Orbital stability of standing waves for semilinear wave equations with indefinite energy

The orbital stability of standing waves for semilinear wave equations is studied in the case that the energy is indefinite and the underlying space domain is bounded or a compact manifold or whole Rn with n?2. The stability is determined by the convexity on ω of the lowest energy d(ω) of standing waves with frequency ω. The arguments rely on the conservation of energy and charge and the construction of suitable invariant manifolds of solution flows.     

The slow wave solution to the FitzHugh-Nagumo equations

This paper gives a new existence proof for a travelling wave solution to the FitzHugh-Nagumo equations, u_t = u_xx +f(u)?w, w _t = ? (u?γw). The proof uses a contraction mapping argument, and also shows that the solution (u, c, w) to the travelling wave equations, where c is the wave speed, converges as ? → 0⁺ to the solution to the equations having ?=0, c=0, and w=0.     

Some results on Hu's conjecture concerning binary trees

Given n weights, w₁, w₂,…, w_n, such that 0?w₁?w₂???w₁, we examine a property of permutation π¹, where π¹=(w₁, w_n, w₂, w_n?1,…), concerning alphabetical binary trees.For each permutation π of these n weights, there is an optimal alphabetical binary tree corresponding to π, we denote it's cost by V(π). There is also an optimal almost uniform alphabetical binary tree, corresponding to π, we denote it's cost by V_u(π).This paper asserts that V_u(π¹)?V_u(π)?V(π) for all π. This is a preliminary result concerning the conjecture of T.C. Hu. Hu's conjecture is V(π¹)?V(π) for all π.     

On the damped nonlinear evolution equation wtt = σ(w)xx − ywt

Initial boundary value problems for the damped nonlinear wave equation w_tt = σ(w)_xx ? yw_t arise in several areas of applied mathematics and, in particular, in studies of shearing flow in a nonlinear viscoelastic fluid; the problems of global existence and nonexistence of smooth solutions have been extensively studied in the strictly hyperbolic case σ′(δ) ? ε > 0, ?δ?R¹ as well as in the case where σ′(0) > 0 and the initial data are chosen so small that σ′(w) > 0 for as long as a smooth solution w(x, t) exists. In this paper the global nonexistence problem is studied for the cases σ′(0) = 0 and σ′(0) > 0 but σ′(δ) < 0 for ¦δ¦ sufficiently large and growth estimates which are valid on the maximal interval of existence of a sufficiently smooth solution are derived.     

Coexistence of a pulse and multiple spikes and transition layers in the standing waves of a reaction-diffusion system

We establish the existence of standing waves with one pulse, multiple spikes and transition layers in the nonlinear reaction-diffusion system

u_t=f(u,w)+u_xx,w_t=?²g(u,w)+w_xx,x∈R,     

Bounds for the number of meeting edges in graph partitioning

Let G be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that G admits a bipartition such that each vertex class meets edges of total weight at least (w ₁?Δ₁)/2+2w ₂/3, where wi is the total weight of edges of size i and Δ₁ is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph G (i.e., multi-hypergraph), we show that there exists a bipartition of G such that each vertex class meets edges of total weight at least (w ₀?1)/6+(w ₁?Δ₁)/3+2w ₂/3, where w ₀ is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with m edges, except for K ₂ and K _1,3, admits a tripartition such that each vertex class meets at least [2m/5] edges, which establishes a special case of a more general conjecture of Bollobás and Scott.     

Stability switches in a system of linear differential equations with diagonal delay

This paper deals with the stability problem of a delay differential system of the form x^′(t)=-ax(t-τ)-by(t), y^′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if ; and from instability to stability to instability if . As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka-Volterra systems.     

On the solution of w-stable sets

The notion of m-stable sets was introduced in Peris and Subiza (2013) for abstract decision problems. Since it may lack internal stability and fail to discriminate alternatives in cyclic circumstances, we alter this notion, which leads to an alternative solution called w-stable set. Subsequently, we characterize w-stable set and compare it with other solutions in the literature. In addition, we propose a selection procedure to filter out more desirable w-stable sets.     

Local analytic solutions of the generalized Dhombres functional equation II

We study local analytic solutions f of the generalized Dhombres functional equation f(zf(z))=φ(f(z)), where φ is holomorphic at w₀≠0, f is holomorphic in some open neighborhood of 0, depending on f, and f(0)=w₀. After deriving necessary conditions on φ for the existence of nonconstant solutions f with f(0)=w₀ we describe, assuming these conditions, the structure of the set of all formal solutions, provided that w₀ is not a root of 1. If |w₀|≠1 or if w₀ is a Siegel number we show that all formal solutions yield local analytic ones. For w₀ with 0<|w₀|<1 we give representations of these solutions involving infinite products.     

Constructive existence theorems for problems of Thomas-Fermi Type

A minimal positive solution of the Thomas-Fermi problem ? = λt^?1/2 w^3/2, w(0) = 1, w(1) = w(1) is shown to exist for each λ > 0. It is proved that all positive solutions, for a given value of λ, are strictly ordered and that the minimal positive solution w_λ is a decreasing function of λ. Upper and lower analytic bounds for w _λ are given and these bounds are shown to initiate sequences of Picard and Newton iterates which converge monotonically to w _λ. A comparative analysis of the efficiency of the iteration schemes is presented. The methods used are of a general nature and can be applied to a variety of nonlinear boundary value problems of convex type [14].     

