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The influence of two nonlinear terms on the orbital stability of solitary wave solutions to the generalized symmetric regularized-long-wave(gsrlw) equation is investigated in this paper. Based on the general conclusion to judge the orbit stability of solitary wave solution to the equation, the stable and unstable wave velocity intervals of solitary wave solutions to the gsrlw equation with two low order nonlinear terms are given. By appropriate transformation and scaling, the complexity caused by two high-order nonlinear terms is overcome, and the stable and unstable wave velocity intervals of solitary wave solutions to the gsrlw equation with high-order nonlinear terms are also obtained. Last, the influences of the coefficients and the order of the nonlinear terms on the stability of solitary wave solutions are studied.

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrodinger equation with combined power-type nonlinearities (FNLS) i ∂ t ψ − ( − Δ ) s ψ + a | ψ | p 1 ψ + | ψ | p 2 ψ = 0 . We prove that when p 2 = 4 s N and a ( p 1 − 4 s N ) 0 , there exist the standing waves of (FNLS), which are orbitally stable. When a = 0 and 4 s N p 2 4 s N − 2 s , we present a new, simpler method to study the strong instability of standing waves. When a = − 1 , 0 p 1 p 2 and 4 s N ≤ p 2 4 s N − 2 s , or a = 1 and 4 s N ≤ p 1 p 2 4 s N − 2 s , or a = 1 , 0 p 1 4 s N p 2 4 s N − 2 s and ∂ λ 2 S ω ( u ω λ ) | λ = 1 ≤ 0 , we deduce that the ground state standing waves of (FNLS) are strongly unstable by blow-up.

In this paper, we study the orbital stability of peakons and periodic peakons for a nonlinear quartic Camassa-Holm equation. We first verify that the QCHE has global peakon and periodic peakon solutions. Then by the invariants of the equation and controlling the extrema of the solution, we prove that the shapes of the peakons and periodic peakons are stable under small perturbations in the energy space.

We consider the third order Benjamin-Ono equation on the torus ∂ t u = ∂ x ( − ∂ x x u − 3 2 u H ∂ x u − 3 2 H ( u ∂ x u ) + u 3 ) . We prove that for any t ∈ R , the flow map continuously extends to H r , 0 s ( T ) if s ≥ 0 , but does not admit a continuous extension to H r , 0 − s ( T ) if 0 s 1 2 . Moreover, we show that the extension is weakly sequentially continuous in H r , 0 s ( T ) if s > 0 , but is not weakly sequentially continuous in L r , 0 2 ( T ) . We then classify the traveling wave solutions for the third order Benjamin-Ono equation in L r , 0 2 ( T ) and study their orbital stability.

The dynamical stability of the periodic peaked solitons for a generalized Camassa–Holm equation is studied in this paper. Its generalization version is known to admit a single-peaked soliton soluti...

We prove the long time orbital stability of the plane wave solutions to the nonlinear Schrodinger equation (NLS) in the defocusing ( λ = 1 ) or focusing ( λ = − 1 ) case, i u t = − Δ u + λ | u | 2 ...

It is well-known that peakons in the Camassa–Holm equation are H 1 -orbitally stable thanks to conserved quantities and properties of peakons as constrained energy minimizers. By using the method of characteristics, we prove that piecewise C 1 perturbations to peakons grow in time in spite of their stability in the H 1 -norm. We also show that the linearized stability analysis near peakons contradicts the H 1 -orbital stability result, hence passage from linear to nonlinear theory is false in H 1 .

We examine the stability of the elliptic solutions of the focusing nonlinear Schrodinger equation (NLS) with respect to subharmonic perturbations. Using the integrability of NLS, we discuss the spe...

In this paper, we present the first result concerning the orbital stability of periodic traveling waves for the modified Kawahara equation. Our method is based on the Fourier expansion of the periodic wave in order to know the behaviour of the nonpositive spectrum of the associated linearized operator around the periodic wave combined with a recent development which significantly simplifies the obtaining of orbital stability results., Comment: 8 pages. arXiv admin note: text overlap with arXiv:1902.04402 To appear in Applicable Analysis

Agraïments: This work was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) and the WWTF project MA09-003 "The flowbeneath a surface water wave" of the Vienna Science and Technology Fund. The authors gratefully acknowledge helpful suggestions by Prof. Gasull. We study the orbital stability of solitary traveling wave solutions of an equation for surface water waves of moderate amplitude in the shallow water regime. Our approach is based on a method proposed by Grillakis, Shatah and Strauss in 1987 [1], and relies on a reformulation of the evolution equation in Hamiltonian form. We deduce stability of solitary waves by proving the convexity of a scalar function, which is based on two nonlinear functionals that are preserved under the flow.

