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In this paper, we consider the generalized quintic-septic Camassa-Holm (gqsCH) equation, which is actually an extension of the quintic CH equation and the septic CH equation. We first prove the existence of the single peakon. Then, by constructing certain Lyapunov functionals, we prove the stability of peakons in the energy space H^1(\mathbb {R})-norm. Finally, we also prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space by using energy argument, combining the method of the orbital stability of single peakons with monotonicity of the local energy norm. [ABSTRACT FROM AUTHOR]

In this work, the dynamical stability of periodic peaked solitary wave for the higher-order modified Camassa-Holm equation is studied. As a generalization of the modified Camassa-Holm equation, it possesses the periodic peakons. However, it has a higher-order nonlinear item compared with the modified Camassa-Holm equation. By the aid of conservation laws and auxiliary functions, we successfully demonstrate that under small perturbations, the periodic peakons are orbital stable in the energy space. [ABSTRACT FROM AUTHOR]

We study the existence and orbital stability of standing waves for a class of Schrödinger equations with L 2 -critical nonlinearity. By considering a minimization problem with bounded potentials, the existence of standing waves is shown by means of the concentration-compactness argument. Moreover, we establish the orbital stability of standing waves. [ABSTRACT FROM AUTHOR]

We analyze the stability of smooth solitary waves in the modified Camassa-Holm equation, a quasilinear, integrable model for shallow water wave propagation. Through phase portrait analysis, we identify a unique smooth solitary wave within a certain range of the dispersive parameter. Using variational methods, we prove the orbital stability of this wave under small disturbances, solving a minimization problem with constraints. We strengthen the H 1 ∩ W 1 , 4 stability result in Li and Liu (2021) [8]. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence and stability of standing waves with prescribed mass for the Hartree-Fock type nonlocal elliptic equation with rotation$ \begin{equation*} \begin{cases} i\partial_{t}u = -\frac{1}{2}\Delta u+V(x)u-\Omega L_{z}u+\lambda u-\mu(J_{\alpha}\ast|u|^{p})|u|^{p-2}u,\ (t,x)\in \mathbb{R}\times\mathbb{R}^{N},\\ u(0,x) = u_{0}(x), \end{cases} \end{equation*} $where $ N\geq3 $, $ \mu>0 $, $ V(x) = \frac{1}{2}\sum\limits_{j = 1}^{N}\gamma_{j}^{2}x_{j}^{2} $, $ L_{z} $ is the angular momentum operator with a rotational speed $ \Omega>0 $, $ \gamma_{j}>0 $ is the trap frequencies, and the constant $ \lambda $ is unknown Lagrange multiplier. We establish the existence and orbital stability results of prescribed mass standing waves for the equation under two distinct scenarios: one in the subcritical case where $ \frac{N+\alpha}{N}

In this paper, we study the existence and orbital stability of pseudo peakons for a new fifth order CH type equation with cubic nonlinearity. Firstly, we derive the equation via an conserved quantity with quartic nonlinearities, then we show such equation admits pseudo peakons, and finally we prove that the pseudo peakons are orbitally stable under small perturbations in the energy space. [ABSTRACT FROM AUTHOR]

Resonant orbits (ROs) within the Earth–Moon system exhibit periodic motion characterized by a simple integer relationship with the lunar period. These orbits hold significant applications for space situational awareness and the design of transfer orbits in cislunar space. However, due to the expansive scale and extended periods associated with Earth–Moon ROs, the resulting eclipse durations often exceed spacecraft power and thermal system capabilities. This paper conducts an analysis of shadow distribution and eclipse durations within the Earth–Moon 3:2 resonant orbit family. The investigation utilizes relative velocity with respect to the lunar shadow, highlighting the imperative need for eclipse mitigation. The study then proceeds to derive eclipse mitigation methods, specifically employing cross-track maneuver and phasing maneuver techniques. These strategies are applied to mitigate and avoid both Earth and lunar shadows, considering different initial solar phases and eclipse occurrence locations. Simulation results conducted within the circular restricted three-body problem (CRTBP) indicate that the eclipse mitigation strategy via cross-track maneuver incurs a lower Δ V cost, remaining within the range of 20 m/s. Furthermore, the stability of the resonant orbits post cross-track maneuver is examined. The study presents the maximum allowable cross-track maneuver Δ V s which maintain orbit stable in five years for the entire resonant orbit family. Ultimately, the paper recommends a 2-hour maximum allowable eclipse duration for spacecraft in Earth–Moon 3:2 resonant orbits, offering valuable insights for the future design of missions within these orbits. The eclipse mitigation and avoidance strategies, along with the analytical methods proposed herein, can be readily extended to other resonant orbit families. [ABSTRACT FROM AUTHOR]

