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Traveling wave solutions are a class of invariant solutions which are critical for shallow water wave equations. In this paper, traveling wave solutions for two perturbed KP-MEW equations with a local delay convolution kernel are examined. The model equation is reduced to a planar near-Hamiltonian system via geometric singular perturbation theorem, and the qualitative properties of the corresponding unperturbed system are analyzed by using dynamical system approach. The persistence of the bounded traveling wave solutions for the perturbed KP-MEW equations with delay is investigated. By using a criterion for the monotonicity of ratio of two Abelian integrals and Melnikov’s method, the existence of kink (anti-kink) wave solutions and periodic wave solutions of the model equation are established. The result shows that the delayed KP-MEW equations with positive perturbation and the one with negative perturbation exhibit completely diverse dynamical properties. These new findings greatly enrich the understanding of dynamical properties of the traveling wave solutions of perturbed nonlinear wave equations with local delay convolution kernel. Numerical experiments further confirm and illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system. [ABSTRACT FROM AUTHOR]

In this paper, we construct the solutions to the following nonlinear Schrödinger system - ϵ 2 Δ u + P (x) u = μ 1 u p + β u p - 1 2 v p + 1 2 in R N , - ϵ 2 Δ v + Q (x) v = μ 2 v p + β u p + 1 2 v p - 1 2 in R N , where 3 < p < + ∞ , N ∈ { 1 , 2 } , ϵ > 0 is a small parameter, the potentials P, Q satisfy 0 < P 0 ≤ P (x) ≤ P 1 and Q(x) satisfies 0 < Q 0 ≤ Q (x) ≤ Q 1 . We construct the solution for attractive and repulsive cases. When x 0 is a local maximum point of the potentials P and Q and P (x 0) = Q (x 0) , we construct k spikes concentrating near the local maximum point x 0 . When x 0 is a local maximum point of P and x ¯ 0 is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point x 0 and m spikes of v concentrating at the local maximum point x ¯ 0 when x 0 ≠ x ¯ 0. This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case N = 3 , p = 3 . [ABSTRACT FROM AUTHOR]

In this paper we consider the following focusing mass-critical Hartree equation with a defocusing perturbation and harmonic potential i ∂ t ψ = - Δ ψ + | x | 2 ψ - (| x | - 2 ∗ | ψ | 2 ) ψ + ε | ψ | p - 2 ψ , in R + × R N , where N ≥ 3 , 2 < p < 2 ∗ = 2 N / (N - 2) and ε > 0 . We mainly focus on the normalized ground state solitary waves of the form ψ (t , x) = e i μ t u ε , ρ (x) , where u ε , ρ (x) is radially symmetric-decreasing and ∫ R N | u ε , ρ | 2 d x = ρ . Firstly, we prove the existence and nonexistence of normalized ground states under the L 2 -subcritical, L 2 -critical ( p = 4 / N + 2 ) and L 2 -supercritical perturbations. Secondly, we characterize perturbation limit behaviors of ground states u ε , ρ as ε → 0 + and find that the ε -blow-up phenomenon happens for ρ ≥ ρ c = ‖ Q ‖ L 2 2 , where Q is a positive radially symmetric ground state of - Δ u + u - (| x | - 2 ∗ | u | 2) u = 0 in R N . We prove that ∫ R N | ∇ u ε , ρ (x) | 2 d x ∼ ε - 4 N (p - 2) + 4 for ρ = ρ c and 2 < p < 2 ∗ , while ∫ R N | ∇ u ε , ρ | 2 d x ∼ ε - 4 N (p - 2) - 4 for ρ > ρ c and 4 / N + 2 < p < 2 ∗ , and obtain two different blow-up profiles corresponding to two limit equations. Finally, we study the limit behaviors as ε → + ∞ , which corresponds to a Thomas–Fermi limit. The limit profile is given by the Thomas–Fermi minimizer u TF = μ TF - | x | 2 + 1 p - 2 , where μ TF is a suitable Lagrange multiplier with exact value. Moreover, we obtain a sharp vanishing rate for u ε , ρ that ‖ u ε , ρ ‖ L ∞ ∼ ε - N N (p - 2) + 4 as ε → + ∞ . [ABSTRACT FROM AUTHOR]