We study orbital stability of solitary wave solutions of the Kudryashov–Sinelshchikov equation in describing the pressure waves in a mixture liquid. Our approach is based on the theories of orbital stability presented by Grillakis, Shatah and Strauss, and relies on the sign of scalar function. By analyzing and proving conditions for stability or instability of solitary wave solution to the Kudryashov–Sinelshchikov equation in terms of parameters, the conditions in stability of solitary wave solution to the Kudryashov–Sinelshchikov equation are given.

We study the Cauchy problem for an inhomogeneous Gross-Pitaevskii equation. We first derive a sharp threshold for global existence and blow up of the solution. Then we construct and classify finite time blow up solutions at the minimal mass threshold. Additionally, using variational techniques, we study the existence, the orbital stability and instability of standing waves., 23 pages

In this paper, we study the theory of forward attractors for non-autonomous impulsive semidynamical systems. Moreover, we investigate some types of stability of the global attractor as orbital stability, asymptotic stability and stability in the sense of Lyapunov-Barbashin. We present an example to illustrate the theory.

In this paper we will continue the analysis of two dimensional Schr\"odinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the stability features.

In this paper, we study the existence and stability of normalized standing waves for the nonlinear Schrödinger equation on a general starlike graph with potentials. Under general assumptions on the graph and the potential, we show the existence of orbitally stable standing waves when the nonlinearity is L2-critical and supercritical. Neste artigo, estudamos a existência e estabilidade de ondas estacionárias normalizadas para a equação de Schrödinger não linear em um gráfico geral semelhante a uma estrela com potenciais. Sob suposições gerais sobre o gráfico e o potencial, mostramos a existência de ondas estacionárias orbitalmente estáveis quando a não linearidade é L2 crítica e supercrítica.

This paper investigates the orbital stability and instability of solitary waves for the generalized Boussinesq equation with two nonlinear terms. Firstly, according to the theory of Grillakis–Shatah–Strauss orbital stability, we present the general results to judge orbital stability of the solitary waves. Further, we deduce the explicit expression of discrimination d′′(c) to judge the stability of the two solitary waves, and give the stable wave speed interval. Moreover, we analyze the influence of the interaction between two nonlinear terms on the stable wave speed interval, and give the maximal stable range for the wave speed. Finally, some conclusions are given in this paper.

In this paper, the orbital stability of dn periodic solutions for the generalized symmetric regularized-long-wave equation with two nonlinear terms is investigated. First, the existence of dn periodic solution to the equation is obtained. Then, according to the Floquet theory and Lame equation, the spectral properties of corresponding linear operators are given. Last, according to the classical theory of stability, the orbital stability of the dn periodic solution for the generalized symmetric regularized-long-wave equation is proved to be stable under the perturbation of period L .

In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schrodinger-Kirchhoff equation. The presence of a mixed term gives us more dispersion, and consequently, a different scenario for the stability of solitary waves in contrast with the corresponding nonlinear Schrodinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.

Considered here is the stability problem of solitary traveling waves with non-zero boundary of an equation describing the free surface waves of moderate amplitude in the shallow water regime. We employ a transform to convert the stability problem of solitary waves with non-zero boundary to the same one of solitary waves vanishing at infinity for a new equation, such that the abstract stability theorem proposed by Grillakis et al. can work on it. We then show that the solitary traveling waves with non-zero boundary are orbitally stable under certain parameter conditions.

Hopping is a highly dynamic motion requiring precise input over brief moments of ground contact in order to achieve desired performance. While this problem has been approached from multiple perspectives, this work provides a comparative analysis of two robot models. The first model uses an actuator to store energy in a spring and release it during the ground phase, while the second uses an actuator to move an additional mass vertically to generate force on the spring. In the first model, analytic expressions are used to find the desired controllers, while trajectory optimization is used in the latter. Orbital stability of each model under the conditions of uncertain damping and poor estimation of the hop height is examined. To this end, Poincare analysis is used to give a metric of stability in the presence of different initial conditions and parameter uncertainty. Simulations show that the first model converges quickly to a point near the desired height determined by the amount of uncertain damping present. The second model is less robust to uncertainty, but is be made to converge to a desired height with the addition of PD control around the optimal trajectory. This robustness is improved with different gains in the controller. In experiments performed on hardware for the second model, stability is observed through convergence to a periodic orbit within several hops.

