Showing papers in "Nonlinearity in 2007"
TL;DR: In this article, the existence of global strong solutions of the primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and the bottom boundaries including the varying bottom topography was proved.
Abstract: We prove the existence of global strong solutions of the primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and the bottom boundaries including the varying bottom topography. Previously, the existence of global strong solutions was known in the case of the Neumann boundary conditions in a cylindrical domain (Cao and Titi 2007 Ann. Math. 166 245–67).
192 citations
TL;DR: In this article, the authors developed and tested two new computational algorithms for predicting the mean linear response of a chaotic dynamical system to small changes in external forcing via the fluctuation-dissipation theorem (FDT): the short-time FDT (ST-FDT), and the hybrid Axiom A FDT, and the results showed that these two algorithms are numerically stable for all times, but less accurate for short times.
Abstract: In a recent paper the authors developed and tested two novel computational algorithms for predicting the mean linear response of a chaotic dynamical system to small changes in external forcing via the fluctuation-dissipation theorem (FDT): the short-time FDT (ST-FDT), and the hybrid Axiom A FDT (hA-FDT). Unlike the earlier work in developing fluctuation-dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, these two new methods are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. These two algorithms take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian (qG-FDT) approximation of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. It has been discovered that the ST-FDT algorithm is an extremely precise linear response approximation for short response times, but numerically unstable for longer response times. On the other hand, the hA-FDT method is numerically stable for all times, but is less accurate for short times. Here we develop blended linear response algorithms, by combining accurate prediction of the ST-FDT method at short response times with numerical stability of qG-FDT and hA-FDT methods at longer response times. The new blended linear response algorithms are tested on the nonlinear Lorenz 96 model with 40 degrees of freedom, chaotic behaviour, forcing, dissipation, and mimicking large-scale features of real-world geophysical models in a wide range of dynamical regimes varying from weakly to strongly chaotic, and to fully turbulent. The results below for the blended response algorithms have a high level of accuracy for the linear response of both mean state and variance throughout all the different chaotic regimes of the 40-mode model. These results point the way towards the potential use of the blended response algorithms in operational long-term climate change projection.
176 citations
TL;DR: For continuous self-maps of compact metric spaces, a preliminary study of stronger forms of sensitivity formulated in terms of 'large' subsets of. Mainly as mentioned in this paper consider'syndetic sensitivity' and 'cofinite sensitivity'.
Abstract: For continuous self-maps of compact metric spaces, we initiate a preliminary study of stronger forms of sensitivity formulated in terms of 'large' subsets of . Mainly we consider 'syndetic sensitivity' and 'cofinite sensitivity'. We establish the following. (i) Any syndetically transitive, non-minimal map is syndetically sensitive (this improves the result that sensitivity is redundant in Devaney's definition of chaos). (ii) Any sensitive map of [0,1] is cofinitely sensitive. (iii) Any sensitive subshift of finite type is cofinitely sensitive. (iv) Any syndetically transitive, infinite subshift is syndetically sensitive. (v) No Sturmian subshift is cofinitely sensitive. (vi) We construct a transitive, sensitive map which is not syndetically sensitive.
150 citations
TL;DR: In this article, a system of semilinear parabolic equations with a free boundary was studied in a predator-prey ecological model. And the conditions for the existence and uniqueness of a classical solution were obtained.
Abstract: This article is concerned with a system of semilinear parabolic equations with a free boundary, which arises in a predator–prey ecological model The conditions for the existence and uniqueness of a classical solution are obtained The evolution of the free boundary problem is studied It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time
149 citations
TL;DR: In this article, the links among the different approaches and the limitations of these approaches are not fully understood, providing: (a) analysis of the theoretical models; (b) discussion of the rigorous mathematical results; (c) identification of the physical mechanisms underlying the validity of theoretical predictions, for a wide range of phenomena.