Strong solutions to a nonlinear fluid structure interaction system

In this paper, we prove the existence of local-in-time smooth solutions to the nonlinear fluid structure interaction model first introduced in [J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969] and considered in [V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007, pp. 55-82; V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J. 57 (3) (2008) 1173-1207]. In particular, the strong solutions here are obtained given initial datum for the Navier-Stokes equation in the space H¹, and initial data for the wave equation w₀ and w₁ in the spaces H²(Ωe) and H¹(Ωe) respectively.     

Approximation schemes for the subset-sum problem: Survey and experimental analysis

Given a set of n positive integers w₁,…,w_n and a positive integer W, the Subset-Sum Problem is to find that subset whose sum is closest to, without exceeding, W. The most important heuristics for this problem are approximation schemes based on a worst-case analysis. We survey them and experimentally analyze their statistical behaviour on a large number of test problems.     

The failure of the Hardy inequality and interpolation of intersections

The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=?1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L _p(w₀), N∩L_p(w₁)), whereN is the linear space which consists of all functions with the integral equal to 0. We calculate theK-functional for the couple (N∩L _p(w₀),N∩L _p (w₁)), which turns out to be essentially different from theK-functional for (L _p(w₀), L_p(w₁)), even for the case whenN∩L _p(w_i) is dense inL _p(w_i) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.     

Solving a combinatorial problem with network flows

In this paper we present an algorithm based on network flow techniques which provides a solution for a combinatorial problem. Then, in order to provide all the solutions of this problem, we make use of an algorithm that given the bipartite graphG=(V ₁ ∪V ₂,E, w) outputs the enumeration of all bipartite matchings of given cardinalityv and costc.     

The non-linear Schrödinger equation with a periodic δ-interaction

We study the existence and stability of space-periodic standing waves for the space-periodic cubic nonlinear Schrödinger equation with a point defect determined by a space-periodic Dirac distributionat the origin. This equation admits a smooth curve of positive space-periodic solutions with a profile given by the Jacobi elliptic function of dnoidal type. Via a perturbationmethod and continuation argument, we prove that in the case of an attractive defect the standing wave solutions are stable in H _per ¹ ([?π, π]) with respect to perturbations which have the same space-periodic as the wave itself. In the case of a repulsive defect, the standing wave solutions are stable in the subspace of even functions of H _per ¹ ([?π, π]) and unstable in H _per ¹ ([?π, π]) with respect to perturbations which have the same space-periodic as the wave itself.     

A remark on linearly unstable standing wave solutions to NLS

We obtain an optimal growth estimate of a semigroup generated by a linearized operator around a standing wave solution nonlinear Schrödinger equations in two-dimension. Using the growth estimate of the semigroup, we prove that a linearly unstable standing wave solution is orbitally unstable and that instability of the standing wave solution is mainly caused by a mode of an eigenfunction associated with the rightmost (or the leftmost) eigenvalues of the linearized operator. Our result is obtained by using the method of Yajima and Cuccagna that proved L^p$L^p$-boundedness of the wave operator.     

Stability of travelling wave solutions of the active Josephson junction transmission line

The behavior of the Josephson line, which is a type of active pulse transmission line, is governed by a partial differential equation which is similar to the sine-Gordon equation. This equation has two solitary travelling wave solutions with different propagation speeds c₁ and c₂, 0 < c₁ < c₂, and a one-parameter family of spatially periodic travelling wave solutions whose propagation speeds range over the intervals (0, c₁) and (c₂, + ∞). First we prove the existence of these solutions. Second we consider the stability of these solutions by linearized stability analysis. It is shown that the slow solitary solution is stable in the sense of linearized stability and that the fast solitary solution is unstable. It is shown also that the periodic solution with the speed c, 0 < c < c₁, is stable in the sense of linearized stability and that the periodic solution with the speed c, c₂ < c < c₄, is unstable, where c₄ is a certain point in (c₂, + ∞).     

An equation admitting infinite true contact transformations

The linear wave equation is shown to possess the unique property that if w_n is a true contact transformation admitted by the wave equation, i.e., w_n is not linear in the first derivatives of the dependent variable, then so is ∑_nw_n. We comment of the physical implications.     

Series solutions and a perturbation formula for the extended Rayleigh problem of hydrodynamic stability

We generalize Tollmien’s solutions of the Rayleigh problem of hydrodynamic stability to the case of arbitrary channel cross sections, known as the extended Rayleigh problem. We prove the existence of a neutrally stable eigensolution with wave number k _s ?>?0; it is also shown that instability is possible only for 0?<?k?<?k _s and not for k?>?k _s . Then we generalize the Tollmien–Lin perturbation formula for the behavior of c _i, the imaginary part of the phase velocity as the wave number k →k _s ? to the extended Rayleigh problem and subsequently, we use this formula to demonstrate the instability of a particular shear flow.     

Stability and instability of periodic standing wave solutions for some Klein–Gordon equations

In the present paper we show some results concerning the orbital stability of dnoidal standing wave solutions and orbital instability of cnoidal standing wave solutions to the following Klein–Gordon equation:

u_tt−u_xx+u−|u|²u=0.