In this work the orbital dynamics of Earth satellites about the geosynchronous altitude are explored, with primary goal to assess current mitigation guidelines as well as to discuss the future exploitation of the region. A thorough dynamical mapping was conducted in a high-definition grid of orbital elements, enabled by a fast and accurate semi-analytical propagator, which considers all the relevant perturbations. The results are presented in appropriately selected stability maps to highlight the underlying mechanisms and their interplay, that can lead to stable graveyard orbits or fast re-entry pathways. The natural separation of the long-term evolution between equatorial and inclined satellites is discussed in terms of post-mission disposal strategies. Moreover, we confirm the existence of an effective cleansing mechanism for inclined geosynchronous satellites and discuss its implications in terms of current guidelines as well as alternative mission designs that could lead to a sustainable use of the geosynchronous orbital region., Comment: Accepted for publication in Celestial Mechanics and Dynamical Astronomy

In this work, we establish necessary and sufficient conditions for the uniform and orbital stability of a special class of sets on impulsive dynamical systems. The results are achieved by means of Lyapunov functions.

This paper is devoted to studying the dynamical stability of periodic peaked solitary waves for the generalized modified Camassa-Holm equation. The equation is a generalization of the modified Camassa-Holm equation and it possesses the Hamiltonian structure shared by the modified Camassa-Holm equation. The equation admits the periodic peakons. It is shown that the periodic peakons are dynamically stable under small perturbations in the energy space.

This paper is concerned with orbital stability of the smooth solitary wave with nonzero asymptotic value for the mCH equationUnder the parametric conditions a > 0 and , an interesting phenomenon is discovered, that is, for the stability there exist three bifurcation wave speedssuch that the following conclusions hold. When wave speed belongs to the interval (c1, c2) for , the smooth solitary wave is orbitally stable.When wave speed belongs to the interval (c2, c3) for , the smooth solitary wave is orbitally unstable.When wave speed belongs to the interval (c1, c3) for , the smooth solitary wave is orbitally unstable.

This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations utt−uxx+u+αuv+β|u|2u=0,vtt−vxx=(|u|2)xx, where α ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech-type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, α = 1,β = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.

The present study is concerned with the stability of solitary waves for the generalized two-component Camassa-Holm system derived formally as a model in the shallow-water waves. Using the property of almost monotonicity and the local coercivity of the solitary-wave solution, it is shown that the train of N N -smooth solitary waves of this system is dynamically stable to perturbations in energy space with a range of parameters.

This paper discusses the existence and stability of solitary-wave solutions of a general higher-order Benjamin-Bona-Mahony (BBM) equation, which involves pseudo-differential operators for the linear part. One of such equations can be derived from water-wave problems as second-order approximate equations from fully nonlin- ear governing equations. Under some conditions onthe symbols of pseudo-differential operators and the nonlinear terms, it is shown that the general higher-orderBBM equa- tion has solitary-wave solutions. Moreover, under slightly more restrictive conditions, the set of solitary-wave solutions is orbitally stable. Here, the equation has a nonlinear part involving the polynomials of solution and its derivatives with different degrees (not homogeneous), which has not been studied before. Numerical stability and in- stability of solitary-wave solutions for some special fifth-order BBM equations are also given.

In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation $~u_{tt} -u_{xx}+a u_{xxxx}-bu_{xxtt} = - (|u|^{p-1}u)_{xx}~$ for $~p>1$, $~a\geq b>0$. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms $u_{xxxx}$ and $u_{xxtt}$. We obtain an explicit condition in terms of $a$, $b$ and $p$ on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation ($b=0$), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide both analytical and numerical results on the variation of the stability region of wave velocities with $a$, $b$ and $p$ and then state explicitly the conditions under which the traveling waves are orbitally stable., 16 pages, 4 figures

In this paper, we identify necessary and sufficient conditions for the existence of appropriately localized waves for the inhomogeneous semi-linear Schrodinger equation driven by the subLaplacian dispersion operators ( − Δ ) s , 0 s ≤ 1 . We construct these waves and we establish sharp asymptotics, both at the singularity 0 and for large values. We show the non-degeneracy of these waves. Finally, we provide spectral and orbital stability classification, under slightly more restrictive assumptions.

