In this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equationswithout any periodicity assumptions on p(t), q(t) and f , providing that f (t, x) grows superlinearly both at origin and at infinity or is an odd function with respect to x ∈ R, and satisfies some additional assumptions.

Abstract. In this paper, we derive the conditions for the existence of stationary solutions (i.e., nonconstant steady states) of a volume-filling chemotaxis model with logistic growth over a bounded domain subject to homogeneous Neumann boundary conditions. At the same time, we show that the same system without the chemotaxis term does not admit pattern formations. Moreover, based on an explicit formula for the stationary solutions, which is derived by asymptotic bifurcation analysis, we establish the stability criteria and find a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue. We show that all bifurcations except the one at the first location of the bifurcation parameter are unstable, and if the pattern is stable, then its principal wave mode must be a positive integer which minimizes the bifurcation parameter. For a special case where the carrying capacity is one half, we find a necessary and sufficient condition for the stability of pattern solutions. Numerical simulations are presented, on the one hand, to illustrate and fit our analytical results and, on the other hand, to demonstrate a variety of interesting spatio-temporal patterns, such as chaotic dynamics and the merging process, which motivate an interesting direction to pursue in the future. 1. Introduction. The process of generation of spontaneous patterns (i.e., selforganization) involves many instances where symmetry is broken or a more symmetric state develops into a less symmetric one. It is striking that many of these breaks in symmetry have no external trigger, but are instigated internally. Hence a natural question arises: how can this process be done? The significant progress toward this question was made in 1952 by Alan Turing, whose paper [29] was considered one of the most influential works in theoretical biology. Turing thought about a simple model with two morphogens (chemical species): a short-range activator and a long-range inhibitor. Both morphogens diffuse in space, but at different rates, and the inhibitor diffuses faster than the activator. The combination of local strong activation and longrange inhibition is able to instigate a spatial patterning process. Turing proposed a mathematical model based on a couple of partial differential equations encapsulating these elements, known as a reaction-diffusion model (see, e.g., [18]). Turing's revolutionary contribution was that passive diffusion could interact with chemical reaction

We study possibilities to suppress the transverse modulational instability (MI) of dark-soliton stripes in two-dimensional (2D) Bose-Einstein condensates (BECs) and self-defocusing bulk optical waveguides by means of quasi-1D structures. Adding an external repulsive barrier potential (which can be induced in BEC by a laser sheet, or by an embedded plate in optics), we demonstrate that it is possible to reduce the MI wavenumber band, and even render the dark-soliton stripe completely stable. Using this method, we demonstrate the control of the number of vortex pairs nucleated by each spatial period of the modulational perturbation. By means of the perturbation theory, we predict the number of the nucleated vortices per unit length. The analytical results are corroborated by the numerical computation of eigenmodes of small perturbations, as well as by direct simulations of the underlying Gross-Pitaevskii/nonlinear Schrödinger equation.

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This paper is devoted to investigating the propagation speed and direction of the bistable traveling wave for the Lotka-Volterra competition model with bistable nonlinearity. A range-estimation of the bistable wave speed is provided by comparing the bistable traveling wave with the monostable ones in the system. Comparison theorems on the speed and direction of wave propagation are first established by the method of upper and lower solutions. Through the construction of novel upper and lower solutions, we obtain some new criteria that explicitly determine the speed sign of the bistable traveling wave. Based on these results, we can predict which species will win in the competition for the same resources.

Earlier work has shown that ring dark solitons in two-dimensional Bose-Einstein condensates are generically unstable. In this work, we propose a way of stabilizing the ring dark soliton via a radial Gaussian external potential. We investigate the existence and stability of the ring dark soliton upon variations of the chemical potential and also of the strength of the radial potential. Numerical results show that the ring dark soliton can be stabilized in a suitable interval of external potential strengths and chemical potentials. We also explore different proposed particle pictures considering the ring as a moving particle and find, where appropriate, results in very good qualitative and also reasonable quantitative agreement with the numerical findings.