Stability of Standing Waves for the Nonlinear Fractional Schrödinger Equation

We study the standing waves of the nonlinear fractional Schrödinger equation. We obtain that when \(0<\gamma <2s\), the standing waves are orbitally stable; when \(\gamma =2s\), the ground state solitary waves are strongly unstable to blow-up.     

The Linear Stability of Shock Waves for the Nonlinear Schrödinger–Inviscid Burgers System

We investigate the coupling between the nonlinear Schrödinger equation and the inviscid Burgers equation, a system which models interactions between short and long waves, for instance in fluids. Well-posedness for the associated Cauchy problem remains a difficult open problem, and we tackle it here via a linearization technique. Namely, we establish a linearized stability theorem for the Schrödinger–Burgers system, when the reference solution is an entropy-satisfying shock wave to Burgers equation. Our proof is based on suitable energy estimates and on properties of hyperbolic equations with discontinuous coefficients. Numerical experiments support and expand our theoretical results.     

Stability of Traveling Waves of Nonlinear Schrödinger Equation with Nonzero Condition at Infinity

We study effective elastic behavior of the incompatibly prestrained thin plates, where the prestrain is independent of thickness and uniform through the plate’s thickness h. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric G, and seek the limiting behavior as \({h \to 0}\). We first establish that when the energy per volume scales as the second power of h, the resulting \({\Gamma}\) -limit is a Kirchhoff-type bending theory. We then show the somewhat surprising result that there exist non-immersible metrics G for whom the infimum energy (per volume) scales smaller than h². This implies that the minimizing sequence of deformations carries nontrivial residual three-dimensional energy but it has zero bending energy as seen from the limit Kirchhoff theory perspective. Another implication is that other asymptotic scenarios are valid in appropriate smaller scaling regimes of energy. We characterize the metrics G with the above property, showing that the zero bending energy in the Kirchhoff limit occurs if and only if the Riemann curvatures R₁₂₁₃, R₁₂₂₃ and R₁₂₁₂ of G vanish identically. We illustrate our findings with examples; of particular interest is an example where \({G_{2 \times 2}}\), the two-dimensional restriction of G, is flat but the plate still exhibits the energy scaling of the Föppl–von Kármán type. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for \({G = Id_{3} + \gamma n \otimes n}\) given in terms of the inhomogeneous unit director field distribution \({ n \in \mathbb{R}^3}\).     

Standing Waves with a Critical Frequency for Nonlinear Schrödinger Equations

This paper is concerned with the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation with E being a critical frequency in the sense that . We show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of V and whose amplitude goes to 0 as . Moreover, depending upon the local behaviour of the potential function V(x) near the minimum points, the limiting profile of the standing-wave solutions will be shown to exhibit quite different characteristic features. This is in striking contrast with the non-critical frequency case which has been extensively studied in recent years.     

Segregated and Synchronized Vector Solutions for Nonlinear Schrödinger Systems

We consider the following nonlinear Schrödinger system in ${\mathbb{R}^3}$ $$\left\{\begin{array}{ll}-\Delta u + P(|x|)u = \mu u^{2}u + \beta v^2u,\quad x \in \mathbb{R}^3,\\-\Delta v + Q(|x|)v = \nu v^{2}v + \beta u^2v,\quad x \in \mathbb{R}^3,\end{array}\right.$$ where P(r) and Q(r) are positive radial potentials, ${\mu > 0, \nu > 0}$ and ${\beta \in \mathbb{R}}$ is a coupling constant. This type of system arises, in particular, in models in Bose–Einstein condensates theory. We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type, and in the attractive case we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Depending upon the system being repulsive or attractive, our results exhibit distinct characteristic features of vector solutions.     

Floquet’s Theorem and Stability of Periodic Solitary Waves

This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in L²_per[0, p]{L^{2}_{\rm per}[0, \pi]} . We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.     

Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential

The nonlinear Schr?dinger equation with attractive quintic nonlinearity in periodic potential in 1D, modeling a dilute-gas Bose–Einstein condensate in a lattice potential, is considered and one family of exact stationary solutions is discussed. Some of these solutions have an analog neither in the linear Schr?dinger equation nor in the integrable nonlinear Schr?dinger equation. Their stability is examined analytically and numerically.     

Stability and Extinction for Lotka–Volterra Systems with Infinite Delay

The paper deals with the global stability of a saturated equilibrium \({x^{\ast} = (x_1^{\ast}, \ldots, x_n^{\ast}) \ge 0}\) for a multi-species Lotka–Volterra model with infinite delay. We assume the existence of non-delayed diagonal intraspecific competition terms, but relax the usual conditions of dominance which permit to control the infinite delay effect. Our results also apply to non-autonomous Lotka–Volterra systems without saturated equilibria, by studying the stability of their limiting equations. Some known results in the literature are improved and generalized.     

Nonlinear Wave-Wave Interaction and Stability Criterion for Parametrically Coupled Nonlinear Schrödinger Equations

A rigorous mathematical reduction of the procedure widely usedfor studying a class of the nonlinear problems with perturbations,namely the method of the multiple scales, is used. A profound analysis,which provides an approach for deriving a coupled nonlinearSchrödinger equations. The investigation has been achieved byperturbing the nonlinear dynamical system about the linear dynamicalproblem. Modulated wavetrains are described to all orders ofapproximation. Moreover, we extend our approach to deal with equationshaving periodic terms. Two types of simultaneous nonlinearSchrödinger equations are derived. One type is valid at thenon-parametric system and the second type represents a modification forthe first type which is governed the non-resonance case. Two parametriccoupled nonlinear Schrödeinger equations are derived to govern thesecond-sub-harmonic resonance. In addition other two coupled equationsare found for the third-sub-harmonic resonance case. These systems ofequations control the stability behavior at the parametric resonancecases. The stability criteria for the several types of coupled nonlinearSchrödinger equations are studied. These criteria are achieved by atemporal periodic perturbation.     