The aim of this work is to present new spectral tools for studying the orbital stability of standing waves solutions for the nonlinear Schrödinger equation (NLS) with power nonlinearity on looping edge graphs, namely, a graph consisting of a circle with several half-lines attached at a single vertex. The main novelty of this paper is at least twofold: by considering δ -type boundary conditions at the vertex, the extension theory of Krein &von Neumann, and a splitting eigenvalue method, we identify the Morse index and the nullity index of a specific linearized operator around of a priori positive single-lobe state profile for every positive power, this information will be main for a local stability study; and so via a bifurcation analysis on the phase plane we build at least two families of positive single-lobe states and we study the stability properties of these in the subcritical, critical, and supercritical cases. Our results recover some spectral studies in the literature associated to the NLS on looping edge graphs which were obtained via variational techniques. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equation with focusing inverse-power potential and the Sobolev critical nonlinearity. By considering the corresponding local minimization problem, we show that the existence of ground state solutions. Then, we prove that the solution of this equation exists globally when the initial data φ sufficiently close to the ground states. Based on these results, we show that the set of ground states is orbitally stable. [ABSTRACT FROM AUTHOR]

In this paper, the orbital instability theorem is introduced for Hamiltonian partial differential equation (PDE) systems. Our specific focus lies on the Schrödinger system featuring quadratic nonlinearity, and we apply the theorem to analyze its behavior. Our theorem establishes the abstract instability theorem for a specific class of Hamiltonian PDE systems. We consider the energy functional to be of class C 2 rather than C 3 , particularly when the second derivative of the energy exhibits multiple degenerate kernels. Using this theorem, we provide a comprehensive classification of the stability and instability of the semitrivial solution within the Hamiltonian PDE system featuring quadratic nonlinearity. This classification resolves an open problem previously posed by Colin et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:2211–2226, 2009), specifically in cases of homogeneous nonlinearity. Additionally, we present proof of instability results for synchronous solutions of Hamiltonian PDE systems. We believe that this abstract theorem constitutes a novel contribution with potential applicability in various situations beyond those specifically discussed in this paper. [ABSTRACT FROM AUTHOR]

Considered herein is the orbital stability of the periodic peaked solitons for the higher-order modified Camassa–Holm equation. This equation can be viewed as a natural higher-order generalization of the modified Camassa–Holm equation, and admits a single peaked soliton and multi-peakons. We first show that the equation possesses the periodic peakons. Furthermore, it is proved that the periodic peakons are dynamically stable under small perturbations in the energy space by utilizing the inequalities with the maximum and minimum of the solutions related to the first two conservation laws. [ABSTRACT FROM AUTHOR]

This paper studies traveling waves with nonzero wave speed (angular traveling waves) of the high‐dimensional Boussinesq equation that have not been studied before. We analyze the properties of these waves and demonstrate that, unlike the unique stationary solution, they lack positivity, radial symmetry, and exponential decay. By employing variational and geometric approaches, along with perturbation theory, we establish the orbital (in)stability and strong instability of these traveling waves. [ABSTRACT FROM AUTHOR]

Orbital evolution of model circumlunar objects moving in circular orbits is analyzed to determine areas optimal for the deployment of new satellite systems moving in orbits of this kind. The areas are determined in which the objects not only remain in their orbits for a long time, but also keep preset initial parameters and hence are the most suitable for the deployment of long-living satellite systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of concentrated traveling-rotating helical vortex for the 3D incompressible Euler equations in an infinite pipe. The solutions are obtained by maximization of the energy over the set of rearrangments of a fixed bounded function with compact support, and tends asymptotically to singular helical vortex filament evolving by the binormal curvature flow. We also show nonlinear stability of maximizers under L p perturbation for p ≥ 2. [ABSTRACT FROM AUTHOR]