Abstract: The fluctuations in nonequilibrium systems are under intense theoretical and experimental investigation. Topical 'fluctuation relations' describe symmetries of the statistical properties of certain observables, in a variety of models and phenomena. They have been derived in deterministic and, later, in stochastic frameworks. Other results first obtained for stochastic processes, and later considered in deterministic dynamics, describe the temporal evolution of fluctuations. The field has grown beyond expectation: research works and different perspectives are proposed at an ever faster pace. Indeed, understanding fluctuations is important for the emerging theory of nonequilibrium phenomena, as well as for applications, such as those of nanotechnological and biophysical interest. However, the links among the different approaches and the limitations of these approaches are not fully understood. We focus on these issues, providing: (a) analysis of the theoretical models; (b) discussion of the rigorous mathematical results; (c) identification of the physical mechanisms underlying the validity of the theoretical predictions, for a wide range of phenomena.
125 citations
TL;DR: In this paper, it was shown that a generic area-preserving map from the Newhouse region is universal in the sense that its iterations approximate the dynamics of any other area preserving map with arbitrarily good accuracy.
Abstract: We show that maps with homoclinic tangencies of arbitrarily high orders and, as a consequence, with arbitrarily degenerate periodic orbits are dense in the Newhouse regions in spaces of real-analytic area-preserving two-dimensional maps and general real-analytic two-dimensional maps (the result was earlier known only for the space of smooth non-conservative maps). Based on this, we show that a generic area-preserving map from the Newhouse region is 'universal' in the sense that its iterations approximate the dynamics of any other area-preserving map with arbitrarily good accuracy. In fact, we show that every dynamical phenomenon which occurs generically in any open set of symplectic diffeomorphisms of a two-dimensional disc, or in any open set of finite-parameter families of such diffeomorphisms, can be encountered at a perturbation of any area-preserving two-dimensional map with a homoclinic tangency.
91 citations
TL;DR: In this article, the authors investigated the recurrence property of irrational rotations and showed that for a fixed y, the result is a metric inhomogeneous Diophantine approximation in an almost everywhere sense.
Abstract: We investigate the recurrence property of irrational rotations. Let T be the rotation by an irrational θ on the unit circle. We show that for a fixed y This result is a metric inhomogeneous Diophantine approximation in an almost everywhere sense.
71 citations
TL;DR: In this article, a construction of p-adic Gibbs measures which depend on weights λ is given, and an investigation of such measures is reduced to the examination of an infinite-dimensional recursion equation.
Abstract: In this paper we consider the countable state p-adic Potts model on the Cayley tree. A construction of p-adic Gibbs measures which depends on weights λ is given, and an investigation of such measures is reduced to the examination of an infinite-dimensional recursion equation. By studying the derived equation under some condition concerning weights, we prove the absence of a phase transition. Note that the condition does not depend on values of the prime p, and the analogous fact is not true when the number of spins is finite. For the homogeneous model it is shown that the recursive equation has only one solution under that condition on weights. This means that there is only one p-adic Gibbs measure μλ. The boundedness of the measure is also established. Moreover, the continuous dependence of the measure μλ on λ is proved. At the end we formulate a one limit theorem for μλ.
63 citations
TL;DR: In this paper, the authors proved the existence of a real solution y(x, T) with no poles on the real line of the following fourth order analogue of the Painleve I equation: x = T y - (1/6y(3) + 1/24(y(x)(2) +2yy(xx)) + 1 /240y(xxxx)).
Abstract: We establish the existence of a real solution y( x, T) with no poles on the real line of the following fourth order analogue of the Painleve I equation: x = T y - (1/6y(3) + 1/24(y(x)(2) +2yy(xx)) + 1/240y(xxxx)). This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for y( x, T) as x -> +/-infinity.
60 citations
TL;DR: In this article, an explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the stable periodic solutions is derived, using the theory of normal form and centre manifold.
Abstract: The dynamics of a scalar delay differential equation, which includes Mackey–Glass equations, are investigated. We prove that a sequence of Hopf bifurcations occurs at the equilibrium as the delay increases. An explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions is derived, using the theory of normal form and centre manifold. The global existence of multiple periodic solutions is established using a global Hopf bifurcation result given by Wu (1998 Trans. Am. Math. Soc. 350 4799) and a Bendixson criterion for higher dimensional ordinary differential equations given by Li and Muldowney (1993 J. Diff. Eqns 106 27).
