We describe a general N-solitonic solution of the focusing nonlinear Schrodinger equation in the presence of a condensate by using the dressing method. We give the explicit form of one- and two-solitonic solutions and study them in detail as well as solitonic atoms and degenerate solutions. We distinguish a special class of solutions that we call regular solitonic solutions. Regular solitonic solutions do not disturb phases of the condensate at infinity by coordinate. All of them can be treated as localized perturbations of the condensate. We find a broad class of superregular solitonic solutions which are small perturbations at a certain moment of time. Superregular solitonic solutions are generated by pairs of poles located on opposite sides of the cut. They describe the nonlinear stage of the modulation instability of the condensate and play an important role in the theory of freak waves.

In this paper we present a quasi-static formulation of a phase-field model for a pressurized crack in a poroelastic medium. The mathematical model represents a linear elasticity system with a fading Gassman tensor as the crack grows, that is coupled with a variational inequality for the phase-field variable containing an entropy inequality. We introduce a novel incremental approximation that decouples displacement and phase-field problems. We establish convergence to a solution of the quasi-static problem, including Rice's condition, when the time discretization step goes to zero. Numerical experiments confirm the robustness and efficiency of this approach for multidimensional test cases.

We consider the regularity criteria for the incompressible Navier- Stokes equations connected with one velocity component. Based on the method from Cao and Titi (2008 Indiana Univ. Math. J. 57 2643-61) we prove that the weak solution is regular, provided u(3) is an element of L-t(0, T; L-s(R-3)), 2/t + 3/s 10/3 or provided del u3 is an element of L-t (0, T; L-s(R-3)), 2/t + 3/s <= 19/12 + 1/2s if s is an element of (30/19, 3] or 2/t + 3/s <= 3/2 + 3/4s if s is an element of (3, infinity]. As a corollary, we also improve the regularity criteria expressed by the regularity of partial derivative p/partial derivative x(3) or partial derivative u(3)/partial derivative x(3).

In this paper, we study a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator. As a particular case, we consider the following diffusion problem {partial derivative(t)u + M ([u](s)(2) )(-Delta)s u = vertical bar u vertical bar(p-2)u in Omega x R+, partial derivative(t)u = partial derivative u/partial derivative t, u(x,t) = 0 in (R-N \ Omega) x R+, u(x, 0) = u(0)(x) in Omega, where [u](s) is the Gagliardo seminorm of u, Omega subset of R-N is a bounded domain with Lipschitz boundary, (-Delta)(s) isthe fractional-Laplacian with 0 R+ is the initial function, and M : R-0(+)-> R-0(+) is continuous. Under some appropriate conditions, the local existence of nonnegative solutions is obtained by employing the Galerkin method. Then, by virtue of a differential inequality technique, we prove that the local nonnegative solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give an estimate for the lower and upper bounds of the blow-up time. The main novelty is that our results cover the degenerate case, that is, the coefficient of (-Delta)(s) could be zero at the origin.

We develop the inverse scattering transform (IST) method for the Degasperis-Procesi equation. The spectral problem is an sl(3) Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST, such as the construction of fundamental analytic solutions, the formulation of a Riemann-Hilbert problem, and the implementation of the dressing method, are presented.

The focusing nonlinear Schrodinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media; MI is considered the main physical mechanism for the appearance of rogue (anomalous) waves (RWs) in nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for periodic initial perturbations of the unstable background solution of NLS exciting just one of the unstable modes. We distinguish two cases. In the case in which only the corresponding unstable gap is theoretically open, the solution describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and the nonlinear RW stages are described by the 1-breather Akhmediev solution, whose parameters, different at each RW appearence, are always given in terms of the initial data through elementary functions. If the number of unstable modes is >1, this uniform in t dynamics is sensibly affected by perturbations due to numerics and/or real experiments, provoking O(1) corrections to the result. In the second case in which more than one unstable gap is open, a detailed investigation of all these gaps is necessary to get a uniform in t dynamics, and this study is postponed to a subsequent paper. It is however possible to obtain the elementary description of the first nonlinear stage of MI, given again by the Akhmediev 1-breather solution, and how perturbations due to numerics and/or real experiments can affect this result. Since the solution of the Cauchy problem is given in terms of different elementary functions in different time intervals, obviously matching in the corresponding overlapping regions, an alternative approach, based on matched asymptotic expansions, is suggested and presented in a separate paper.

