Global bifurcation of travelling waves in discrete nonlinear Schrödinger equations

The discrete nonlinear Schrödinger equations of n sites are studied with periodic boundary conditions. These equations have n branches of standing waves that bifurcate from zero. Travelling waves appear as a symmetry-breaking from the standing waves for different amplitudes. The bifurcation is proved using the global Rabinowitz alternative in subspaces of symmetric functions. Applications to the Schrödinger and Saturable lattices are presented.     

Spectrally Stable Encapsulated Vortices for Nonlinear Schr?dinger Equations

Summary. A large class of multidimensional nonlinear Schrodinger equations admit localized nonradial standing-wave solutions that carry nonzero intrinsic angular momentum. Here we provide evidence that certain of these spinning excitations are spectrally stable. We find such waves for equations in two space dimensions with focusing-defocusing nonlinearities, such as cubic-quintic. Spectrally stable waves resemble a vortex (nonlocalized solution with asymptotically constant amplitude) cut off at large radius by a kink layer that exponentially localizes the solution. For the evolution equations linearized about a localized spinning wave, we prove that unstable eigenvalues are zeroes of Evans functions for a finite set of ordinary differential equations. Numerical computations indicate that there exist spectrally stable standing waves having central vortex of any degree.     

Standing Waves of the Inhomogeneous Klein-Gordon Equations with Critical Exponent

This paper is concerned with the standing wave in the inhomogeneous nonlinear Klein- Gordon equations with critical exponent. Firstly, we obtain the existence of standing waves associated with the ground states by using variational calculus as well as a compactness lemma. Next, we establish some sharp conditions for global existence in terms of the characteristics of the ground state. Then, we show that how small the initial data are for the global solutions to exist. Finally, we prove the instability of the standing wave by combining the former results.     

Orbital stability of standing waves of two-component Bose–Einstein condensates with internal atomic Josephson junction

In this paper, we prove existence, symmetry and uniqueness of standing waves for a coupled Gross–Pitaevskii equations modeling component Bose–Einstein condensates BEC with an internal atomic Josephson junction. We will then address the orbital stability of these standing waves and characterize their orbit.     

Stability of the standing waves for a class of coupled nonlinear Klein-Gordon equations

This paper deals with the standing waves for a class of coupled nonlinear Klein-Gordon equations with space dimension N ≥ 3, 0 〈 p, q 〈 2/N-2 and p ＋ q 〈 4/N. By using the variational calculus and scaling argument, we establish the existence of standing waves with ground state, discuss the behavior of standing waves as a function of the frequency ω and give the sufficient conditions of the stability of the standing waves with the least energy for the equations under study.     

On ground states for the 2D Schrödinger equation with combined nonlinearities and harmonic potential

We consider the nonlinear Schrödinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a framework includes physical models, and ensures that finite energy solutions are global in time. We address the questions of the existence and the orbital stability of the set of standing waves. Given the mathematical features of the equation (external potential and inhomogeneous nonlinearity), the set of parameters for which standing waves exist in unclear. In the two-dimensional case, we adapt the method of fundamental frequency solutions, introduced by the second author in the higher-dimensional case without potential. This makes it possible to describe accurately the set of fundamental frequency standing waves and ground states, and to prove its orbital stability.     

Analysis of bifurcation in a system of n coupled oscillators with delays

A n-coupled BVP oscillators system with delays is considered. By choosing the delays as the bifurcating parameters, some results of the Hopf bifurcations occurring at the zero equilibrium as the delays increase are exhibited. Using the symmetric functional differential equation theories of Wu [Jianhong Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (12) (1998) 4799–4838], the multiple Hopf bifurcations are obtained, and their spatio-temporal patterns: mirror-reflecting waves, standing waves, and discrete waves are demonstrated. Finally, computer simulations are performed to illustrate the analytical results found.     

Self-similar problem of Cauchy-Poisson waves at an inclined shore : PMM vol. 37, n6, 1973, pp. 1035–1039

We consider a two-dimensional problem concerning Cauchy-Poisson waves at an inclined shore in the case of an initial disturbance concentrated near the shore edge. We study the behavior of the solution near the shore and at large distances from it.Numerous investigations, devoted to the study of standing and progressive waves on an inclined shore, are described in [1]. A two-dimensional problem concerning nonstationary waves on a shore with an angle of inclination γ = π/2n, where n is an integer, was analyzed in [2, 3]. We consider below a case in which the angle of inclination is commensurable with λ/2, subject to the condition that the initial disturbance is concentrated in the vicinity of the shore edge, so that the problem may be considered self-similar.     

Existence, Uniqueness, and Construction of the Solution of a System of Ordinary Functional Differential Equations, with Application to the Design of Perfectly Focusing Symmetric Lenses

In the design of perfectly focusing symmetric lenses, one isled, in a natural way, to a set offunctional differential equations;that is, differential equations involving composites of unknownfunctions, with initial conditions prescribed on the lens axis.This paper concentrateson those features of the equations whichmake them uniquely solvable. They are: (i) a contractivenessproperty of the equations near the axis; (ii) a uniform retardationin the arguments of thecomposite functions away from the axis.The second and third sections of this paper generalize and formalizethese properties and provide proofs of existence, uniqueness,and continuous dependence on the data for solutions of suchgeneralized systems of functional differential equations. Becauseof the lens context which motivates our study, the problem inwhich the contractiveness property (i) above holds is calledthe ‘local’ problem, and the problem in which thearguments of composite functions are uniformly retarded is calledthe ‘global’ problem. In the final section of thepaper we apply the general results of the preceding sectionsto prove existence and uniqueness of perfectly focusing symmetriclenses up to distances from the lens axis at which various typesof breakdown, discussed in the text, may occur.     

