Discrete breathers and modulational instability in a discrete $$\varvec{\phi ^{4}}$$ nonlinear lattice with next-nearest-neighbor couplings

The properties of discrete breathers and modulational instability in a discrete \(\phi ^{4}\) nonlinear lattice which includes the next-nearest-neighbor coupling interaction are investigated analytically. By using the method of multiple scales combined with a quasi-discreteness approximation, we get a dark-type and a bright-type discrete breather solutions and analyze the existence conditions for such discrete breathers. It is found that the introduction of the next-nearest-neighbor coupling interactions will influence the existence condition for the bright discrete breather. Considering that the existence of bright discrete breather solutions is intimately linked to the modulational instability of plane waves, we will analytically study the regions of discrete modulational instability of plane carrier waves. It is shown that the shape of the region of modulational instability changes significantly when the strength of the next-nearest-neighbor coupling is sufficiently large. In addition, we calculate the instability growth rates of the \(q=\pi \) plane wave for different values of the strength of the next-nearest-neighbor coupling in order to better understand the appearance of the bright discrete breather.     

Families of rational solutions of the <Emphasis Type="Italic">y</Emphasis>-nonlocal Davey–Stewartson II equation

The y-nonlocal Davey–Stewartson II equation is an extension of the usual DS II equation involving a partially parity-time-symmetric potential only with respect to the spatial variable y. By using the Hirota bilinear method, families of n-order rational solutions are obtained, which include lumps in the (x, y)-plane and the (y, t)-plane, growing-and-decaying line waves in the (x, t)-plane, and hybrid solutions of interacting line rogue waves and lumps in the (x, y)-plane.     

Stability of Periodic Solutions¶of Conservation Laws with Viscosity:¶Analysis of the Evans Function

Nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling-wave phenomena, including homoclinic (pulse-type) and periodic solutions along with the standard heteroclinic (shock, or front-type) solutions. Here, we investigate stability of periodic traveling waves within the abstract Evans-function framework established by R. A. Gardner. Our main result is to derive a useful stability index analogous to that developed by Gardner and Zumbrun in the traveling-front or -pulse context, giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period T as the traveling wave; moreover, we show that the periodic-stability index has an interpretation analogous to that of the traveling-front or -pulse index in terms of well-posedness of an associated Riemann problem for an inviscid medium, now to be interpreted as allowing a wider class of measure-valued solutionsor, alternatively, in terms of existence and nonsingularity of a local “mass map” from perturbation mass to potential time-asymptotic T-periodic states. A closely related calculation yields also a complementary long-wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period NT, N → ∞. We augment these analytical results with numerical investigations analogous to those carried out by Brin in the traveling-front or -pulse case, approximating the spectrum of the linearized operator about the wave.The stability index and long-wave stability criterion are explicitly evaluable in the same planar, Hamiltonian cases as is the index of Gardner and Zumbrun, and together yield rigorous results of instability similar to those obtained previously for pulse-type solutions; this is established through a novel dichotomy asserting that the two criteria are in certain cases logically exclusive. In particular, we obtain results bearing on the nature and mechanism for formation of highly oscillatory Turing-like patterns observed numerically by Frid and Liu and ?ani? and Peters in models of multiphase flow. Specifically, for the van der Waals model considered by Frid and Liu, we show instability of all periodic waves such that the period increases with amplitude in the one-parameter family of nearby periodic orbits, and in particular of large- and small-amplitude waves; for the standard, double-well potential, this yields instability of all periodic waves.Likewise, for a quadratic-flux model like that considered by ?ani? and Peters, we show instability of large-amplitude waves of the type lying near observed patterns, and of all small-amplitude waves; our numerical results give evidence that intermediate-amplitude waves are unstable as well. These results give support for an alternative mechanism for pattern formation conjectured by Azevedo, Marchesin, Plohr, and Zumbrun, not involving periodic waves.     

Consequences of the Translation Invariance on the Darcy Free Convection Flow Past a Vertical Surface

It is shown that the governing equation for the stream function of the Darcy free convection boundary layer flows past a vertical surface is invariant under arbitrary translations of the transverse coordinate y. The consequences of this basic symmetry property on the solutions corresponding to a prescribed surface temperature distribution T _w (x) are investigated. It is found that starting with a “primary solution” which describes the temperature boundary layer on an impermeable surface, infinitely many “translated solutions” can be generated which form a continuous group, the “translation group” of the given primary solution. The elements of this group describe free convection boundary layer flows from permeable counterparts of the original surface with a transformed temperature distribution \({\tilde {T}_w \left( x \right)}\), when simultaneously a suitable lateral suction/injection of the fluid is applied. It turns out in this way that several exact solutions discovered during the latter few decades are in fact not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.     