59 citations
TL;DR: In this article, a method for the study of steady-state nonlinear modes for the Gross-Pitaevskii equation (GPE) is described, based on the exact statement about the coding of the steady state solutions of GPE which vanish as x → + ∞ by reals.
Abstract: A method for the study of steady-state nonlinear modes for the Gross–Pitaevskii equation (GPE) is described. It is based on the exact statement about the coding of the steady-state solutions of GPE which vanish as x → +∞ by reals. This allows us to fulfil the demonstrative computation of nonlinear modes of GPE, i.e. the computation which allows us to guarantee that all nonlinear modes within a given range of parameters have been found. The method has been applied to GPE with quadratic and double-well potentials, for both repulsive and attractive nonlinearities. The bifurcation diagrams of nonlinear modes in these cases are represented. The stability of these modes has been discussed.
TL;DR: In this paper, the dynamical stability of reflectionless N-solitons for a large class of integrable systems is considered and it is shown that these solitons are realized as a local minimum of an appropriate Lyapunov function, and are hence dynamically stable.
Abstract: The dynamical stability of reflectionless N-solitons for a large class of integrable systems is considered. The underlying eigenvalue problem is the Zakharov–Shabat problem on for any r ≥ 1. Physical examples of interest include the vector nonlinear Schrodinger equation and the integrable (r + 1)-wave interaction problem. It is shown herein that under appropriate assumptions that these solitons are realized as a local minimum of an appropriate Lyapunov function, and are hence dynamically stable.
TL;DR: In this article, it was shown that the speed of propagation is given by, as e → 0, for a large class of cut-off functions for the Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cutoff.
Abstract: The Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597–604) to model N-particle systems in which concentrations less than e = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold e, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by , as e → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction–diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of ccrit on e as well as for the universality of the corresponding leading-order coefficient (π2).
TL;DR: In this paper, the authors studied blow-up behavior of solutions of the 6th-order thin film equation 0, p>1, q > 0, q < 1, q = 0.
Abstract: We study blow-up behaviour of solutions of the sixth-order thin film equation 0,\quad
p>1,\end{eqnarray*}
\] SRC=http://ej.iop.org/images/0951-7715/20/8/002/non236915ude001.gif/> containing an unstable (backward parabolic) second-order term. By a formal matched expansion technique, we show that, for the first critical exponent where N is the space dimension, the free-boundary problem (FBP) with zero-height, zero-contact-angle, zero-moment, and zero-flux conditions at the interface admits a countable set of continuous branches of radially symmetric self-similar blow-up solutions where T > 0 is the blow-up time. We also study the Cauchy problem (CP) in RN × R+ and show that the corresponding self-similar family {uk(x, t)} is countable and consists of solutions of maximal regularity, which are oscillatory near the interfaces. Actually, we show that compactly supported oscillatory blow-up profiles for the CP exist for all n (0, nh), where is a heteroclinic bifurcation point for the ordinary differential equation involved. The FBP ceases to exist before, at .
TL;DR: In this article, it was shown that in three dimensions, a nonlinear wave equation has a countable family of regular spherically symmetric self-similar solutions, with p = 7 being an odd integer.
Abstract: We prove that in three space dimensions a nonlinear wave equation utt − Δu = up, with p≥ 7 being an odd integer, has a countable family of regular spherically symmetric self-similar solutions.
TL;DR: In this article, the authors studied the long-time effective dynamics of the pseudo-relativisitic Hartree equation and established a (long-time) stability result for solutions describing boson stars that move under the influence of an external gravitational field.
Abstract: We study solutions close to solitary waves of the pseudo-relativistic Hartree equation describing boson stars under the influence of an external gravitational field. In particular, we analyse the long-time effective dynamics of such solutions. In essence, we establish a (long-time) stability result for solutions describing boson stars that move under the influence of an external gravitational field. The proof of our main result tackles difficulties that are absent when deriving similar results on effective solitary wave motions for nonlinear Schrodinger equations or nonlinear wave equations. This is due to the fact that the pseudo-relativisitic Hartree equation does not exhibit Galilean or Lorentz covariance.
