A chimera state is a spatio-temporal pattern in a network of identical coupled oscillators in which synchronous and asynchronous oscillation coexist. This state of broken symmetry, which usually coexists with a stable spatially symmetric state, has intrigued the nonlinear dynamics community since its discovery in the early 2000s. Recent experiments have led to increasing interest in the origin and dynamics of these states. Here we review the history of research on chimera states and highlight major advances in understanding their behaviour.

A nonlocal nonlinear Schrodinger (NLS) equation was recently introduced and shown to be an integrable infinite dimensional Hamiltonian evolution equation. In this paper a detailed study of the inverse scattering transform of this nonlocal NLS equation is carried out. The direct and inverse scattering problems are analyzed. Key symmetries of the eigenfunctions and scattering data and conserved quantities are obtained. The inverse scattering theory is developed by using a novel left-right Riemann-Hilbert problem. The Cauchy problem for the nonlocal NLS equation is formulated and methods to find pure soliton solutions are presented; this leads to explicit time-periodic one and two soliton solutions. A detailed comparison with the classical NLS equation is given and brief remarks about nonlocal versions of the modified Korteweg-de Vries and sine-Gordon equations are made.

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

This paper focuses on the following scalar field equation involving a fractional Laplacian: (-Delta)(alpha)u = g(u) in R-N, where N >= 2, alpha is an element of (0, 1), (-Delta)(alpha) stands for the fractional Laplacian. Using some minimax arguments, we obtain a positive ground state under the general Berestycki-Lions type assumptions.

Two new integrable nonlocal Davey-Stewartson equations are introduced. These equations provide two-spatial dimensional analogues of the integrable, nonlocal nonlinear Schro-dinger equation introduced in Ablowitz and Musslimani (2013 Phys. Rev. Lett. 110 064105). Furthermore, like the latter equation, they also possess a PT symmetry and, as it is well known, this symmetry is important for the occurence of such equations in nonlinear optics. A method for solving the initial value problem of these integrable equations is discussed. It is shown that the technique used for constructing these novel integrable equations has general validity; as an illustrative example, an additional two-dimensional integrable generalization of the nonlocal nonlinear Schrodinger is also presented.

In this paper, we investigate the multiplicity of solutions for a p-Kirchhoff system driven by a nonlocal integro-differential operator with zero Dirichlet boundary data. As a special case, we consider the following fractional p-Kirchhoff system {(Sigma(k)(i=1) [u(i)](s,p)(p))(theta-1) (-Delta)(p)(s)u(j)(x) = lambda(j)vertical bar u(j)vertical bar(q-2)u(j) + Sigma(i not equal j) beta(ij)vertical bar u(i)vertical bar(m)vertical bar u(j)vertical bar(m-2)u(j) in Omega, u(j) = 0 in R-N\Omega, where [u(j)](s,p) = (integral integral(2N)(R) vertical bar u(j)(x)-u(j)(y)vertical bar(p)/vertical bar x-y vertical bar(N+ps)dxdy)(1/p), j = 1,2, ..., k, k >= 2, theta >= 1, Omega is an open bounded subset of R-N with Lipschitz boundary partial derivative Omega, N > ps with s is an element of (0, 1), (-Delta)(p)(s) is the fractional p-Laplacian, lambda(j) > 0 and beta(ij) = beta(ji) for i not equal j, j= 1, 2, ..., k. When 1 0 for all 1 0 for all 1 <= i < j <= k, the existence of infinitely many solutions is obtained by applying the symmetric mountain pass theorem. To our best knowledge, our results for the above system are new in the study of Kirchhoff problems.

When computing a trajectory of a dynamical system, influence of noise can lead to large perturbations which can appear, however, with small probability. Then when calculating approximate trajectories, it makes sense to consider errors small on average, since controlling them in each iteration may be impossible. Demand to relate approximate trajectories with genuine orbits leads to various notions of shadowing (on average) which we consider in the paper. As the main tools in our studies we provide a few equivalent characterizations of the average shadowing property, which also partly apply to other notions of shadowing. We prove that almost specification on the whole space induces this property on the measure center which in turn implies the average shadowing property. Finally, we study connections among sensitivity, transitivity, equicontinuity and (average) shadowing.