Multiple Brake Orbits and Homoclinics in Riemannian Manifolds

Let (M, g) be a complete Riemannian manifold, \({\Omega\subset M}\) an open subset whose closure is homeomorphic to an annulus. We prove that if ?Ω is smooth and it satisfies a strong concavity assumption, then there are at least two distinct geodesics in \({\overline\Omega=\Omega\cup\partial\Omega}\) starting orthogonally to one connected component of ?Ω and arriving orthogonally onto the other one. Using the results given in Giambò et al. (Adv Differ Equ 10:931–960, 2005), we then obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. Under a further symmetry assumption, the result is improved by showing the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinic orbits. In our proof we shall use recent deformation results proved in Giambò et al. (Nonlinear Anal Ser A: Theory Methods Appl 73:290–337, 2010).     

Traveling Wave Phenomena in a Kermack–McKendrick SIR Model

We study the existence and nonexistence of traveling waves of a general diffusive Kermack–McKendrick SIR model with standard incidence where the total population is not constant. The three classes, susceptible S, infected I and removed R, are all involved in the traveling wave solutions. We show that the minimum wave speed of traveling waves for the three-dimensional non-monotonic system can be derived from its linearizaion at the initial disease-free equilibrium. The proof in this paper is based on Schauder fixed point theorem and Laplace transform. Our study provides a promising method to deal with high dimensional epidemic models.     

Stationary Solutions and Connecting Orbits for <Emphasis Type="Italic">p</Emphasis>-Laplace Equation

We deal with one dimensional p-Laplace equation of the form

$$\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, \end{aligned}$$

under Dirichlet boundary condition, where \(p>2\) and \(f:[0,l]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with \(f(x,0)=0\). We will prove that if there is at least one eigenvalue of the p-Laplace operator between \(\lim _{u\rightarrow 0} f(x,u)/|u|^{p-2}u\) and \(\lim _{|u|\rightarrow +\infty } f(x,u)/|u|^{p-2}u\), then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.

     

Entire Solutions to Equivariant Elliptic Systems with Variational Structure

We consider the system Δu ? W _u (u) = 0, where \({u : \mathbb{R}^n \to \mathbb{R}^n}\) , for a class of potentials \({W : \mathbb{R}^n \to \mathbb{R}}\) that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial G-equivariant entire solutions connecting the global minima of W along certain directions at infinity.     

On the orders of the best approximations of integrals of functions by integrals of rank <Emphasis Type="Italic">σ</Emphasis>

We study the values e _σ(f) of the best approximation of integrals of functions from the spaces L _p (A, dμ) by integrals of rank σ. We determine the orders of the least upper bounds of these values as σ → ∞ in the case where the function ? is the product of two nonnegative functions one of which is fixed and the other varies on the unit ball U _p (A) of the space L _p (A, dμ). We consider applications of the obtained results to approximation problems in the spaces S _p ^? .     

Mixed Boundary Value Problems for the Stationary Navier–Stokes System in Polyhedral Domains

The authors consider boundary value problems for the Navier–Stokes system in a polyhedral domain, where different boundary conditions (in particular, Dirichlet, Neumann, slip conditions) are arbitrarily combined on the faces of the polyhedron. They prove existence and regularity theorems for weak solutions in weighted (and nonweighted) L _p Sobolev and Hölder spaces with sharp integrability and smoothness parameters.     

Surface waves under constrained deformation

The possibility of the existence of surface waves in a range of velocities greater than the velocity of transverse waves, but smaller than the velocity of longitudinal waves is shown. It turns out that, in the boundary value problem for an elastic half-space in this velocity range, there are the surface waves whose velocity is constant and equal to \(\sqrt 2 \) b, where b is the velocity of transverse waves. These waves as well as the Rayleigh surface waves have no dispersion. Their velocity is specified only by the elastic constants and density of the medium. It is also shown that the existence of such a velocity is possibly related to the velocity of surface waves that appear as unloading waves under constrained deformation.     