This paper deals with positive radially symmetric solutions of the Neumann boundary value problem for the fully parabolic chemotaxis system, {u(t) = Delta u - del . (u del chi(v)) in Omega x (0, infinity), tau v(t) = Delta v - v + u in Omega x (0, infinity), in a ball Omega subset of R-2 with general sensitivity function chi(v) satisfying chi' > 0 and decaying property chi'(s) -> 0 (s -> infinity), parameter tau is an element of (0, 1] and nonnegative radially symmetric initial data. It is shown that if tau is an element of (0, 1] is sufficiently small, then the problem has a unique classical radially symmetric solution, which exists globally and remains uniformly bounded in time. Especially, this result establishes global existence of solutions in the case chi(v) = chi(0) log v for all chi(0) > 0, which has been left as an open problem.

We consider the Fisher-KPP equation with a non-local saturation effect defined through an interaction kernel phi(x) and investigate the possible differences with the standard Fisher-KPP equation. Our first concern is the existence of steady states. We prove that if the Fourier transform (phi) over cap(xi) is positive or if the length sigma of the non-local interaction is short enough, then the only steady states are u equivalent to 0 and u equivalent to 1. Next, we study existence of the travelling waves. We prove that this equation admits travelling wave solutions that connect u = 0 to an unknown positive steady state u(infinity)(x), for all speeds c >= c*. The travelling wave connects to the standard state u(infinity)(x) = 1 under the aforementioned conditions: (phi) over cap(xi) > 0 or sigma is sufficiently small. However, the wave is not monotonic for sigma large.

Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott-Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed.

We develop a detailed analysis of edge bifurcations of standing waves in the nonlinear Schrodinger (NLS) equation on a tadpole graph (a ring attached to a semi-infinite line subject to the Kirchhoff boundary conditions at the junction). It is shown in the recent work [7] by using explicit Jacobi elliptic functions that the cubic NLS equation on a tadpole graph admits a rich structure of standing waves. Among these, there are different branches of localized waves bifurcating from the edge of the essential spectrum of an associated Schrodinger operator. We show by using a modified Lyapunov-Schmidt reduction method that the bifurcation of localized standing waves occurs for every positive power nonlinearity. We distinguish a primary branch of never vanishing standing waves bifurcating from the trivial solution and an infinite sequence of higher branches with oscillating behavior in the ring. The higher branches bifurcate from the branches of degenerate standing waves with vanishing tail outside the ring. Moreover, we analyze stability of bifurcating standing waves. Namely, we show that the primary branch is composed by orbitally stable standing waves for subcritical power nonlinearities, while all nontrivial higher branches are linearly unstable near the bifurcation point. The stability character of the degenerate branches remains inconclusive at the analytical level, whereas heuristic arguments based on analysis of embedded eigenvalues of negative Krein signatures support the conjecture of their linear instability at least near the bifurcation point. Numerical results for the cubic NLS equation show that this conjecture is valid and that the degenerate branches become spectrally stable far away from the bifurcation point.

General soliton solutions to a nonlocal nonlinear Schrodinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions are considered via the combination of Hirota's bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. First, general N-soliton solutions with zero boundary conditions are constructed. Starting from the tau functions of the two-component KP hierarchy, it is shown that they can be expressed in terms of either Gramian or double Wronskian determinants. On the contrary, from the tau functions of single component KP hierarchy, general soliton solutions to the nonlocal NLS equation with nonzero boundary conditions are obtained. All possible soliton solutions to nonlocal NLS with Parity (PT)-symmetry for both zero and nonzero boundary conditions are found in the present paper.

In this paper, a class of neural networks with time-varying delays are investigated for the first time using a periodically intermittent control technique. First, some new and useful stabilization criteria and synchronization conditions based on p-norm are derived by introducing multi-parameters and using the Lyapunov functional technique. For infinity-norm, using the analysis technique, some novel conditions ensuring exponential stability and synchronization are also obtained. It is worth noting that the methods used in this paper are totally different from the corresponding previous works and the obtained conditions are less conservative. Particularly, the traditional assumptions on control width and time delay are removed in this paper. Finally, some numerical simulations are given to verify the theoretical results.

In this paper we study relations of various types of sensitivity between a t.d.s. (X, T) and t.d.s. (M(X), T-M) induced by (X, T) on the space of probability measures. Among other results, we prove that F -sensitivity of (M(X), TM) implies the same of (X, T) and the converse is also true when F is a filter. We show that (X, T) is multi-sensitive if and only if so is (M(X), TM) and that (X, T) is F -sensitive if and only if (M-n(X), T-M) is F -sensitive (for some/all n is an element of N). We finish the paper providing an example of a minimal syndetically sensitive t.d.s. or a Li-Yorke sensitive t.d.s. such that induced t.d.s. fails to be sensitive.