The aim of this paper is to establish the multiplicity of weak solutions for a Kirchhoff-type problem driven by a fractional p-Laplacian operator with homogeneous Dirichlet boundary conditions: {M(integral integral(R2N) vertical bar u(x) - u(y)vertical bar(p)/vertical bar x - y vertical bar(N+ps)dxdy)(-Delta)(p)(s)u(x) = f(x, u) in Omega u=0 in R-N/Omega where Omega is an open bounded subset of R-N with Lipshcitz boundary partial derivative Omega, (-Delta)(p)(s) is the fractional p-Laplacian operator with 0 < s < 1 < p < N such that sp < N, M is a continuous function and f is a Caratheodory function satisfying the Ambrosetti-Rabinowitz-type condition. When f satisfies the suplinear growth condition, we obtain the existence of a sequence of nontrivial solutions by using the symmetric mountain pass theorem; when f satisfies the sublinear growth condition, we obtain infinitely many pairs of nontrivial solutions by applying the Krasnoselskii genus theory. Our results cover the degenerate case in the fractional setting: the Kirchhoff function M can be zero at zero.

Our discovery of multi-rogue wave (MRW) solutions in 2010 completely changed the viewpoint on the links between the theory of rogue waves and integrable systems, and helped explain many phenomena which were never understood before. It is enough to mention the famous Three Sister waves observed in oceans, the creation of a regular approach to studying higher Peregrine breathers, and the new understanding of 2 + 1 dimensional rogue waves via the NLS-KP correspondence. This article continues the study of the MRW solutions of the NLS equation and their links with the KP-I equation started in a previous series of articles (Dubard et al 2010 Eur. Phys. J. 185 247-58, Dubard and Matveev 2011 Natural Hazards Earth Syst. Sci. 11 667-72, Matveev and Dubard 2010 Proc. Int. Conf. FNP-2010 (Novgorod, StPetersburg) pp 100-101, Dubard 2010 PhD Thesis). In particular, it contains a discussion of the large parametric asymptotics of these solutions, which has never been studied before.

The aim of this review is to introduce the reader to some of the physical notions and the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyse some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g. the linear or the nonlinear limit or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.

We consider here solutions of the nonlinear fractional Schrodinger equation epsilon(2s) (-Delta)(s) u + V (x) u = u(p). We show that concentration points must be critical points for V. We also prove that if the potential V is coercive and has a unique global minimum, then ground states concentrate suitably at such a minimal point as epsilon tends to zero. In addition, if the potential V is radial and radially decreasing, then the minimizer is unique provided epsilon is small.

This paper studies the global asymptotic stability of neural networks of neutral type with mixed delays. The mixed delays include constant delay in the leakage term (i.e. 'leakage delay'), time-varying delays and continuously distributed delays. Based on the topological degree theory, Lyapunov method and linear matrix inequality (LMI) approach, some sufficient conditions are derived ensuring the existence, uniqueness and global asymptotic stability of the equilibrium point, which are dependent on both the discrete and distributed time delays. These conditions are expressed in terms of LMI and can be easily checked by the MATLAB LMI toolbox. Even if there is no leakage delay, the obtained results are less restrictive than some recent works. It can be applied to neural networks of neutral type with activation functions without assuming their boundedness, monotonicity or differentiability. Moreover, the differentiability of the time-varying delay in the non-neutral term is removed. Finally, two numerical examples are given to show the effectiveness of the proposed method.

In this paper we study the existence of infinitely many weak solutions for equations driven by nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. A model for these operators is given by the fractional Laplacian -(-Delta)(s)u(x) := integral(Rn) u(x + y) + u(x - y) - 2u(x)/vertical bar y vertical bar(n+2s) dy, x is an element of R-n where s is an element of (0, 1) is fixed. We consider different superlinear growth assumptions on the nonlinearity, starting from the well-known Ambrosetti-Rabinowitz condition. In this framework we obtain three different results about the existence of infinitely many weak solutions for the problem under consideration, by using the Fountain Theorem. All these theorems extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

This paper investigates the synchronization of coupled chaotic systems with time delay in the presence of parameter mismatches by using intermittent linear state feedback control. Quasi-synchronization criteria are obtained by means of a Lyapunov function and the differential inequality method. Numerical simulations on the chaotic systems are presented to demonstrate the effectiveness of the theoretical results.