We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem for the L 2 supercritical case. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2 , which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surprising" and completely different from the results obtained on the entire space and on balls. We will also show that the nature of the domain is crucial for the existence and stability of standing waves. As a foretaste, it is well-known that in the supercritical case these waves are unstable in R N. In this paper, we will show that in the exterior domain they are strongly stable. [ABSTRACT FROM AUTHOR]

We consider solitary wave solutions of a two-component Novikov system, which is a coupled Camassa-Holm type system with cubic nonlinearity. Inspired by the methods established by Constantin and Strauss in [6-7], we prove that the smooth solitary waves and non-smooth peakons to the system are both orbitally stable. [ABSTRACT FROM AUTHOR]

• A new spherical harmonic model of coma gas density is proposed. • An improved orbital model about a comet with coma gas, SRP, and non-spherical gravity is established. • A global grid method is used to search stable orbits around a comet. • The heliotropism and spatial distribution of stable orbits are analyzed. As a frontier field of deep-space explorations, the study of comets is of great value to planetary scientific research, resource exploration, and planetary defense. To date, only the ESA's Rosetta spacecraft successfully orbited the comet 67P. The orbital dynamical environment around comets is more complex than that of other small bodies, making the orbital stability around comets an important issue. The parameters of Rosetta's target, comet 67P/Churyumov-Gerasimenko, are used in this study of the orbital dynamics around a comet. The most unique perturbation around a comet is the aerodynamic force caused by the coma gas. However, the current numerical models are too complicated for analytical studies, and the current spherical harmonic model is too simple to reflect the actual distribution of the coma gas field accurately. To overcome the disadvantages of current models, a method is proposed to reduce the numerical model to a simplified analytical model. Then, the simplified model is fitted with a new spherical harmonic model, which is suitable for both numerical simulations and analytical studies. By considering perturbative effects of coma gas, solar radiation pressure (SRP), and non-spherical gravity of the comet nucleus, the orbital dynamical model is then established. Next, with a global search method, stable orbits around 67P/Churyumov-Gerasimenko are searched, and their characteristics are analyzed. It is found that the perturbative effect of the coma gas reduces the stability of the orbits. As the distance between the comet and the Sun increases, the stable orbital region will gradually shift from low inclinations to higher ones, and all of the stable orbits are heliotropic. According to the results, some candidate exploration orbits around the comet are finally identified. [ABSTRACT FROM AUTHOR]

We consider a normal autonomous system of differential equations with a small parameter, which has a critical linear approximation at zero value of the parameter. We introduce the concept of orbital stability with respect to the parameter; according to this concept, the closeness of the right semitrajectories is achieved not only due to the proximity of initial values of solutions, but also due to the smallness of the parameter. We examine the problem of branching of a stable periodic solution with a period close to the period of solutions of the corresponding linear homogeneous system. Sufficient conditions for the solvability of the problem are established. Our reasonings are based on the properties of the first homogeneous nonlinear approximation of the monodromy operator. [ABSTRACT FROM AUTHOR]

The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter ε The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale ε -dependent topology reduces to the classical W¹;∞-topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regularity estimates, that the global-in-time analysis is also spatially multiscale due to the coexistence of nontrivial slow parts with fast shock-layer parts, that the limiting smooth spectral problem (in fast variables) has no spectral gap and that uniformity requires a very precise and unusual design of the phase shift encoding orbital stability. In particular, our analysis builds a phase that somehow interpolates between the hyperbolic shock location prescribed by the Rankine–Hugoniot conditions and the nonuniform shift arising merely from phasing out the nondecaying 0-mode, as in the classical stability analysis for fronts of reaction–diffusion equations. [ABSTRACT FROM AUTHOR]

Consideration herein is the stability issue of peaked solitary wave solution for the modified Camassa-Holm-Novikov equation, which is derived from the shallow water theory. This wave configuration accommodates the ordered trains of the modified Camassa-Holm-Novikov-peaked solitary solution. With the application of conservation laws and the monotonicity property of the localized energy functionals, we prove the orbital stability of this wave profile in the H1(R){H}^{1}\left({\mathbb{R}}) energy space according to the modulation argument.