This paper is devoted to establishing the convergence rate of the hypersonic similarity for the inviscid steady irrotational Euler flow over a two-dimensional Lipschitz slender wedge in B V ∩ L 1 space. The rate we established is the same as the one predicted by Newtonian-Busemann law (see (3.29) in [2, p 67] for more details) as the incoming Mach number M ∞ → ∞ for a fixed hypersonic similarity parameter K. The hypersonic similarity, which is also called the Mach-number independence principle, is equivalent to the following Van Dyke's similarity theory: For a given hypersonic similarity parameter K, when the Mach number of the flow is sufficiently large, the governing equations after the scaling are approximated by a simpler equation, that is called the hypersonic small-disturbance equation. To achieve the convergence rate, we approximate the curved boundary by piecewisely straight lines and find a new Lipschitz continuous map P h such that the trajectory can be obtained by piecing together the Riemann solutions near the approximated boundary. Next, we derive the L 1 difference estimates between the approximate solutions U h , ν (τ) (x , ·) to the initial-boundary value problem for the scaled equations and the trajectories P h (x , 0) (U 0 ν) by piecing together all the Riemann solvers. Then, by the uniqueness and the compactness of P h and U h , ν (τ) , we can further establish the L 1 estimates of order τ 2 between the solutions to the initial-boundary value problem for the scaled equations and the solutions to the initial-boundary value problem for the hypersonic small-disturbance equations, if the total variations of the initial data and the tangential derivative of the boundary are sufficiently small. Based on it, we can further establish a better convergence rate by considering the hypersonic flow past a two-dimensional Lipschitz slender wing and show that for the length of the wing with the effect scale order O (τ - 1) , that is, the L 1 convergence rate between the two solutions is of order O (τ 3 2) under the assumption that the initial perturbation has compact support. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global existence of mild solutions as well as uniform h-stability for non-autonomous evolution equations in a Banach space, where the operators in linear part (possibly unbounded) depend on time t and generate an evolution family. Under some estimates on the perturbation terms, we discuss the global uniform h-stability, the uniform h-stability and the global practical uniform h-stability using Gronwall integral inequalities. Some examples are provided to show the validity of the obtained results. [ABSTRACT FROM AUTHOR]

The paper investigates the robustness of pullback attractors and pullback exponential attractors of the non-autonomous Kirchhoff wave models with strong nonlinear damping: u tt - (1 + ϵ ‖ ∇ u ‖ 2 ) Δ u - σ (‖ ∇ u ‖ 2) Δ u t + f (u) = g (x , t) , where ϵ ∈ [ 0 , 1 ] is an extensibility parameter. It shows that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: 1 ≤ p < p ∗ ∗ ≡ N + 4 (N - 4) + , (i) the related process has a pullback attractor A ϵ in natural energy space H = (H 0 1 ∩ L p + 1) × L 2 for each ϵ , and it is upper semicontinuous on the perturbation ϵ ; (ii) the related process has a partially strong pullback exponential attractor M ϵ for each ϵ , and it is Hölder continuous on ϵ ∈ [ 0 , 1 ] . These results deepen and extend those in recent literatures (Chueshov in J Diff Equ 252:1229–1262, 2012; Ding et al. in Appl Math Lett 76:40–45, 2018; Wang and Zhong in Discrete Contin Dyn Syst 7:3189–3209, 2013). The main novelty of this paper is that it provides a new method to investigate the upper semicontinuity of pullback attractors and the stability of pullback exponential attractors in supercritical nonlinearity case. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stability problem on perturbation near a physically steady state solution of the 3D generalized incompressible magnetohydrodynamic system in Lei-Lin space. The global stability and analytic estimates for small perturbation are established by the semigroup method in the critical space χ 1 - 2 α (R 3) with 1 2 ≤ α ≤ 1 , where linear terms from perturbation incur much difficulty. By introducing a diagonalization process we successfully eliminate the linear terms. Then, by virtue of the analytic estimates for a solution, the temporal decay rate (1 + t) - (5 4 α - 1) of the global solution is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying the existence or non-existence of solutions for a class of elliptic problems involving the N-Laplacian operator in the whole space. The nonlinearity considered involves critical Trudinger–Moser growth. Our approach is non-variational, and in this way we can address a wide range of problems not yet contained in the literature. Even W 1 , N (R N) ↪ L ∞ (R N) failing, we establish ‖ u ‖ L ∞ (R N) ≤ C ‖ u ‖ W 1 , N (R N) Θ (for some Θ > 0 ), when u is a solution. To conclude, we explore some asymptotic properties. [ABSTRACT FROM AUTHOR]

