We consider the five-component Meinhardt-Gierer model for mutually exclusive patterns and segmentation which was proposed in [11]. We prove rigorous results on the existence and stability of mutually exclusive spikes which are located in different positions for the two activators. Sufficient conditions for existence and stability are derived, which depend in particular on the relative size of t...

We consider the following Gierer-Meinhardt system in R 1 : A 0 (?1) = A 0 (1) = H 0 (?1) = H 0 (1) = 0; where (p; q; r; s) satisses 1 < qr (s + 1)(p ? 1) < +1; 1 < p < +1: We rigorously show that the existence and stability of symmetric and asymmetric N?peaked steady-states can be reduced to computing two matrices in terms of the coeecients D; N; p; q; r; s. The main techniques are the Liapunov...

First we develop the theory of reduced instability in order to analyze stability of bifurcating solutions via the Liapunov{Schmidt reduction. Next, this theory is generalized to cover sideband instabilities using a Floquet ansatz. The method is applied to roll patterns in the Swift{Hohenberg equation which serves as a model for pattern formation. We analyze the stability problem of small rolls ...

We consider the linear stability and structural stability of non-ground state traveling waves of a pair of coupled nonlinear Schrr odinger equations (CNLS) which describe the evolution of co-propagating polarized pulses in the presence of birefringence. Viewing the CNLS equation as a Hamiltonian perturbation of the Manakov equations, we nd parameter regimes in which there are two stable familie...

Following the authors’ earlier work in [9, 10], we show that the nonlinear eigenvalue problem introduced in [10] can be recast in the language of bifurcation theory as a perturbation of a linearized eigenvalue problem. Solutions of this nonlinear eigenvalue problem correspond to time periodic solutions of the compressible Euler equations that exhibit the simplest possible periodic structure ide...

This paper develops the Liapunov-Schmidt procedure for systems with additional constraints such as having a first integral, being Hamiltonian, or being a gradient system. Similar developments for systems with symmetry, including reversibility, are well known, and the method of this paper augments and is consistent with that approach. One of the results states that the bifurcation equation for H...

We present a study on the critical time step for the numerical integration based on the Runge-Kutta method of the monodromy matrix (the fundamental matrix solution) associated with a set of n rst-order linear ordinary diierential equations with periodic coeecients. By applying the Liapunov-Schmidt method, for any dimension n and systems which are perturbations of autonomous systems, we give an ...

Recently, skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on two-component systems of nonlinear Schrödinger equations (NLSE) describing a binary mixture of Bose-Einstein condensates (cf. [2] and [25]). Besides, half-skyrmions characterized by half-integer topological charges can also be found in the nonlinear σ model which is a mo...

We introduce a new method for the analysis of sideband instabilities which are important for periodic patterns appearing in systems close to the instability threshold. The method relies on a two{fold application of the Liapunov{Schmidt reduction procedure, a rst application to the nonlinear bifurcation problem and a second application to the linear spectral problem. We obtain rigorous result on...