In this article, we conduct a comprehensive investigation into the global existence, blow-up and stability of standing waves for a $ L^{2} $-critical Schrödinger-Choquard equation with harmonic potential. First, by taking advantage of the ground-state solutions and scaling techniques, we obtain some criteria for the global existence and blow-up of the solutions. Second, in terms of the refined compactness argument, scaling techniques and the variational characterization of the ground state solution to the Choquard equation with $ p_{2} = 1+\frac{2+\alpha}{N} $, we explore the limiting dynamics of blow-up solutions to the $ L^{2} $-critical Choquard equation with $ L^{2} $-subcritical perturbation, including the $ L^{2} $-mass concentration and blow-up rate. Finally, the orbital stability of standing waves is investigated in the presence of $ L^{2} $-subcritical perturbation, focusing $ L^{2} $-critical perturbation and defocusing $ L^{2} $-supercritical perturbation by using variational methods. Our results supplement the conclusions of some known works.

In this paper, we generalized the results of the following chemotaxis model with the nonlinear degenerate viscosity $ \begin{equation*} \begin{cases} u_{t} -\chi (uv)_{x} = D(u^{m})_{xx}, \\ v_{t} -u_{x} = 0, \end{cases} \end{equation*} $ by introducing the following general initial perturbation $ \begin{equation*} \begin{split} \int_{-\infty}^{+\infty}\kappa(Z_0|\tilde{Z})dx where $ \kappa $ is the relative entropy function defined in Eq (2.24). We further employed the relative entropy method by choosing the specific shift function. According to the estimates with the cutoff version, and overcoming the complexity caused by the porous media diffusion, the nonlinear orbital stability of traveling waves was established under small amplitude and general perturbations.

This paper investigates a generalized Camassa–Holm equation with cubic nonlinearities (alias the mCH-Novikov equation), which is a generalization of some special equations. The mCH-Novikov equation possesses well-known peaked solitary waves that are called peakons. The peakons were proved orbital stable by Chen et al. in [Stability of peaked solitary waves for a class of cubic quasilinear shallow-water equations. Int Math Res Not. 2022;1–33]. We mainly prove the orbital stability of the multi-peakons in the mCH-Novikov equation. In this paper, it is proved that the sum of N fully decoupled peaks is orbitally stable in the energy space by using energy argument, combining the orbital stability of single peakons and local monotonicity of the method. [ABSTRACT FROM AUTHOR]

We prove the orbital stability of soliton solutions for 2D Maxwell–Lorentz system with extended charged particle. The solitons corresponds to the uniform motion and rotation of the particle.We reduce the corresponding Hamilton system by the canonical transformation via transition to a comoving frame. The solitons are the critical points of the reduced Hamiltonian. The key point of the proof is a lower bound for the Hamiltonian. [ABSTRACT FROM AUTHOR]

We are interested in the following semilinear elliptic problem: - Δ u + λ u = u p - 1 , x ∈ T , u > 0 , u = 0 on ∂ T , ∫ T u 2 d x = c where T = { x ∈ R N : 1 < | x | < 2 } is an annulus in R N , N ≥ 2 , p > 1 is Sobolev-subcritical, searching for conditions (about c, N and p) for the existence of positive radial solutions. We analyze the asymptotic behavior of c as λ → + ∞ and λ → - λ 1 to get the existence, non-existence and multiplicity of normalized solutions. Additionally, based on the properties of these solutions, we extend the results obtained in Pierotti et al. in Calc Var Partial Differ Equ 56:1–27, 2017. In contrast of the earlier results, a positive radial solution with arbitrarily large mass can be obtained when N ≥ 3 or if N = 2 and p < 6 . Our paper also includes the demonstration of orbital stability/instability results. [ABSTRACT FROM AUTHOR]