TL;DR: In this article, the spectrum of quantized open baker's maps admits a fractal repeller, and it is shown that the exponent appearing in the fractal Weyl law is not related with the information dimension, but rather the Hausdorff dimension of the repeller.
Abstract: We study the spectrum of quantized open maps as a model for the resonance spectrum of quantum scattering systems. We are particularly interested in open maps admitting a fractal repeller. Using the 'open baker's map' as an example, we numerically investigate the exponent appearing in the fractal Weyl law for the density of resonances; we show that this exponent is not related with the 'information dimension', but rather the Hausdorff dimension of the repeller. We then consider the semiclassical measures associated with the eigenstates: we prove that these measures are conditionally invariant with respect to the classical dynamics. We then address the problem of classifying semiclassical measures among conditionally invariant ones. For a solvable model, the 'Walsh-quantized' open baker's map, we manage to exhibit a family of semiclassical measures with simple self-similar properties.
TL;DR: In this paper, the authors present a general method to produce computer assisted proofs of the existence of choreographies in the N-body problem, which allows them to verify rigorously numerical data from computer simulations.
Abstract: We present a general method to produce computer assisted proofs of the existence of choreographies in the N-body problem. This method allows us to verify rigorously numerical data from computer simulations. As an example we use it to prove the existence of nonsymmetric choreographies with six and seven bodies. The method provides estimates for the initial conditions and for the monodromy matrix of the choreography. These data are used to show linear stability of the Eight solution restricted to the plane and zero angular momentum motions.
TL;DR: In this paper, a general local well-posedness existence theory is given for a physical class of data (roughly H1) via fixed point methods, and a Lyapunov energy functional is constructed, which is locally convex about the uniform porosity state.
Abstract: An outstanding problem in Earth science is understanding the method of transport of magma in the Earth's mantle. Two proposed methods for this transport are percolation through porous rock and flow up conduits. Under reasonable assumptions and simplifications, both means of transport can be described by a class of degenerate nonlinear dispersive partial differential equations of the form: where (z, 0) > 0 and (z, t) → 1 as z → ±∞.Although we treat arbitrary n and m, the exponents are physically expected to be between 2 and 5 and 0 and 1, respectively.In the case of percolation, the magma moves via the buoyant ascent of a less dense phase, treated as a fluid, through a denser, porous phase, treated as a matrix. In contrast to classical porous media problems where the matrix is fixed and the fluid is compressible, here the matrix is deformable, with a viscous constitutive relation, and the fluid is incompressible. Moreover, the matrix is modelled as a second, immiscible, compressible fluid to mimic the process of dilation of the pores. Flow via a conduit is modelled as a viscously deformable pipe of magma, fed from below.Analogue and numerical experiments suggest that these equations behave akin to KdV and BBM; initial conditions evolve into a collection of solitary waves and dispersive radiation. As → 0, the equations become degenerate. A general local well-posedness existence theory is given for a physical class of data (roughly H1) via fixed point methods. The strategy requires positive lower bounds on (z, t). The key to global existence is the persistence of these bounds for all time. Furthermore, we construct a Lyapunov energy functional, which is locally convex about the uniform porosity state, ≡ 1, and prove (global in time) nonlinear dynamic stability of the uniform state for any m and n. For data which are large perturbations of the uniform state, we prove global in time well-posedness for restricted ranges of m and n. This includes, for example, the case n = 4,m = 0, where an appropriate uniform in time lower global on can be proved using the conservation laws. We compare the dynamics with that of other problems and discuss open questions concerning a larger range of exponents, for which we conjecture global existence.
TL;DR: In this paper, the existence of cocoon bifurcations for the Michelson system where and is a parameter, based on the theory given in (Dumortier et al 2006 Nonlinearity 19 305−28).
Abstract: We prove the existence of cocoon bifurcations for the Michelson system where and is a parameter, based on the theory given in (Dumortier et al 2006 Nonlinearity 19 305–28). The main difficulty lies in the verification of the (topological) transversality of some invariant manifolds in the system. The proof is computer-assisted and combines topological tools including covering relations and the smooth ones using the cone conditions. These new techniques developed in this paper will have broader applicability to similar global bifurcation problems.
