink waves and anti-kink waves. Their implicit expressions are obtained. For some concrete data, the graphs of the implicit functions are displayed, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.Comment: 14 pages, 10 figure

Exact traveling wave solutions to the Klein–Gordon equation using the novel (G′/G)-expansion method

AbstractThe novel (G′/G)-expansion method is one of the powerful methods that appeared in recent times for establishing exact traveling wave solutions of nonlinear partial differential equations. Exact traveling wave solutions in terms of hyperbolic, trigonometric and rational functions to the cubic nonlinear Klein–Gordon equation via this method are obtained in this article. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It is shown that the novel (G′/G)-expansion method is a simple and valuable mathematical tool for solving nonlinear evolution equations (NLEEs) in applied mathematics, mathematical physics and engineering

Numerical bifurcation analysis for multi-section semiconductor lasers

We investigate the dynamics of a multi-section laser resembling a delayed feedback experiment where the length of the cavity is comparable to the length of the laser. Firstly, we reduce the traveling-wave model with gain dispersion (a hyperbolic system of partial differential equations) to a system of ordinary differential equations (ODEs) describing the semiflow on a local center manifold. Then, we analyse the dynamics of the system of ODEs using numerical continuation methods (AUTO). We explore the plane of the two parameters feedback phase and feedback strength to obtain a complete bifurcation diagram for small and moderate feedback strength. This diagram allows to understand the roots of a variety of nonlinear phenomena like, e. g., self-pulsations, excitability, hysteresis or chaos, and to locate them in the parameter plane

Peakon, Cuspon, Compacton, and Loop Solutions of a Three-Dimensional 3DKP(3, 2) Equation with Nonlinear Dispersion

We study peakon, cuspon, compacton, and loop solutions for the three-dimensional Kadomtsev-Petviashvili equation (3DKP(3,2) equation) with nonlinear dispersion. Based on the method of dynamical systems, the 3DKP(3,2) equation is shown to have the parametric representations of the solitary wave solutions such as peakon, cuspon, compacton, and loop solutions. As a result, the conditions under which peakon, cuspon, compacton, and loop solutions appear are also given

Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues

Stochastic analysis of nonlinear dynamics and feedback control for gene regulatory networks with applications to synthetic biology

The focus of the thesis is the investigation of the generalized repressilator model (repressing genes ordered in a ring structure). Using nonlinear bifurcation analysis stable and quasi-stable periodic orbits in this genetic network are characterized and a design for a switchable and controllable genetic oscillator is proposed. The oscillator operates around a quasi-stable periodic orbit using the classical engineering idea of read-out based control. Previous genetic oscillators have been designed around stable periodic orbits, however we explore the possibility of quasi-stable periodic orbit expecting better controllability. The ring topology of the generalized repressilator model has spatio-temporal symmetries that can be understood as propagating perturbations in discrete lattices. Network topology is a universal cross-discipline transferable concept and based on it analytical conditions for the emergence of stable and quasi-stable periodic orbits are derived. Also the length and distribution of quasi-stable oscillations are obtained. The findings suggest that long-lived transient dynamics due to feedback loops can dominate gene network dynamics. Taking the stochastic nature of gene expression into account a master equation for the generalized repressilator is derived. The stochasticity is shown to influence the onset of bifurcations and quality of oscillations. Internal noise is shown to have an overall stabilizing effect on the oscillating transients emerging from the quasi-stable periodic orbits. The insights from the read-out based control scheme for the genetic oscillator lead us to the idea to implement an algorithmic controller, which would direct any genetic circuit to a desired state. The algorithm operates model-free, i.e. in principle it is applicable to any genetic network and the input information is a data matrix of measured time series from the network dynamics. The application areas for readout-based control in genetic networks range from classical tissue engineering to stem cells specification, whenever a quantitatively and temporarily targeted intervention is required

Bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion

We study the stability and bifurcation structure of spatially extended patterns arising in nonlin- ear optical resonators with a Kerr-type nonlinearity and anomalous group velocity dispersion, as described by the Lugiato-Lefever equation. While there exists a one-parameter family of patterns with different wavelengths, we focus our attention on the pattern with critical wave number k c arising from the modulational instability of the homogeneous state. We find that the branch of solutions associated with this pattern connects to a branch of patterns with wave number $2k_c$ . This next branch also connects to a branch of patterns with double wave number, this time $4k_c$ , and this process repeats through a series of 2:1 spatial resonances. For values of the detuning parameter approaching $\theta = 2$ from below the critical wave number $k_c$ approaches zero and this bifurcation structure is related to the foliated snaking bifurcation structure organizing spatially localized bright solitons. Secondary bifurcations that these patterns undergo and the resulting temporal dynamics are also studied.Comment: 13 pages, 13 figure