Given a heavy rigid body with one fixed point, we investigate the problem of orbital stability of its periodic motions. Based on the analysis of the linearized system of equations of perturbed motion, the orbital instability of the pendulum rotations is proved. In the case of pendulum oscillations, a transcendental situation occurs, when the question of stability cannot be solved using terms of an arbitrarily high order in the expansion of the Hamiltonian of the equations of perturbed motion. It is proved that the pendulum oscillations are orbitally unstable for most values of the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with solutions to the following nonlinear Schrödinger equation with combined inhomogeneous nonlinearities, - Δ u + λ u = μ | x | - b | u | q - 2 u + | x | - b | u | p - 2 u in R N , under the L 2 -norm constraint ∫ R N | u | 2 d x = c > 0 , where N ≥ 1 , μ = ± 1 , 2 < q < p < 2 (N - b) / (N - 2) + , 0 < b < min { 2 , N } and the parameter λ ∈ R appearing as Lagrange multiplier is unknown. In the mass subcritical case, we establish the compactness of any minimizing sequence to the minimization problem given by the underlying energy functional restricted on the constraint. As a consequence of the compactness of any minimizing sequence, orbital stability of minimizers is derived. In the mass critical and supercritical cases, we investigate the existence, radial symmetry and orbital instability of solutions. Meanwhile, we consider the existence, radial symmetry and algebraical decay of ground states to the corresponding zero mass equation with defocusing perturbation. In addition, dynamical behaviors of solutions to the Cauchy problem for the associated dispersive equation are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlocal double phase problem involving the fractional p-Laplacian (- Δ) p α u + μ u β , q p - q (- Δ) q β u = λ u p - 2 u + f (x , u) x ∈ Ω , u = 0 x ∈ R N \ Ω , where 0 < β < α < 1 , 1 < q ≤ p < N α , μ , λ ∈ R , f ∈ C (Ω × R) . Based on the fractional Gagliardo–Nirenberg inequality, we first give the definition of the L p -mass critical exponent. Then the existence of normalized ground state solutions no matter whether the nonlinearity f is L p -mass subcritical, critical, or supercritical is discussed by using variational methods. In particular, in the supercritical case, we use the eigenvalue of perturbed fractional p-Laplace operator to characterize the conditions for existence of normalized ground states solutions. Finally, we investigate the orbital stability of ground state set of the problem. Our results are new even in the p-Laplacian case. [ABSTRACT FROM AUTHOR]

In this paper, we generalized the results of the following chemotaxis model with the nonlinear degenerate viscosity... by introducing the following general initial perturbation... where is the relative entropy function defined in Eq (2.24). We further employed the relative entropy method by choosing the specific shift function. According to the estimates with the cutoff version, and overcoming the complexity caused by the porous media diffusion, the nonlinear orbital stability of traveling waves was established under small amplitude and general perturbations. [ABSTRACT FROM AUTHOR]

In this article, we consider the global existence and stability issues of the nonlinear Schrödinger equation with partial confinement. First, by establishing some new cross-invariant manifolds and variational problems, a new sharp criterion of global existence is derived in the $ L^{2} $-critical and $ L^{2} $-supercritical cases. Then, the existence of orbitally stable standing waves is obtained in the $ L^{2} $-subcritical and $ L^{2} $-critical cases by taking advantage of the profile decomposition technique. Our work extends and complements some earlier results.

In this work, we show some results concerning the orbital stability of dnoidal wave solutions to some Benjamin-Bona-Mahony equations (BBM equations henceforth). First, by the standard argument, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type. Then, we show that this type of solutions are orbitally stable by perturbations with the same period L. The major tools to obtain these results are the Grillaks, Shatah and Strauss' general theory in the periodic case. The results in the present paper extend some previous stability results for the BBM equations.