We consider the aggregation equation rho(t) - del . (rho del K * rho) = 0 in R-n, where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials for which the equilibria are of finite density and compact support. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. In particular, we consider a potential for which the corresponding equilibrium solutions are of uniform density inside a ball of R-n and zero outside. For such a potential, various explicit calculations can be carried out in detail. In one dimension we fully solve the temporal dynamics, and in two or higher dimensions we show the global stability of this steady state within the class of radially symmetric solutions. Finally, we solve the following restricted inverse problem: given a radially symmetric density. (rho) over bar that is zero outside some ball of radius R and is polynomial inside the ball, construct an interaction potential K for which (rho) over bar is the steady-state solution of the corresponding aggregation equation. Throughout the paper, numerical simulations are used to motivate and validate the analytical results.

This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion {u(t) = del. (D(u)del u - chi del . (u del v) - xi del . (u del w) + mu u(1 - u - w), x is an element of Omega, t > 0, v(t) = Delta v - v + u, x is an element of Omega, t > 0, w(t) = -vw x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a bounded smooth domain Omega subset of R-n, n = 2,3, 4, where chi, xi and mu are given nonnegative parameters. The diffusivity D(u) is assumed to satisfy D(u) >= delta um(-1) for all u > 0 with some delta > 0. It is proved that for sufficiently regular initial data global bounded solutions exist whenever m> 2- 2/n. For the case of non- degenerate diffusion (i.e. D(0) > 0) the solutions are classical; for the case of possibly degenerate diffusion (D(0) >= 0), the existence of bounded weak solutions is shown.

In this paper, we consider a susceptible-infected-susceptible (SIS) reaction-diffusion model, where the rates of disease transmission and recovery are assumed to be spatially heterogeneous and temporally periodic and the total population number is constant. We introduce a basic reproduction number R-0 and establish threshold-type results on the global dynamics in terms of R-0. In particular, we obtain the asymptotic properties of R-0 with respect to the diffusion rate d(I) of the infected individuals, which exhibit the delicate influence of the time-periodic heterogeneous environment on the extinction and persistence of the infectious disease. Our analytical results suggest that the combination of spatial heterogeneity and temporal periodicity tends to enhance the persistence of the disease.

We consider the multidimensional aggregation equation u(t) - del. (u del K * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun. Math. Sci. 274 717-35), whereas for C-2 kernels there is no finite-time blow-up. We prove, under mild monotonicity assumptions on the kernel K, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded nonnegative initial data. When the Osgood condition is violated, we present a new proof of finite time blow-up that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally, we conclude with a discussion of similarity solutions for the case K( x) = vertical bar x vertical bar and some open problems.

In this paper, the exponential stability problem is investigated for a class of Cohen-Grossberg-type bidirectional associative memory neural networks with time-varying delays. By using the analysis method, inequality technique and the properties of an M-matrix, several novel sufficient conditions ensuring the existence, uniqueness and global exponential stability of the equilibrium point are derived. Moreover, the exponential convergence rate is estimated. The obtained results are less restrictive than those given in the earlier literature, and the boundedness and differentiability of the activation functions and differentiability of the time-varying delays are removed. Two examples with their simulations are given to show the effectiveness of the obtained results.

We study a system of three partial differential equations modelling the spatio-temporal behaviour of two competitive populations of biological species both of which are attracted chemotactically by the same signal substance. More precisely, we consider the initial-boundary value problem for {u(t) = d(1)Delta u - chi(1)del . (u del w) + mu(1)u(1 - u - a(1)v), x is an element of Omega, t > 0, v(t) = d(2)Delta v - chi(2)del . (v del w) + mu(2)v(1 - a(2)u - v), x is an element of Omega, t > 0, -Delta w + lambda w = u + v, x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-n, n >= 1, with smooth boundary. When 0 <= a(1) < 1 and 0 <= a(2) < 1, this system possesses a uniquely determined spatially homogeneous positive equilibrium (u(star), v(star)). We show that given any such a(1) and a(2) and any positive diffusivities d(1) and d(2) and cross-diffusivities chi(1) and chi(2), this steady state is globally asymptotically stable within a certain nonempty range of the logistic growth coefficients mu(1) and mu(2).

We are interested in the attractive Gross-Pitaevskii (GP) equation in R-2, where the external potential V(x) vanishes on m disjoint bounded domains Omega(i) subset of R-2 (i = 1, 2, . . . , m) and V(x) -> infinity as vertical bar x vertical bar -> infinity, that is, the union of these Omega(i) is the bottom of the potential well. By establishing some delicate estimates on the associated energy functional of the GP equation, we prove that when the interaction strength a approaches some critical value a*, the ground states concentrate and blow up at the center of the incircle of some Omega(j) with the largest inradius. Moreover, under some further conditions on V(x), we show that the ground states of the GP equations are unique and radially symmetric at least for almost every a is an element of (0, a*).