In order to achieve flexible and smooth walking, we must accomplish subtasks (e. g., loading response, forward propulsion or swing initiation) within a gait cycle. To evaluate subtasks within a gait cycle, the analysis of muscle synergies may be effective. In the case of walking, extracted sets of muscle synergies characterize muscle patterns that relate to the subtasks within a gait cycle. Although previous studies have reported that the muscle synergies of individuals with disorders reflect impairments, a way to investigate the instability in the activations of muscle synergies themselves has not been proposed. Thus, we investigated the local dynamic stability and orbital stability of activations of muscle synergies across various walking speeds using maximum Lyapunov exponents and maximum Floquet multipliers. We revealed that the local dynamic stability in the activations decreased with accelerated walking speeds. Contrary to the local dynamic stability, the orbital stability of the activations was almost constant across walking speeds. In addition, the increasing rates of maximum Lyapunov exponents were different among the muscle synergies. Therefore, the local dynamic stability in the activations might depend on the requirement of motor output related to the subtasks within a gait cycle. We concluded that the local dynamic stability in the activation of muscle synergies decrease as walking speed accelerates. On the other hand, the orbital stability is sustained across broad walking speeds.

We study the standing waves of the nonlinear fractional Schrodinger equation. We obtain that when $$0

In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: $ u_{tt}-Lu_{xx}=B(\pm |u|^{p-1}u)_{xx}$, $ p>1$. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operators $L$ and $B$. Members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations. For instance, all Boussinesq-type equations and the so-called double-dispersion equation are members of the class. We first establish the existence of traveling wave solutions to the nonlocal wave equations considered. We then obtain results on the orbital stability or instability of traveling waves. For the case $L=I$, corresponding to a class of Klein-Gordon-type equations, we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow-up., 40 pages

In present work, in order to avoid the spread of disease, the impulse control strategy is implemented to keep the density of infections at a low level. The SIR epidemic model with resource limitation including a nonlinear impulsive function and a state-dependent feedback control scheme is proposed and analyzed. Based on the qualitative properties of the corresponding continuous system, the existence and stability of positive order-k ( $k\in\mathbf{Z}^{+}$ ) periodic solution are investigated. By using the Poincare map and the geometric method, some sufficient conditions for the existence and stability of positive order-1 or order-2 periodic solution are obtained. Moreover, the sufficient conditions which guarantee the nonexistence of order-k ( $k\geq3$ ) periodic solution are given. Some numerical simulations are carried out to illustrate the feasibility of our main results.

We have investigated the dynamical stability of the proposed companions orbiting the Algol type short-period eclipsing binary SW Lyncis (Kim et al. 2010). The two candidate companions are of stellar to sub-stellar nature, and were inferred from timing measurements of the system's primary and secondary eclipses. We applied well-tested numerical techniques to accurately integrate the orbits of the two companions and to test for chaotic dynamical behaviour. We carried out the stability analysis within a systematic parameter survey varying both the geometries and orientation of the orbits of the companions, as well as their masses. In all our numerical integrations we found that the proposed SW Lyn multi-body system is highly unstable on time-scales on the order of 1000 years. Our results cast doubt on the interpretation that the timing variations are caused by two companions. This work demonstrates that a straightforward dynamical analysis can help to test whether a best-fit companion-based model is a physically viable explanation for measured eclipse timing variations. We conclude that dynamical considerations reveal that the propsed SW Lyncis multi-body system most likely does not exist or the companions have significantly different orbital properties as conjectured in Kim et al. (2010)., 9 pages, 6 figures, 2 tables. Submitted to and accepted by JASS -- Journal for Astronomy and Space Sciences (using JKAS LaTeX style file)

In this paper we study the orbital stability of the peaked solitons to the Novikov equation, which is an integrable Camassa–Holm type equation with cubic nonlinearity. We show that the shapes of these peaked solitons are stable under small perturbations in the energy space.

We consider an epidemic-species hybrid dynamical system. The disease is spread among the prey only and the infected prey can reproduce virus. The predator only eats the infected prey. Mathematical analyses are given for the system with regard to the existence of equilibria, local stability, Hopf bifurcation, and the orbital stability of the Hopf bifurcating limit cycle. We further analyse the system under impulsive releasing of virus and predator.

The aim of this paper is to provide a proof of the existence and (conditional) orbital stability for a two-parameter family of solitary-wave solutions to a coupled system of logarithmic nonlinear Schrodinger–Korteweg–de Vries equations. We also obtain the existence of global solutions for initial data in the energy space H ℂ 1 ( R ) × H 1 ( R ) and in an appropriate Orlicz space.