TL;DR: In this article, the dynamics of a periodic chain of coupled overdamped particles under the influence of noise were considered, where each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise.
Abstract: We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. The system shows a metastable behaviour, which is characterized by the location and stability of its equilibrium points. We show that as the coupling strength increases, the number of equilibrium points decreases from 3N to 3. While for weak coupling, the system behaves like an Ising model with spin-flip dynamics, for strong coupling (of the order N2), it synchronizes, in the sense that all particles assume almost the same position in their respective local potential most of the time. We derive the exponential asymptotics for the transition times, and describe the most probable transition paths between synchronized states, in particular for coupling intensities below the synchronization threshold. Our techniques involve a centre-manifold analysis of the desynchronization bifurcation, with a precise control of the stability of bifurcating solutions, allowing us to give a detailed description of the system's potential landscape.
TL;DR: In this article, the magnetization dynamics in soft ferromagnetic films with small damping were investigated, and the authors proved the existence of a travelling wave solution under a small constant forcing.
Abstract: We investigate the magnetization dynamics in soft ferromagnetic films with small damping. In this case, the gyrotropic nature of Landau–Lifshitz–Gilbert dynamics and the shape anisotropy effects from stray-field interactions effectively lead to a wave-type dynamics for the in-plane magnetization. We apply this result to study the motion of Neel walls in thin films and prove the existence of a travelling wave solution under a small constant forcing.
TL;DR: For connected self-similar sets in the plane, it was shown in this paper that a finite overlap implies OSC, and that there are Cantor sets with arbitrary small dimensions which do not fulfil the OSC.
Abstract: Even though the open set condition (OSC) is generally accepted as the right condition to control overlaps of self-similar sets, it seems unclear how it relates to the actual size of the overlap. For connected self-similar sets in the plane, we prove that a finite overlap implies OSC. On the other hand, there are Cantor sets with arbitrary small dimensions which do not fulfil the OSC.
TL;DR: In this paper, the authors studied the Hamiltonian structure of the second Painleve hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation, and gave explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates.
Abstract: In this paper we study the Hamiltonian structure of the second Painleve hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The nth element of the hierarchy is a nonlinear ODE of order 2n in the independent variable z depending on n parameters denoted by t1, ..., tn−1 and αn. We introduce new canonical coordinates and obtain Hamiltonians for the z and t1, ..., tn−1 evolutions. We give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates.
TL;DR: In this paper, the symmetry properties of the reversible planar maps were used to improve the homoclinic orbit approach without assuming small perturbations to prove rigorously the existence of bright and dark soliton solutions of the discrete nonlinear Schrodinger equations with various nonlinearities in one-dimensional lattices.
Abstract: By using the symmetry properties of the reversible planar maps, instead of the Melnikov analysis, we improve the homoclinic orbit approach without assuming small perturbations to prove rigorously the existence of bright and dark soliton solutions of the discrete nonlinear Schrodinger equations with various nonlinearities in one-dimensional lattices. Our approach is valid both for small and large coupling constants. The latter case is inaccessible to the anti-integrability method.
TL;DR: In this paper, the amplitude equations for stochastic partial differential equations with quadratic nonlinearities were derived under the assumption that the noise acts only on the stable modes and for an appropriate scaling between the distance from bifurcation and the strength of the noise.
Abstract: In this paper we derive rigorously amplitude equations for stochastic partial differential equations with quadratic nonlinearities, under the assumption that the noise acts only on the stable modes and for an appropriate scaling between the distance from bifurcation and the strength of the noise. We show that, due to the presence of two distinct timescales in our system, the noise (which acts only on the fast modes) gets transmitted to the slow modes and, as a result, the amplitude equation contains both additive and multiplicative noise. As an application we study the case of the one-dimensional Burgers equation forced by additive noise in the orthogonal subspace to its dominant modes. The theory developed in the present paper thus allows us to explain theoretically some recent numerical observations on stabilization with additive noise.
TL;DR: It is shown that a reduced description in the form of an effective Langevin equation characterized by a double-well potential is quantitatively successful in describing the high-dimensional stochastic dynamics of a single scalar 'coarse' variable.
