Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearity

We are concerned with the following Schrödinger system with coupled quadratic nonlinearity − ε 2 Δ v + P ( x ) v = μ v w , x ∈ R N , − ε 2 Δ w + Q ( x ) w = μ 2 v 2 + γ w 2 , x ∈ R N , v > 0 , w > 0 , v , w ∈ H 1 R N , $$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right),\end{array}\right. \end{equation}$$ which arises from second-harmonic generation in quadratic media. Here ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive function potentials. By applying reduction method, we prove that if x 0 is a non-degenerate critical point of Δ(P + Q) on some closed N − 1 dimensional hypersurface, then the system above has a single peak solution (vε , wε ) concentrating at x 0 for ε small enough.

CYCLICITY OF PERIOD ANNULUS OF A QUADRATIC REVERSIBLE SYSTEM WITH DEGENERATE CRITICAL POINT

This paper is concerned with the bifurcation of limit cycles from the period annulus of a quadratic reversible system. The outer boundary of the period annulus contains a degenerate critical point. The exact upper bound of the number of limit cycles is given. Our result shows that the conjecture on the cyclicity of (r4) system is correct.

On the breakup of air bubbles in a Hele-Shaw cell

We study the problem of breakup of an air bubble in a Hele-Shaw cell. In particular, we propose some sufficient conditions of breakup of the bubble, and ways to find the contraction points of its parts. We also study regulated contraction of a pair of bubbles (in which the rates of air extraction from the bubbles are controlled) and study various asymptotic questions (such as the asymptotics of contraction of a bubble to a degenerate critical point, and asymptotics of contraction of a small bubble in the presence of a big bubble)

Stability of a boundary spike layer for the Gierer–Meinhardt system

We consider the activator-inhibitor Gierer–Meinhardt reaction-diffusion system of biological pattern formation in a closed bounded domain. The existence and stability of a boundary apike-layer solution to the Gierer–Meinhardt model, and it, so-called shadow limit, is analysed. In the limit of small activator diffusivity, together with a large inhibitor diffusivity, an equilibrium boundary spike-layer solution is constructed that concentrates at a non-degenerate critical point P of the boundary. By non-degenerate we mean that every principal curvature of the boundary has a local maximum at P, and hence the mean curvature at the boundary has a local maximum at P. Rigorous results for the stability of such a boundary spike-layer solution are given.

Semi-classical Integrability, Hyperbolic Flows and the Birkhoff Normal Form

We prove that a Hamiltonianp∈C∞(T*Rn) is locally integrable near a non-degenerate critical point ρ0of the energy, provided that the fundamental matrix at ρ0has rationally independent eigenvalues, none purely imaginary. This is done by using Birkhoff normal forms, which turn out to be convergent in theC∞sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the complex case, showing that whenpis holomorphic near ρ0∈T*Cn, then Repbecomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge,i.e., pmay not be integrable. These normal forms also hold in the semi-classical frame.

SUFFICIENT CONDITIONS FOR A CENTER AT A COMPLETELY DEGENERATE CRITICAL POINT

Consider the two-dimensional autonomous systems of differential equations of the form [Formula: see text] where P3(x, y) and Q3(x, y) are homogeneous polynomials of degree 3, and P4(x, y) and Q4(x, y) are homogeneous polynomials of degree 4. The origin is a completely degenerate critical point of this system. In this work we give sufficient conditions in order to have a center at the origin.

Uniqueness and critical spectrum of boundary spike solutions

We study the properties of single boundary spike solutions for the following singularly perturbed problem It is known that at a non-degenerate critical point of the mean curvature function H(P), there exists a single boundary spike solution. In this paper, we show that the single boundary spike solution is unique and moreover it has exactly (N − 1) small eigenvalues. We obtain the exact asymptotics of the small eigenvalues in terms of H(P).