In this paper, we study the Cauchy problem for the two-component Dullin–Gottwald–Holm system with the initial data ( u 0 , ρ 0 ) ∈ B p , r s ( R ) × B p , r s − 1 ( R ) , 1 ≤ p , r ≤ + ∞ , and s > max { 1 + 1 p , 3 2 , 2 − 1 p } , which generalizes some previous local well-posed results in Sobolev spaces. Then we prove that all the smooth solitary wave solutions are orbitally stable under small disturbances.

We present a detailed dynamical analysis of the orbital stability of the two circumbinary planets recently proposed to orbit the evolved eclipsing binary star system NSVS 14256825. As is the case for other recently proposed circumbinary planetary systems detected through the timing of mutual eclipses between the central binary stars, the proposed planets do not stand up to dynamical scrutiny. The proposed orbits for the two planets are extremely unstable on timescales of less than a thousand years, regardless of the mutual inclination between the planetary orbits. For the scenario where the planetary orbits are coplanar, a small region of moderate stability was observed, featuring orbits that were somewhat protected from destabilisation by the influence of mutual 2:1 mean-motion resonance between the orbits of the planets. Even in this stable region, however, the systems tested typically only survived on timescales of order 1 million years, far shorter than the age of the system. Our results suggest that, if there are planets in the NSVS 14256825 system, they must move on orbits dramatically different to those proposed in the discovery work. More observations are clearly critically required in order to constrain the nature of the suggested orbital bodies., Comment: Accepted for publication in MNRAS

In this paper, we consider the orbital stability of smooth solitary wave solutions of the generalized Camassa–Holm equation. By constructing the functional extremum problem and using the orbital stability theory presented by Grillakis, Shatah, Strauss and Bona, and Souganidis, we show that the solitary wave solutions of the generalized Camassa–Holm equation are orbitally stable or unstable as determined by the sign of a discriminant. The conclusions presented by the previous authors, such as Hakkaev and Kirchev, Constantin and Strauss, can be considered as a special case of our results.

We consider a derivative nonlinear Schrodinger equation with a general nonlinearity. This equation has a two-parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, on the velocity. We illustrate these results with numerical simulations.

In this paper we investigate the orbital stability of solitary waves to the (generalized) Kawahara equation (gKW) which is a fifth order dispersive equation. For some values of the power of the nonlinearity, we prove the orbital stability in the energy space H 2 (R) of two branches of even solitary waves of gKW by combining the well-known spectral method introduced by Benjamin [3] with continuity arguments. We construct the first family of even solitons by applying the implicit function theorem in the neighborhood of the explicit solitons of gKW found by Dey et al. [8]. The second family consists of even travelling waves with low speeds. They are solutions of a constraint minimization problem on the line and rescaling of perturbations of the soliton of gKdV with speed 1.

The present paper deals with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves. Our method can be seen as an extension to spatially periodic waves of the theory of solitary waves recently developed in [25] . Firstly, our main result do not depend on the parametrization of the periodic wave itself. Secondly, motived by the well known orbital stability criterion for solitary waves, we show that the same criterion holds for periodic waves. In addition, we show that the positiveness of the principal entries of the Hessian matrix related to the “energy surface function” are also sufficient to obtain the stability. Consequently, we establish the orbital stability of periodic waves for several nonlinear dispersive models. We believe our method can be applied in a wide class of evolution equations; in particular it can be extended to regularized dispersive wave equations.

The stability of the Camassa-Holm (periodic) peakons in the dynamics of an integrable shallow-water-type system is investigated. A variational approach with the use of the Lyapunov method is presented to prove the variational characterization and the orbital stability of these wave patterns. In addition, a sufficient condition for the global existence of strong solutions is given. Finally, a local-in-space wave-breaking criterion is illustrated in the periodic setting.

In this paper, we investigate the properties of solitonic structures arising in quadratic media. First, we recall the derivation of systems governing the interaction process for waves propagating in such media and we check the local and global well-posedness of the corresponding Cauchy problem. Then, we look for stationary states in the context of normal or anomalous dispersion regimes, that lead us to either elliptic or non-elliptic systems and we address the problem of orbital stability. Finally, some numerical experiments are carried out in order to compute localized states for several regimes and to study dynamic stability as well as long-time asymptotics.

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting.

Analogous to the Camassa–Holm equation, the Dullin–Gottwald–Holm equation also possesses peaked solitary waves, which are called peakons. We prove in this paper the stability of ordered trains of peakons for the Dullin–Gottwald–Holm equation.