Using theoretical arguments, we prove the numerically well-known fact that the eigenvalues of all localized stationary solutions of the cubic-quintic (2+1) -dimensional nonlinear Schrödinger equation exhibit an upper cutoff value. The existence of the cutoff is inferred using Gagliardo-Nirenberg and Hölder inequalities together with Pohozaev identities. We also show that, in the limit of eigenvalues close to zero, the eigenstates of the cubic-quintic nonlinear Schrödinger equation behave similarly to those of the cubic nonlinear Schrödinger equation.

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In this work we consider a model problem describing one phase flow through a thin porous layer made of weakly permeable porous blocks separated by thin fissures. The flow is modeled by a linear parabolic equation considered in a bounded 2D domain with high contrast coefficients. The problem involves three small parameters: the first one characterizes the periodicity of the distribution of the blocks in the layer, the second one stands for the thickness of the layer, the third one characterizes the volume fraction of the fissure part in the layer. Using the notion of two-scale convergence, we derive the homogenized models which govern the global behavior of the flow when the small parameters tend to zero. The global models essentially depend on the relation between the small parameters.

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Abstract.We study the nonlinear dynamics of globally coupled nonidentical oscillators in the framework of two order parameter (mean field and amplitude-frequency correlator) reduction. The main result of the paper is the exact solution of a corresponding nonlinear system on a two-dimensional invariant manifold. We present a complete classification of phase portraits and bifurcations, obtain explicit expressions for invariant manifolds (a limit cycle among them) and derive analytical solutions for arbitrary initial data and different regimes.