The motion of a heavy rigid body with a fixed point in a uniform gravitational field is considered. It is assumed that the principal moments of inertia of the body for the fixed point satisfy the condition of Goryachev–Chaplygin; i.e., they are in the ratio 1 : 4 : 1. In contrast to the integrable case of Goryachev–Chaplygin, no additional restrictions are imposed on the position of the center of mass of the body. The problem of orbital stability of pendulum periodic motions of the body is investigated. In the neighborhood of periodic motions, local variables are introduced and equations of perturbed motion are obtained. On the basis of a linear analysis of stability, the orbital instability of pendulum rotations for all values of the parameters has been proven. It has been established that, depending on the values of the parameters, pendulum oscillations can be both orbitally unstable and orbitally stable in a linear approximation. For pendulum oscillations that are stable in the linear approximation, based on the methods of KAM theory, a nonlinear analysis is performed and rigorous conclusions about the orbital stability are obtained. [ABSTRACT FROM AUTHOR]

We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment. The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential; such a coupling lead to challenging issues compared to the usual nonlinear Schrödinger equations. The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals. We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime. In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter. [ABSTRACT FROM AUTHOR]

This paper studies the stability of traveling wave solutions of the second-order Camassa–Holm equation. Applying the pseudo-conformal transformation, the solutions near the travelling wave solutions of the second-order C–H equation are decomposed: λ 1 2 t u t , y + x t = η t , y + Q y . The uniform boundedness of the difference term is proved by smoothing technique, and the orbital stability of the traveling wave solution of the second-order C–H equation is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two conservative Crank–Nicolson type finite difference schemes and a Chebyshev collocation scheme for the nonlinear Schrödinger equation with a Dirac delta potential in 1D. The key to the proposed methods is to transform the original problem into an interface problem. Different treatments on the interface conditions lead to different discrete schemes and it turns out that a simple discrete approximation of the Dirac potential coincides with one of the conservative finite difference schemes. The optimal H 1 error estimates and the conservative properties of the finite difference schemes are investigated. Both Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time, and the first-order/second-order convergence rates in space, depending on the approximation of the interface condition. Furthermore, the Chebyshev collocation method has been established by the domain-decomposition techniques, and it is numerically verified to be second-order convergent in time and spectrally accurate in space. Numerical examples are provided to support our analysis and study the orbital stability and the motion of the solitary solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of periodic peakons for a generalized-μ-Camassa–Holm (μQCH) equation with quartic nonlinearity. The equation is a μ-version of a generalized Camassa–Holm equation with quartic nonlinearities. First, we derive such equation via variational principle throw a modified Lagrangian, then we show the μQCH equation admits periodic peakons, and finally we prove that the periodic peakons of μQCH equation are orbitally stable under small perturbations in the energy space. [ABSTRACT FROM AUTHOR]

We consider two nonlinear equations, the localized induction equation and the cubic nonlinear Schrodinger equation, and prove that the solvability of certain initial-boundary value problems for each equation is equivalent through the generalized Hasimoto transformation. As an application, we prove the orbital stability of plane wave solutions of the nonlinear Schr\o"dinger equation based on stability estimates obtained for the localized induction equation by the author in a paper in preparation. As far as the author knows, this is the first time that the analysis of the localized induction equation, along with the Hasimoto transformation, has provided new insight for the nonlinear Schr\"odinger equation. [ABSTRACT FROM AUTHOR]

A general method is presented for constructing a nonlinear canonical transformation, which makes it possible to introduce local variables in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. This method can be used for investigating the behavior of the Hamiltonian system in the vicinity of its periodic trajectories. In particular, it can be applied to solve the problem of orbital stability of periodic motions. [ABSTRACT FROM AUTHOR]

We consider a dispersive equation of Schrödinger type with a nonlinearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with Lee-Huang-Yang correction. Mathematically, it is seen to be mass supercritical and energy subcritical with a sign-indefinite nonlinearity. For the corresponding initial value problem, we prove global in-time existence of strong solutions in the energy space. Furthermore, we prove the existence and uniqueness (up to symmetries) of nonlinear ground states and the orbital stability of the set of energy minimizers. We also show that for the corresponding model in 1D a stronger stability result is available. [ABSTRACT FROM AUTHOR]

The generalized modified Camassa-Holm equation possesses well-known peaked solitary waves that are called peakons. Their stability has been established by Z. Guo et al. (J Differ Equ 266:7749–7779, 2019) by using the Constantin-Strauss approach. In this paper, using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm, we prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space. [ABSTRACT FROM AUTHOR]