Multi-hump pulses in systems with reflection and phase invariance

We investigate the existence and stability of standing and travelling multi-hump waves in partial differential equations with reflection and phase symmetries. We focus on 2- and 3-pulse solutions that arise near bi-foci and apply our results to the complex cubic-quintic Ginzburg-Landau equation.     

Can Parity‐Time‐Symmetric Potentials Support Families of Non‐Parity‐Time‐Symmetric Solitons?

For the one‐dimensional nonlinear Schrödinger equations with parity‐time (PT) symmetric potentials, it is shown that when a real symmetric potential is perturbed by weak PT‐symmetric perturbations, continuous families of asymmetric solitary waves in the real potential are destroyed. It is also shown that in the same model with a general PT‐symmetric potential, symmetry breaking of PT‐symmetric solitary waves does not occur. Based on these findings, it is conjectured that one‐dimensional PT‐symmetric potentials cannot support continuous families of non‐PT‐symmetric solitary waves.     

Wave Properties of Double One-Dimensional Periodic Sheet Grating

We show that the double one-dimensional periodic sheet gratings always have waveguide properties for acoustic waves. In general, there are two types of pass bands: i.e., the connected sets of frequencies for which there exist harmonic acoustic traveling waves propagating in the direction of periodicity and localized in the neighborhood of the grating. Using numerical-analytical methods, we describe the dispersion relations for these waves, pass bands, and their dependence on the geometric parameters of the problem. The phenomenon is discovered of bifurcation of waveguide frequencies with respect to the parameter of the distance between the gratings that decreases from infinity. Some estimates are obtained for the parameters of frequency splitting or fusion in dependence on the distance between the simple blade gratings forming the double grating. We show that near a double sheet grating there always exist standing waves (in-phase oscillations in the neighboring fundamental cells of the group of translations) localized near the grating. By numerical-analytical methods, the dependences of the standing wave frequencies on the geometric parameters of the grating are determined. The mechanics is described of traveling and standing waves localized in the neighborhood of the double gratings.     

Heteroclinic standing waves in defocusing DNLS equations: variational approach via energy minimization

We study heteroclinic standing waves (dark solitons) in discrete nonlinear Schrödinger equations with defocusing nonlinearity. Our main result is a quite elementary existence proof for waves with monotone and odd profile, and relies on minimizing an appropriately defined energy functional. We also study the continuum limit and the numerical approximation of standing waves.     

Paralinearization of the Dirichlet to Neumann Operator,and Regularity of Three-Dimensional Water Waves

This paper is concerned with a priori C ^∞ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a Diophantine condition are automatically C ^∞. In particular, we prove that the solutions defined by Iooss and Plotnikov are C ^∞. Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.     

微极各向同性弹性板中的Rayleigh-Lamb波

研究了无应力作用条件下，均匀、各向同性、圆柱形微极结构弹性板中波的传播．导出了对称和斜对称模式下波传播的特征方程．对短波这一极端情况，无应力圆板中对称和斜对称模态波的特征方程退化为Pmyle曲表面波频率方程．并得到薄板的计算结果．给出了位移和微转动分量，并绘制了相应图形．给出了若干特殊情况的研究结果及对称和斜对称模态特征方程的图示．     

On the evolution of two-dimensional patterns in an inviscid liquid sheet

We perform an analysis of the pattern formation for a moving sheet of inviscid fluid. The sheet, which is assumed to have an infinite horizontal extent, moves at some prescribed velocity into a passive surrounding gas. The sheet’s thickness is assumed much smaller than the horizontal scale of the fluid motion. By considering a system that is symmetric with respect to the horizontal planes, long scale asymptotics are used to reduce the full governing equations in three dimensions to a set of three coupled nonlinear partial differential equations for the horizontal components of the velocity field and the height of the interface profile. The interfacial conditions consisting of the kinematic and normal stress balance are incorporated into these evolution equations. Investigations are carried out as function of the sole dimensionless parameter, namely the Weber number. A small amplitude stability analysis around the planar gas–liquid interface reveals that wave patterns in the form of traveling plane waves occur subcritically, and are therefore unstable. The reduced evolution equations are solved numerically for fixed values of the Weber number. Since the reduced system of equations is homogeneous, the wave motion is generated by initial conditions. Five initial conditions have been imposed: one-dimensional rolls, two-dimensional squares, two-dimensional hexagons, two-dimensional ridges, and smooth peaks. The ensuing evolution of the liquid sheet’s shape and corresponding flow fields are described by illustrations of the changes in the sheet’s morphology with time.     

Existence and stability of TE modes in a stratified non-linear dielectric

Starting from Maxwell's equations for a stratified optical mediumwith a non-linear refractive index, we derive the equationsfor monochromatic planar TE modes. It is then shown that TEmodes in which the electromagnetic fields are travelling wavescorrespond to solutions of these reduced equations in the formof standing waves. The equations of the paraxial approximationare then formulated and the stability of the travelling wavesis investigated in that context.     

On resonant interactions of gravity‐capillary waves without energy exchange

We consider resonant triad interactions of gravity‐capillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the two‐dimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a two‐dimensional generalization of Wilton ripples.     

Propagating and Standing Rayleigh Waves near Rivet Chains Connecting Kirchhoff Plates

We show that near periodic rivet chains that connect two Kirchhoff plates and are modeled by point Sobolev transmission conditions, Rayleigh waves arise, propagate along the chains, and decay exponentially in the orthogonal direction. Under additional geometric conditions we discover the standing (periodic) waves that carry no energy.