In this paper, the invasive speed selection of the monostable travelling wave for a three-component lattice Lotka–Volterra competition system is studied via the upper and lower solution method, as well as the comparison principle. By constructing several special upper and lower solutions, we establish sufficient conditions such that the linear or nonlinear selection is realized. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$. [ABSTRACT FROM AUTHOR]

In this paper, a classical Stefan problem is studied. It is assumed that a small, time-dependent heat influx is present at the boundary, and that the initial values are small. By using a multiple timescales perturbation approach, it is shown analytically (most likely for the first time in the literature) how the moving interface and its stability are influenced by the time-dependent heat influx at the boundary and by the initial conditions. Accurate approximations of the solution of the problem are constructed, which are valid on long timescales. The constructed approximations turn out to agree very well with solutions of problems for which similarity solutions are available (in numerical form). [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of time periodic solutions for the quantum version of the nonlinear Schrödinger equation. The uniqueness can also be proved by applying the Runge-Kutta-Matsaev’s theory Bai (J. Math. Anal. Appl. 444(1):721–736, 2016) and Phragmén-Lindelöf method Li and Li (Acta Math. Sci. Ser. A. 40(5):1248–1258, 2020). In particular, the steady equation and the quantum effects are taken into account which play an exceptionally important role in time decay estimates, especially in the the case when the equation is periodic with respect to the Schrödinger norm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem $-\Delta u =-u^{-\beta }\chi _{\{u>0\}} + f(u)$ in $\Omega$ with $u=0$ on $\partial \Omega$ , where $0 and $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$ , $N\geq 2$. We are able to solve this problem provided $f$ has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=\lambda s+s^{\frac {N+2}{N-2}}$ and $N\geq 3$. In this case, we are able to obtain a solution for large values of $\lambda$. We replace the singular function $u^{-\beta }$ by a function $g_\epsilon (u)$ which pointwisely converges to $u^{-\beta }$ as $\epsilon \rightarrow 0$. The corresponding energy functional to the perturbed equation $-\Delta u + g_\epsilon (u) = f(u)$ has a critical point $u_\epsilon$ in $H_0^{1}(\Omega)$ , which converges to a non-trivial non-negative solution of the original problem as $\epsilon \rightarrow 0$. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Choquard equations with small perturbation f − Δ u + V (x) u = (I α * | u | p) | u | p − 2 u + f (x) , x ∈ R N. where N ≥ 3 and Iα denotes the Riesz potential. As is known that the above equation has a ground state uα and a bound state vα by fibering maps (see [22] or [23]), our aim is to show that for fixed p ∈ (1 , N N − 2) , uα and vα converge to a ground state and a bound state of the limiting local problem respectively, as α → 0. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the detection of the shape and location of a discontinuous source term from the knowledge of boundary measurements. We propose a non-iterative reconstruction algorithm based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The inverse source problem is formulated as a topology optimization one. A topological sensitivity analysis is derived from an energy-like cost function. The unknown shape of the term source support is reconstructed using a level-set curve of the topological gradient. The efficiency of our algorithm is illustrated by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the multiplicity of solutions for a class of noncooperative Schrödinger systems in R N . The systems involves a variable exponent elliptic operators with critical nonlinearity. By applying the Limit Index Theory developed by Li (Nonlinear Anal 25, 1371–1389, 1995) and utilizing a version of the concentration-compactness principle and the principle of symmetric criticality of Krawcewicz and Marzantowicz, we obtain a sequence of solutions under appropriate assumptions. [ABSTRACT FROM AUTHOR]

In this paper we adopt an alternative, analytical approach to Arnol'd problem [4] about the existence of closed and embedded K-magnetic geodesics in the round 2-sphere S 2 , where K : S 2 → R is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo's theorem [21] and Bottkoll results [7]. [ABSTRACT FROM AUTHOR]

This paper deals with a class of nonlinear scalar field equations with an inhomogeneous perturbation. Two positive solutions were obtained using the variational methods. [ABSTRACT FROM AUTHOR]

In this paper, the Schrödinger operator with a localized potential in a multidimensional cylinder is considered. The boundary of the cylinder is split into three parts, two of which are "sleeves" going to infinity, and the third (central) part is located between them. On the sleeves and the central part, respectively, the Neumann and Dirichlet boundary conditions are posed. We examine the situation where the distance between the sleeves increases. We assume that the same Schrödinger operator in the same cylinder endowed with the Dirichlet condition on the whole boundary has an isolated double eigenvalue. We show that for a sufficiently large distance between the sleeves, this double eigenvalue splits into a pair of resonances of the original operator. For these resonances, we explicitly obtain the first terms of their asymptotic expansions and describe the behavior of the imaginary part of the resonances. [ABSTRACT FROM AUTHOR]