Multipulse Phases in k-Mixtures of Bose–Einstein Condensates

For the system

$-\Delta U_i+ U_i=U_i^3-\beta U_i\sum_{j\neq i}U_j^2,\quad i=1,\dots,k,$

(with k ≧ 3), we prove the existence for β large of positive radial solutions on \({\mathbb R^N}\) . We show that as β →  + ∞, the profile of each component U _i separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing-sign solutions of the scalar equation  ? ΔW  +  W  =  W³. Within an Hartree–Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the k-mixtures of Bose–Einstein condensates.

     

New Periodic Solutions for Newtonian n-Body Problems with Dihedral Group Symmetry and Topological Constraints

In this paper, we prove the existence of a family of new non-collision periodic solutions for the classical Newtonian n-body problems. In our assumption, the \({n=2l \geqq 4}\) particles are invariant under the dihedral rotation group D_l in \({\mathbb{R}^3}\) such that, at each instant, the n particles form two twisted l-regular polygons. Our approach is the variational minimizing method and we show that the minimizers are collision-free by level estimates and local deformations.     

Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity

In this paper, the well-established two-dimensional mathematical model for linear pyroelectric materials is employed to investigate the reflection of waves at the boundary between a vacuum and an elastic, transversely isotropic, pyroelectric material. A comparative study between the solutions of (a) classical thermoelasticity, (b) Cattaneo–Lord–Shulman theory and (c) Green–Lindsay theory equations, characterised by none, one and two relaxation times, respectively, is presented. Suitable boundary conditions are considered in order to determine the reflection coefficients when incident elasto–electro–thermal waves impinge the free interface. It is established that, in the quasi-electrostatic approximation, three different classes of waves: (1) two principally elastic waves, namely a quasi-longitudinal Primary (qP) wave and a quasi-transverse Secondary (qS) wave; and (2) a mainly thermal (qT) wave. The observed electrical effects are, on the other hand, a direct consequence of mechanical and thermal phenomena due to pyroelectric coupling. The computed reflection coefficients of plane qP waves are found to depend upon the angle of incidence, the elastic, electric and thermal parameters of the medium, as well as the thermal relaxation times. The special cases of normal and grazing incidence are also derived and discussed. Finally, the reflection coefficients are computed for cadmium selenide observing the influence of (1) the anisotropy of the material, (2) the electrical potential and (3) temperature variations and (4) the thermal relaxation times on the reflection coefficients.     

Time Periodic Traveling Curved Fronts in the Periodic Lotka–Volterra Competition–Diffusion System

This paper is concerned with time periodic traveling curved fronts for periodic Lotka–Volterra competition system with diffusion in two dimensional spatial space

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u_{1}}{\partial t}=\Delta u_{1} +u_{1}(x,y,t)\left( r_{1}(t)-a_{1}(t)u_{1}(x,y,t)-b_{1}(t)u_{2}(x,y,t)\right) ,\\ \dfrac{\partial u_{2}}{\partial t}=d\Delta u_{2} +u_{2}(x,y,t)\left( r_{2}(t)-a_{2}(t)u_{1}(x,y,t)-b_{2}(t)u_{2}(x,y,t)\right) , \end{array}\right. } \end{aligned}$$

where \(\Delta \) denotes \(\frac{\partial ^{2}}{\partial x^{2} }+ \frac{\partial ^{2}}{\partial y^{2} }\), \(x,y\in {\mathbb {R}}\) and \(d>0\) is a constant, the functions \(r_i(t),a_i(t)\) and \(b_i(t)\) are T-periodic and Hölder continuous. Under suitable assumptions that the corresponding kinetic system admits two stable periodic solutions (p(t), 0) and (0, q(t)), the existence, uniqueness and stability of one-dimensional traveling wave solution \(\left( \Phi _{1}(x+ct,t),\Phi _{2}(x+ct,t)\right) \) connecting two periodic solutions (p(t), 0) and (0, q(t)) have been established by Bao and Wang ( J Differ Equ 255:2402–2435, 2013) recently. In this paper we continue to investigate two-dimensional traveling wave solutions of the above system under the same assumptions. First, we establish the asymptotic behaviors of one-dimensional traveling wave solutions of the system at infinity. Using these asymptotic behaviors, we then construct appropriate super- and subsolutions and prove the existence and non-existence of two-dimensional time periodic traveling curved fronts. Finally, we show that the time periodic traveling curved front is asymptotically stable.