In this paper, we obtain a uniform Darboux transformation for multi-component coupled nonlinear Schrodinger (NLS) equations, which can be reduced to all previously presented Darboux transformations. As a direct application, we derive the single dark soliton and multi-dark soliton solutions for multi-component NLS equations with a defocusing case and a mixed focusing and defocusing case. Some exact single and two-dark solitons of three-component NLS equations are investigated explicitly. The results are meaningful for vector dark soliton studies in many physical systems, such as Bose-Einstein condensate, nonlinear optics, etc.

This work studies the initial value problem for a Camassa-Holm type equation with cubic nonlinearities that has been recently discovered by Vladimir Novikov to be integrable. For s > 3/2, using a Galerkin-type approximation method, it is shown that this equation is well-posed in Sobolev spaces H-s on both the line and the circle with continuous dependence on initial data. Furthermore, it is proved that this dependence is optimal by showing that the data-to-solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions in conjunction with well-posedness estimates.

The global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles. The key conditions for this approach are identified and previous results in the literature are unified and generalized. New existence results are obtained for the population model with or without volume filling.

In this paper we study the existence and multiplicity of solutions for the critical fractional Schrodinger equation epsilon(2 alpha) (-Delta)(alpha)u + V (x)u = vertical bar u vertical bar(2)(-2)(alpha*) u + lambda f (u), x is an element of R-N, where epsilon and lambda are positive parameters, 0 2 alpha, 2(alpha)* = 2N/N-2 alpha is the fractional critical exponent; V is a positive continuous potential satisfying some conditions and f is a continuous subcritical nonlinear term. We prove that the equation has a nonnegative ground state solution and investigate the relation between the number of solutions and the topology of the set where V attains its minimum, for all sufficiently large lambda and small epsilon.

In the framework of the focusing nonlinear Schrodinger equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. The development of the MI leads to the formation of 'integrable turbulence' (Zakharov 2009 Stud. Appl. Math. 122 219-34). We study the time evolution of its major characteristics averaged across realizations of initial data-the condensate solution seeded by small random noise with fixed statistical properties. We observe that the system asymptotically approaches to the stationary integrable turbulence, however this is a long process. During this process momenta, as well as kinetic and potential energies, oscillate around their asymptotic values. The amplitudes of these oscillations decay with time t as t(-3/2), the phases contain the nonlinear phase shift that decays as t(-1/2), and the frequency of the oscillations is equal to the double maximum growth rate of the MI. The evolution of wave-action spectrum is also oscillatory, and characterized by formation of the power-law region similar to vertical bar k vertical bar(-alpha) in the small vicinity of the zeroth harmonic k = 0 with exponent a close to 2/3. The corresponding modes form 'quasi-condensate', that acquires very significant wave action and macroscopic potential energy. The probability density function of wave amplitudes asymptotically approaches the Rayleigh distribution in an oscillatory way. Nevertheless, in the beginning of the nonlinear stage the MI slightly increases the occurrence of rogue waves. This takes place at the moments of potential energy modulus minima, where the PDF acquires 'fat tales' and the probability of rogue waves occurrence is by about two times larger than in the asymptotic stationary state. Presented facts need a theoretical explanation.

We consider a paradigmatic spatially extended model of non-locally coupled phase oscillators which are uniformly distributed within a one-dimensional interval and interact depending on the distance between their sites' modulo periodic boundary conditions. This model can display peculiar spatio-temporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherence-incoherence pattern, or chimera state. For such patterns we formulate a general bifurcation analysis scheme based on a hierarchy of continuum limit equations. This provides the possibility of classifying known coherence-incoherence patterns and of suggesting directions for the search for new ones.