In this paper, we study the differential inclusion associated with the minimal surface system for two-dimensional graphs in R 2 + n . We prove regularity of W 1 , 2 solutions and a compactness result for approximate solutions of this differential inclusion in W 1 , p . Moreover, we make a perturbation argument to infer that for every R > 0, there exists α (R) > 0 such that R-Lipschitz stationary points for functionals α-close in the C2 norm to the area functional are always regular. We also use a counterexample of B. Kirchhem (2003) to show the existence of irregular critical points to inner variations of the area functional. [ABSTRACT FROM AUTHOR]

In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with anisotropic p-Laplacian and matrix-valued $$L^\infty (\varOmega ,\mathbb {R}^{N\times N})$$ -controls in its coefficients and a nonlinear equation of Hammerstein type. Using the direct method in calculus of variations, we prove the existence of an optimal control in considered problem and provide sensitivity analysis for a specific case of considered problem with respect to two-parameter regularization. [ABSTRACT FROM AUTHOR]

This paper considers the propagation of TE‐modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. Using the Floquet–Bloch theory in negative‐order Sobolev spaces, we characterize the precise number of eigenvalues created by the line defect in terms of the band functions of the original periodic background medium for arbitrarily weak defects. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of at least three distinct solutions for a perturbed anisotropic discrete Dirichlet problem. The approach is based on variational methods. We also provide two examples in order to illustrate the results. [ABSTRACT FROM AUTHOR]

We consider reaction–diffusion equations on closed surfaces in R 3 having genus 1. Stable nonconstant stationary solutions are often called patterns. The purpose of this paper is to construct closed surfaces together with patterns having as many critical points as one wants. [ABSTRACT FROM AUTHOR]

In this paper we consider the quasilinear critical problem - Δ p u = λ u q - 2 u + u p ⋆ - 2 u in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N with smooth boundary, - Δ p u : = div (∇ u p - 2 ∇ u) is the p-Laplacian, N ≥ 3 , 1 < p ≤ q < p ⋆ , p ⋆ : = N p / N - p , and λ > 0 is a parameter. We investigate the multiplicity of sign-changing solutions to the problem and find the phenomenon depending on the positive solutions. Precisely, we show that the problem admits infinitely many pairs of sign-changing solutions when a positive solution exists. These results complete those obtained in Schechter and Zou (On the Brézis–Nirenberg problem, Arch Rational Mech Anal 197:337–356, 2010) for the cases p = q = 2 and N ≥ 7 , and in Azorero and Peral (Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues, Commun Partial Differ Equ 12:1389–1430, 1987) for the case of one positive solution. Our approach is based on variational methods combining upper-lower solutions and truncation techniques, and flow invariance arguments. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global existence and stability of strong shock front for one-dimensional piston problem. We use the exothermically reacting Euler equations as a mathematical model to describe the gas motion. Under the assumptions that the total variations of initial data and the perturbation of piston velocity are sufficiently small, we establish the global existence and asymptotic behavior of entropy solutions including a strong shock front without restriction on the strength. A modified fractional wave front tracking scheme is developed and a modified Glimm-type functional is carefully designed. [ABSTRACT FROM AUTHOR]

Recently we have proposed a monostable reaction-diffusion system to explain the Neolithic transition from hunter-gatherer life to farmer life in Europe. The system is described by a three-component system for the populations of hunter-gatherer (H), sedentary farmer (F1) and migratory one (F2). The conversion between F1 and F2 is specified by such a way that if the total farmers F1 + F2 are overcrowded, F1 actively changes to F2, while if it is less crowded, the situation is vice versa. In order to include this property in the system, the system incorporates a critical parameter (say F0) depending on the development of farming technology in a monotonically increasing way. It determines whether the total farmers are either over crowded (F1 + F2 >F0) or less crowded (F1 + F2 0) ([9, 20]). Previous numerical studies indicate that the structure of travelling wave solutions of the system is qualitatively similar to the one of the Fisher-KPP equation, that the asymptotically expanding velocity of farmers is equal to the minimal velocity (say cm(F0)) of travelling wave solutions, and that cm(F0) is monotonically decreasing as F0 increases. The latter result suggests that the development of farming technology suppresses the expanding velocity of farmers. As a partial analytical result to this property, the purpose of this paper is to consider the two limiting cases where F0 = 0 and F0 → ∞, and to prove cm(0)>cm(∞). [ABSTRACT FROM AUTHOR]