     

Stability of Periodic Solutions¶of Conservation Laws with Viscosity:¶Pointwise Bounds on the Green Function

We investigate stability of periodic traveling-wave solutions of systems of conservation laws with viscosity within the abstract Evans function framework established by R. A. Gardner. Our main result, generalizing the work of Zumbrun and Howard in the traveling-front or -pulse setting, is to establish sharp pointwise bounds on the Green function for the linearized evolution equations, provided that an appropriate Evans function condition applies to the linearized operator about the wave. This condition is equivalent to a spectral stability criterion introduced by Schneider in the context of periodic reaction-diffusion waves. An immediate consequence is that strong spectral stability (in the sense of Schneider) implies linearized L¹ → L ^p asymptotic stability for all p > 1. On the other hand, we show that the strict version of Schneider's condition genericallyfails in the conservation law setting, leading to complicated new “metastable” behavior reminiscent of that seen for degenerate, neutrally stable families in the traveling-front or -pulse case. Our results apply also to the reaction-diffusion setting, sharpening (at the linearized level) results obtained by Schneider using weighted-norm and Bloch-decomposition methods.As in the traveling-front or -pulse case, the basic approach is to mimic in the Laplace-transform setting the elementary Fourier-transform analysis of the constant-coefficient case. However, the technical issues involved are rather different in the two cases. Somewhat surprisingly, we find the analogy to the constant-coefficient case to be rather stronger in the periodic-coefficient case, permitting a more standard approach involving the explicit construction of “continuous” spectral measure as in the self-adjoint case. This is equivalent to the method of Zumbrun and Howard in this special case.     

Regularity and Singularities of Optimal Convex Shapes in the Plane

We focus here on the analysis of the regularity or singularity of solutions Ω ₀ to shape optimization problems among convex planar sets, namely:

$J(\Omega_{0})={\rm min} \{J(\Omega), \Omega \quad {\rm convex},\Omega \in \mathcal{S}_{\rm ad}\},$

where \({\mathcal{S}_{\rm ad}}\) is a set of 2-dimensional admissible shapes and \({J:\mathcal{S}_{\rm ad}\rightarrow\mathbb{R}}\) is a shape functional. Our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions, including the infinite dimensional Lagrange multiplier due to the convexity constraint. We prove two types of results:

1. i)
   under a suitable convexity property of the functional J, we prove that Ω ₀ is a W ^2,p -set, \({p\in[1, \infty]}\). This result applies, for instance, with p = ∞ when the shape functional can be written as J(Ω) = R(Ω) + P(Ω), where R(Ω) = F(|Ω|, E _f (Ω), λ₁(Ω)) involves the area |Ω|, the Dirichlet energy E _f (Ω) or the first eigenvalue of the Laplace–Dirichlet operator λ₁(Ω), and P(Ω) is the perimeter of Ω;
    

1. ii)
   under a suitable concavity assumption on the functional J, we prove that Ω ₀ is a polygon. This result applies, for instance, when the functional is now written as J(Ω) = R(Ω) ? P(Ω), with the same notations as above.
    

     

On the Darboux transformation of a generalized inhomogeneous higher-order nonlinear Schrödinger equation

Recently, a paper about the Nth-order rogue waves for an inhomogeneous higher-order nonlinear Schrödinger equation using the generalized Darboux transformation is published. Song et al. (Nonlinear Dyn 82(1):489–500. doi: 10.1007/s11071-015-2170-6, 2015). However, the inhomogeneous equation which admits a nonisospectral linear eigenvalue problem is mistaken for having a constant spectral parameter by the authors. This basic error causes the results to be wrong, especially regarding the Darboux transformation (DT) in Sect. 2 when the inhomogeneous terms are dependent of spatial variable x. In fact, the DT for inhomogeneous equation has an essential difference from the isospectral case, and their results are correct only in the absence of inhomogeneity which was already discussed in detail before. Consequently, we firstly modify the DT based on corresponding nonisospectral linear eigenvalue problem. Then, the nonautonomous solitons are obtained from zero seed solutions. Properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. Finally, the failure of finding breather and rogue wave solutions from this modified DT is also discussed.     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.