ORBITAL STABILITY OF PERIODIC TRAVELING WAVE SOLUTIONS TO THE GENERALIZED ZAKHAROV EQUATIONS

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations {iu ＋uxx = uv ＋ ｜u｜^2u, vtt-vxx=（｜u｜^2）xx.First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19].

Orbital Stability of Solitary Waves for Generalized Zakharov System

In the interaction of laser-plasma the system of Zakharov equation plays an important role.This system attracted many scientists' wide interest and attention.And the formation, evolution and interaction of the Langmuir solutions differ from solutions of the KDV equation. Here we consider the following generalized Zakharov

ORBITAL STABILITY FOR SCHRDINGER SYSTEMS WITH NONAUTONOMOUS COUPLED NONLINEARITIES

We prove the existence and the orbital stability of standing waves for the nonau- tonomous Schrodinger system……under suitable conditions on the coefficient functions a and b. We follow the idea of analyzing the compactness of minimizing sequence of the constrained minimization problems.

Orbital stability of two-component peakons

We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable Novikov equation.We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures.Moreover,we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.

Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential

In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L2-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.

ORBITAL STABILITY OF SOLITARY WAVES FOR GENERALIZED ZAKHAROV SYSTEM

This paper considers the stability of the solitary waves for the generalized Zakharov system. By applying the abstract theory of Grillakis M. et al. and detailed spectral analysis, we obtain the stability of the solitary waves.

Existence,Orbital Stability and Instability of Solitary Waves for Coupled BBM Equations

This paper is concerned with the orbital stability/instability of solitary waves for coupled BBM equations which have Hamiltonian form. The explicit solitary wave solutions will be worked out first. Then by detailed spectral analysis and decaying estimates of solutions for the initial value problem, we obtain the orbital stability/instability of solitary waves.

Orbital Stability of Solitary Waves of Compound KdV-type Equation

In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of ut ＋ auPux ＋ bu2pux ＋ Uzzz = 0 （b 〉 0, p 〉 0）. Our results imply that orbital stability of solitary waves is affected not only by the highest-order nonlinear term bu2pux, but also the nonlinear term auPux. For the case of b 〉 0 and 0 〈 p ≤ 2, we obtain that the positive solitary wave Ul（X - ct） is stable when a 〉 0, while that unstable when a 〈 0. The stability for negative solitary wave u2（x - ct） is on the contrary. In particular, we point that the nonlinear term with coefficient a makes contributes to the stability of the solitary waves when p= 2 and a〉0.

Stability of Schrdinger-Poisson type equations

Variational methods are used to study the nonlinear SchrSdinger-Poisson type equations which model the electromagnetic wave propagating in the plasma in physics. By analyzing the Hamiltonian property to construct a constrained variational problem, the existence of the ground state of the system is obtained. Furthermore, it is shown that the ground state is orbitally stable.

Existence and Stability of Standing Waves with Prescribed L2-Norm for a Class of Schrödinger-Bopp-Podolsky System

In this paper, we look for solutions to the following Schrödinger-Bopp-Podolsky system with prescribed L2-norm constraint , where q ≠ 0, a, ρ> 0 are constants. At first, by the classical minimizing argument, we obtain a ground state solution to the above problem for sufficiently small ρwhen . Secondly, in the case p = 6, we show the nonexistence of positive solutions by using a Liouville-type result. Finally, we argue by contradiction to investigate the orbital stability of standing waves for .

Stability and control of dynamic walking for a five-link planar biped robot with feet

During dynamic walking of biped robots, the underactuated rotating degree of freedom （DOF） emerges between the support foot and the ground, which makes the biped model hybrid and dimension-variant. This paper addresses the asymptotic orbit stability for dimension-variant hybrid systems （DVHS）. Based on the generalized Poincare map, the stability criterion for DVHS is also presented, and the result is then used to study dynamic walking for a five-link planar biped robot with feet. Time-invariant gait planning and nonlinear control strategy for dynamic walking with fiat feet is also introduced. Simulation results indicate that an asymptotically stable limit cycle of dynamic walking is achieved by the proposed method.

Coupled trajectory and attitude stability of displaced orbits

Coupled trajectory and attitude stability of displaced solar orbits is studied by using sailcraft with a kind of two-folding construction with two unequal rectangular plates forming a right angle. Three-dimensional coupled trajectory and attitude equations are developed for the coupled dynamical system, and the results show that all three types of displaced solar orbits widely referenced can be achieved through selecting an appropriate size of the two-folding sail. An anal- ysis of the corresponding linear stability of the trajectory and attitude coupled system is carried out, and both trajectory and attitude linearly stable orbits are found to exist in a small range of parameters, whose non-linear stability is then examined via numerical simulations. Finally, passively stable orbits are found to have weak stability, and such passive means of station-keeping are attractive and useful in practice because of its simplicity.

Stability of Standing Waves for the Nonlinear Schrödinger Equation with Mixed Power-Type and Hartree-Type Nonlinearities

This paper studies the existence of stable standing waves for the nonlinear Schrödinger equation with Hartree-type nonlinearity i∂tψ+Δψ+| ψ |pψ+(| x |−γ∗| ψ |2)ψ=0, (t,x)∈[ 0,T )×ℝN.Where ψ=ψ(t,x)is a complex valued function of (t,x)∈ℝ+×ℝN. The parameters N≥3, 0p4Nand 0γmin{ 4,N }. By using the variational methods and concentration compactness principle, we prove the orbital stability of standing waves.

The Effect of State-Dependent Control for an SIRS Epidemic Model with Varying Total Population

Based on the mechanism of prevention and control of infectious disease, we propose, in this paper, an SIRS epidemic model with varying total population size and state-dependent control, where the fraction of susceptible individuals in population is as the detection threshold value. By the Poincaré map, theory of differential inequalities and differential equation geometry, the existence and orbital stability of the disease-free periodic solution are discussed. Theoretical results show that by state-dependent pulse vaccination we can make the proportion of infected individuals tend to zero, and control the transmission of disease in population.

STABILITY AND INSTABILITY OF SOLITARY WAVES FOR ABSTRACT COMPLEX HAMILTONIAN SYSTEM

This paper is concerned with the orbital stability and orbital instability of solitary waves for some complex Hamiltonian systems in abstract form. Under some assumptions on the spectra of the related operator and the decaying estimates of the semigroup, the sufficient conditions on orbital stability and instability are obtained.

Stability/Instability of Solitary waves with Nonzero Asymptotic Value for a PDE in Microstructural Solid Materials

The present paper deals with results of stability/instability of solitary waves with nonzero asymptotic value for a microstructure PDE. By the exact solitary wave solutions and detailed computations, we set up the explicit expression for the discrimination d′′（c）. Finally, a complete study of orbital stablity/instablity for the explicit exact solutions is given.

THE ORBIT SHIFT TOPOLOGICAL STABILITY OF ANOSOV MAPS

It is shown that Anosov maps are orbit shift topologically stable.

THE NEIGHBORHOOD IN STABILIZING HIGHER PERIOD ORBITS FOR DISCRETE CHAOTIC DYNAMICS

In this paper, variable linear feedback control is used to stabilize unstable higher period orbit in nonlinear discrete chaotic dynamical system. The existence of neighborhood in stabilizing higher period orbits is rigorously proved by functional analysis theory and nonlinear dynamical theory. Numerical simulation is included to support the theoretical analysis in the paper.