This paper concerns ecological invasion phenomenon of species based on the diffusive Lotka–Volterra competition model. We investigate the spreading speed (or the minimal wave speed of traveling waves) selection to the model and concentrate on the conjecture raised by Roques et al. (J Math Biol 71(2):465–489, 2015). By using an abstract implicit function theorem in a weighted functional space coupled with a perturbation technique, we not only prove this conjecture, but also show that the fast decay behavior of the first species is necessary and sufficient for the nonlinear speed selection of the whole system. This may lead to further significant results on the answer to the original Hosono's conjecture, a problem that has been outstanding for more than twenty years. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of elliptic problems of p-Kirchhoff type with critical exponent in bounded domains and new results as regards the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and variational method. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study the long time behavior, in terms of upper semicontinuous property of trajectory attractor of perturbed 3D Navier-Stokes equations by adding a high-order term of hyperdissipation ( l-Lapalcian). We show that the trajectory attractor of the perturbed system converges to the trajectory attractor of the 3D Navier-Stokes system as time tends to infinity and the artificial dissipation parameter approaches zero. [ABSTRACT FROM AUTHOR]

This paper is the first attempt to systematically study properties of the effective Hamiltonian H¯ arising in the periodic homogenization of some coercive but nonconvex Hamilton-Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min-max formulas for a class of nonconvex H¯. Secondly, we analytically and numerically investigate other related interesting phenomena, such as “quasi-convexification” and breakdown of symmetry, of H¯ from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex Hamilton-Jacobi equations. Some conjectures and problems are also proposed. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of multi-bump solutions for the following Kirchhoff equation - a + b ∫ R 3 | ∇ u | 2 d x Δ u + u = (1 - ε q (x)) | u | p - 2 u , x ∈ R 3 , where a , b > 0 , 2 < p < 6 and q (x) ∈ C (R 3 , R +) satisfies some suitable conditions. By using the Lyapunov–Schmidt reduction method, we extend the results in [14] to the Kirchhoff problem. [ABSTRACT FROM AUTHOR]

We investigate the following problem - div (v (x) | ∇ u | m - 2 ∇ u) + V (x) | u | m - 2 u = | x | - θ ∗ | u | b | x | α | u | b - 2 | x | α u + λ | x | - γ ∗ | u | c | x | β | u | c - 2 | x | β u in R N , where b , c , α , β > 0 , θ , γ ∈ (0 , N) , N ≥ 3 , 2 ≤ m < ∞ and λ ∈ R . Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of solutions for the following problem: $$\begin{equation*} \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+\theta h(x,u), \ \ x\in \Omega,\\ u=0, \ \ x\in \partial\Omega, \end{cases} \end{equation*}$$ where α ≥ 2, Ω is a smooth bounded domain in ${\mathbb{R}}$N, θ is a parameter and g, h ∈ C($\bar{\Omega}$ × ${\mathbb{R}}$). Under the assumptions that g(x, u) is odd and locally superlinear at infinity in u, we prove that for any j ∈ $\mathbb{N}$ there exists ϵj > 0 such that if |θ| ≤ ϵj, the above problem possesses at least j distinct solutions. Our results generalize some known results in the literature and are new even in the symmetric situation. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the following nonlinear Schrödinger system in $$\mathbb {R}^{3}$$ where $$\mu _{1}>0$$ , $$\mu _{2}>0$$ , $$2\le p,q\le 5$$ , $$p+q\le 6$$ and $$\beta \in \mathbb {R}$$ is a coupling constant. We show that the nonlinear Schrödinger system has infinitely many non-radial positive segregated solutions. [ABSTRACT FROM AUTHOR]

This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. It is shown that arbitrarily weak perturbations introduce spectrum into the spectral gaps of the background operator. [ABSTRACT FROM AUTHOR]

In this paper we give the complete description of the structure of compact global (forward) attractors for non-autonomous perturbations of discrete autonomous gradient-like dynamical systems under the assumption that the original discrete autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed non-autonomous (in particular-periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular the perturbed systems has the same number of-periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff solutions). It is shown that the compact global (forward) attractor of non-autonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections. [ABSTRACT FROM AUTHOR]