This paper investigates the orbital stability of periodic standing waves for the following coupled Klein-Gordon-Zakharov equations $ \begin{equation*} \left\{ \begin{aligned} &u_{tt}-u_{xx}+u+\alpha uv+\beta|u|^{2}u = 0, \ &v_{tt}-v_{xx} = (|u|^{2})_{xx}, \end{aligned} \right. \end{equation*} $ where $\alpha>0$ and $\beta$ are two real numbers and $\alpha>\beta$. Under some suitable conditions, we show the existence of a smooth curve positive standing wave solutions of dnoidal type with a fixed fundamental period L for the above equations. Further, we obtain the stability of the dnoidal waves for the coupled Klein-Gordon-Zakharov equations by applying the abstract stability theory and combining the detailed spectral analysis given by using Lam\'{e} equation and Floquet theory. When period $L\rightarrow\infty$, dnoidal type will turn into sech-type in the sense of limit. In such case, we can obtain stability of sech-type standing waves. In particular, $\beta = 0$ is advisable, we still can show the the stability of the dnoidal type and sech-type standing waves for the classical Klein-Gordon-Zakharov equations.

In this paper, we establish the existence and stability properties of odd periodic waves related to the Klein–Gordon type equations, which include the well known ϕ4$$ {\phi}^4 $$ and ϕ6$$ {\phi}^6 $$ models. Existence of periodic waves is determined by using a general planar theory of ODE. The spectral analysis for the corresponding linearized operator is established using the monotonicity of the period map combined with an improvement of the standard Floquet theory. Orbital stability in the odd sector of the energy space is proved using exclusively the monotonicity of the period map. The orbital instability of explicit solutions for the ϕ4$$ {\phi}^4 $$ and ϕ6$$ {\phi}^6 $$ models is presented using the abstract approach of instability for general abstract Hamiltonian systems. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the nonlinear fractional Schrödinger equation. We extend the result of Guo and Huo and prove that the Cauchy problem of the nonlinear fractional Schrödinger equation is global well-posed in H 3 2 − γ (R) with 1 2 ≤ γ < 1 . In view of the complexity of the nonlinear fractional Schrödinger equation itself, the local smoothing effect and maximal function estimates are not enough for presenting the global well-posedness for the nonlinear fractional Schrödinger equation. In this paper, we use a suitably iterative scheme and complete the global well-posed result for Equation (R). Moreover, we obtain the orbital stability of standing waves for the above equations via establishing the profile decomposition of bounded sequences in H s (R N) ( 0 < s < 1 ) with N ≥ 2 . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study nonlinear orbital stability of steady vortex rings without swirl, which are special global solutions of the three-dimensional incompressible Euler equations. We prove the existence of orbitally stable steady vortex rings. The proof is based on the classical variational method. [ABSTRACT FROM AUTHOR]

In this article, we deal with orbital stability and Zhukovskiǐ quasi-stability of periodic or recurrent orbits in an impulsive dynamical system defined in the n-dimensional Euclidean space Rn{{\mathbb{R}}}^{n}. We show that for a periodic orbit of an impulsive system, its asymptotically orbital stability is equivalent to the asymptotically Zhukovskiǐ quasi-stability, and for a recurrent orbit, the orbital stability is equivalent to the Zhukovskiǐ quasi-stability.

Vertical transmission refers to the process in which a mother transmits bacteria or viruses to her offspring through childbirth, and this phenomenon takes place commonly in nature. This paper formulates an SIR epidemic model where the impact of vertical transmission and two state-dependent pulse controls are both taken into consideration. Using the Poincare´map, an analogue of Poincare´ criterion and the method of related qualitative analysis, the existence and the stability of a positive order-1 or order-2 periodic solution for the epidemic model are proved. Furthermore, phase diagrams are demonstrated by means of numerical simulations, illustrating the feasibility and correctness of our main results. It can be further implied that the epidemic can be controlled to a certain extent, with vertical transmission reduced and timely state-dependent pulse controls carried out.