Abstract: We study a stochastic nonlocal partial differential equation, arising in the context of modelling spatially distributed neural activity, which is capable of sustaining stationary and moving spatially localized 'activity bumps'. This system is known to undergo a pitchfork bifurcation in bump speed as a parameter (the strength of adaptation) is changed; yet increasing the noise intensity effectively slowed the motion of the bump. Here we study the system from the point of view of describing the high-dimensional stochastic dynamics in terms of the effective dynamics of a single scalar 'coarse' variable, i.e. reducing the dimensionality of the system. We show that such a reduced description in the form of an effective Langevin equation characterized by a double-well potential is quantitatively successful. The effective potential can be extracted using short, appropriately initialized bursts of direct simulation, and the effects of changing parameters on this potential can easily be studied. We demonstrate this approach in terms of (a) an experience-based 'intelligent' choice of the coarse variable and (b) a variable obtained through data-mining direct simulation results, using a diffusion map approach.
TL;DR: For the first critical exponent, where N is the space dimension, the free-boundary problem with zero-contact-angle, zero-moment and zero-flux conditions at the interface admits continuous families (branches) of radially symmetric self-similar solutions defined for all t > 0, and also study the Cauchy problem, for which they construct global similarity solutions of the maximal regularity, these being oscillatory near the interfaces for n (0, nh), where is a 'heteroclinic bifurcation' point for a
Abstract: We continue the study begun in part I (Evans J D, Galaktionov V A and King J R 2007 Unstable sixth-order thin film equation: I. Blow-up of similarity solutions Nonlinearity 20 1799–841) of asymptotic large time behaviour of global solutions of the sixth-order thin film equation 0, \quad p>1,\end{eqnarray*}
\] SRC=http://ej.iop.org/images/0951-7715/20/8/003/non237043ude001.gif/> with bounded integrable initial data. We show that for the first critical exponent, where N is the space dimension, the free-boundary problem with zero-contact-angle, zero-moment and zero-flux conditions at the interface admits continuous families (branches) of radially symmetric self-similar solutions defined for all t > 0, We also study the Cauchy problem, for which we construct global similarity solutions of the maximal regularity, these being oscillatory near the interfaces for n (0, nh), where is a 'heteroclinic bifurcation' point for a related nonlinear ordinary differential equation. We use various concepts based on the branching of sufficiently small solutions from the known eigenfunctions of the linear rescaled operator corresponding to n = 0.
TL;DR: In this paper, the authors analyse a joint action of symbolic dynamics and thermodynamic formalism for stable evolution of dynamical networks and their subnetworks and provide sufficient conditions for stable topology and local evolution.
Abstract: Dynamical networks are characterized by their topology (structure of a network), by interactions between the elements (local subsystems) sitting at the nodes of a network and by intrinsic dynamics (local evolution) of these local subsystems. We analyse a joint action (interplay) of these three factors by the methods of symbolic dynamics and thermodynamic formalism. The main result provides sufficient conditions for stable evolution of dynamical networks and their subnetworks. No restrictions on the topology of a network are considered. Therefore, our results are applicable to dynamical networks with an arbitrary (random, regular, scale-free, small-world, etc) structure.
TL;DR: In this article, the sign of the limit at 0 of the Vakhitov?Kolokolov function is used to determine whether a black soliton solution to a one-dimensional nonlinear Schr?dinger equation is linearly stable or not.
Abstract: In this paper, we prove a criterion to determine if a black soliton solution (which is an odd solution that does not vanish at infinity) to a one-dimensional nonlinear Schr?dinger equation is linearly stable or not. This criterion handles the sign of the limit at 0 of the Vakhitov?Kolokolov function. For some nonlinearities, we numerically compute the black soliton and the Vakhitov?Kolokolov function in order to investigate linear stability of black solitons. We then show that linearly unstable black solitons are also orbitally unstable. In the Gross?Pitaevskii case, we rigorously prove the linear stability of the black soliton. Finally, we numerically study the dynamical stability of these solutions solving both linearized and fully nonlinear equations with a finite differences algorithm.