Erratum to: Floquet’s Theorem and Stability of Periodic Solitary Waves

This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in L²_per[0, p]{L^2_{{\rm per}}[0, \pi]}. We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.     

Pointwise Estimates and Stability for Dispersive–Diffusive Shock Waves

We study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion. Along with the usual Lax shocks, such equations are known to admit undercompressive shocks. We unify the treatment of these two cases by introducing a new wave-tracking method based on “instantaneous projection”, giving improved estimates even in the Lax case. Another important feature connected with the introduction of dispersion is the treatment of a non-sectorial operator. An immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all L ^p norms, p≥ 1 for the Lax case and p > 1 for the undercompressive case. Our approach extends immediately to the case of certain scalar equations of higher order, and would also appear suitable for extension to systems. Accepted May 29, 2000?Published online November 16, 2000     

Stability and Spectral Comparison of a Reaction–Diffusion System with Mass Conservation

We study the global-in-time behavior of solutions to a reaction–diffusion system with mass conservation, as proposed in the study of cell polarity, particularly, the second model of the work by Otsuji et al. (PLoS Comput Biol 3:e108, 2007). First, we show the existence of a Lyapunov function and confirm the global-in-time existence of the solution with compact orbit. Then we study the stability and instability of stationary solutions by using the semi-unfolding-minimality property and the spectral comparison. As a result the dynamics near the stationary solutions is qualitatively characterized by a variational function.     

Gap Estimates for Double-Well Schrödinger Operators and Quantum Stabilization of Anharmonic Crystals

The spectrum of the Schrödinger operator of a one-dimensional quantum anharmonic oscillator of mass m is studied. This spectrum consists of simple (nondegenerate) eigenvalues E _n , $$n\in {\mathbb N}_0$$ such that n ^– E _n + as n + with a certain > 1. The gap parameter =min _n (E _n – E _n-1) is in the center of the study. It is proven that this parameter is a continuous function of m; its small mass and large mass asymptotics are found. The influence of the dependence of on m on the stability of systems of interacting quantum anharmonic oscillators is briefly discussed.     

Riemann–Hilbert method and multi-soliton solutions of the Kundu-nonlinear Schrödinger equation

In this work, we study the Kundu-nonlinear Schrödinger (Kundu-NLS) equation (so-called the extended NLS equation), which can describe the propagation of the waves in dispersive media. A Lax spectral problem is used to construct the Riemann–Hilbert problem, via a matrix transformation. Based on the inverse scattering transformation, the general solutions of the Kundu-NLS equation are calculated. In the reflection-less case, the special matrix Riemann–Hilbert problem is carefully proposed to derive the multi-soliton solutions. Finally, some novel dynamics behaviors of the nonlinear system are theoretically and graphically discussed.

     

Multiplicity Results for some Nonlinear¶Schrödinger Equations with Potentials

Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations with potentials are found by means of a perturbative variational method.     

Transverse Vibrations and Stability of Systems with Gyroscopic Forces∗

Abstract

Rotating shafts and pipes conveying fluid are examples of systems involving gyroscopic forces. The vibration and stability properties of such systems are often of practical interest to structural engineers. In this paper attention is focused on the characteristic curves of gyroscopic conservative systems in an appropriately chosen loading-frequency space. An upper bound to the fundamental frequency is obtained via the concept of a “corresponding nongyroscopic system.” The choice of the parameters and the resulting

characteristic curves shed light on the stabilizing effect of gyroscopic forces. Special emphasis is placed on flutter instability. Three well-defined types of systems are discussed and several examples are analyzed. It is shown that various sequences of stable, divergence, and flutter regions may be exhibited as the loading parameter is increased, and that flutter instability may take place in an otherwise stable region.     

Soliton interactions for coupled nonlinear Schrödinger equations with symbolic computation

Soliton interactions for the coupled nonlinear Schrödinger equations, governing the propagation of envelopes of electromagnetic waves in birefringent optical fibers, are investigated with symbolic computation. Based on the Hirota method, analytic two- and three-soliton solutions for this model are derived. Relevant interaction properties are discussed. Stationary bound vector solitons with the periodic attraction and repulsion are obtained. Soliton intensity could be reduced if the nonlinearity in optical fibers is enlarged, while the soliton period could be prolonged as the group velocity dispersion in the anomalous dispersion regime of optical fibers increases. Through the asymptotic analysis for the two-soliton solutions, interactions between two solitons are proven to be elastic. Besides, parallel soliton transmission systems without soliton interactions are presented. Moreover, interactions between the regular and bound vector solitons are studied. Dual complex structures and triple-soliton bound states are presented. Results could be of certain value to the studies on the soliton control and optical switching technologies.     

Solutions and connections of nonlocal derivative nonlinear Schrödinger equations

Nonlinear Dynamics - All possible nonlocal versions of the derivative nonlinear Schrödinger equations are derived by the nonlocal reduction from the Chen–Lee–Liu equation, the...