We consider the generalized Choquard equation describing trapped electron gas in 3 dimensional case. The study of orbital stability of the energy minimizers (known as ground states) depends essentially in the local uniqueness of these minimizers. In equivalent way one can optimize the Gagliardo--Nirenberg inequality subject to the constraint fixing the $L^2$ norm. The uniqueness of the minimizers for the case $p=2$, i.e. for the case of Hartree--Choquard is well known. The main difficulty for the case $p > 2$ is connected with possible lack of control on the $L^p$ norm of the minimizers.
We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system proving the existence of solutions in which the Dirac wave function is of the form $ϕ(x,ω)e^{-iωt}$, $ω\in(-m,ω_*)$, with some $ω_*>-m$, such that $ϕ_ω\in H^1(\mathbb{R}^3,\mathbb{C}^4)$, $\Vertϕ_ω\Vert^2_{L^2}=O(m-|ω|)$, and $\Vertϕ_ω\Vert_{L^\infty}=O(m-|ω|)$. The method of proof is an implicit function theorem argument based on an identification of the nonrelativistic limit as the ground state of the Choquard equation.
In this paper, we consider the upper critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -Δu=λu+(I_α\ast|u|^{p})|u|^{p-2}u+μ|u|^{q-2}u,\ x\in \mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N),\ \int_{\mathbb{R}^N}|u|^2=a, \end{cases} \end{equation*} where $N\geq 3$, $μ>0$, $a>0$, $λ\in \mathbb{R}$, $α\in (0,N)$, $p=\bar{p}:=\frac{N+α}{N-2}$, $q\in (2,2+\frac{4}{N})$ and $I_α=\frac{C}{|x|^{N-α}}$ with $C>0$. When $μa^{\frac{q(1-γ_q)}{2}}\leq (2K)^{\frac{qγ_q-2\bar{p}}{2(\bar{p}-1)}}$ with $γ_q=\frac{N}{2}-\frac{N}{q}$ and $K$ being some positive constant, we prove (1) Ex…
In this paper, we show the orbital stability of solitons arising in the cubic derivative nonlinear Schrodinger equations. We consider the zero mass case that is not covered by earlier works [8, 3]. As this case enjoys L^2 scaling invariance, we expect the orbital stability in the sense up to scaling symmetry, in addition to spatial and phase translations. For the proof, we are based on the variational argument and extend a similar argument in [21]. Moreover, we also show a self-similar type blow up criteria of solutions with the critical mass 4π.
The Degasperis-Procesi equation is the integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the Desgasperis-Procesi (DP) equation on the real line. %extending our previous work on their spectral stability \cite{LLW}. The main difficulty stems from the fact that the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the $L^2$-norm, which by itself can not bound the higher-order nonlinear terms in th…
We consider the following generalized derivative nonlinear Schrödinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2σ}\partial_xu=0,\ (t,x)\in\mathbb R\times\mathbb R \end{equation*} when $σ\in(0,1)$. The equation has a two-parameter family of solitary waves $$u_{ω,c}(t,x)=Φ_{ω,c}(x)e^{iωt+\frac{ic}2x-\frac i{2σ+2}\int_0^xΦ_{ω,c}(y)^{2σ}dy},$$ with $(ω,c)$ satisfying $ω>c^2/4$, or $ω=c^2/4$ and $c>0$. The stability theory in the frequency region $ω>c^2/4$ was studied previously. In this paper, we prove the stability of the solitary wave solutions in the endpoint case $ω=c^2/4$ …
We study general semilinear scalar-field equations on the real line with variable coefficients in the linear terms. These coefficients are uniformly small, but slowly decaying, perturbations of a constant-coefficient operator. We are motivated by the question of how these perturbations of the equation may change the stability properties of kink solutions (one-dimensional topological solitons). We prove existence of a stationary kink solution in our setting, and perform a detailed spectral analysis of the corresponding linearized operator, based on perturbing the linearized operator around the …
In this paper we consider a three-component system of one dimensional long wave-short wave interaction equations. The system has two-parameter family of solitary wave solutions. We prove orbital stability of the solitary wave solutions using variational methods.
We obtain explicit characterization of spectral and orbital stability of solitary wave solutions to the $\mathbf{U}(1)$-invariant Klein--Gordon equation in one spatial dimension coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave.
We study the orbital instability of solitary waves for a generalized derivative nonlinear Schrödinger equation. We give sufficient conditions for instability of a two-parameter family of solitary waves in a degenerate case.
For the cubic Schrödinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We provide a variational characterization of such solutions, which gives information on the stability through of a condition of Grillakis-Shatah-Strauss type. As an application, we show existence of conditionally orbitally stable solitary waves when: a) the masses are small, for almost every scattering lengths, and b) in the defocusing, weakly interacting case, for any…
This work studies the rotation-generalized Benjamin-Ono equation which is derived from the theory of weakly nonlinear long surface and internal waves in deep water under the presence of rotation. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.
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 consi…
In this paper, we study the stability of smooth solitary waves for the $b$-family of Camassa-Holm equations. We verify the stability criterion analytically for the general case $b>1$ by the idea of the monotonicity of the period function for planar Hamiltonian systems and show that the smooth solitary waves are orbitally stable, which gives a positive answer to the open problem proposed by Lafortune and Pelinovsky [S. Lafortune, D. E. Pelinovsky, Stability of smooth solitary waves in the $b$-Camassa-Holm equation].
In this paper, we consider the stability for line solitary waves of the two dimensional Zakharov-Kuznetsov equation on $\mathbb{R}\times\mathbb{T}_L$ which is one of a high dimensional generalization of Korteweg-de Vries equation , where $\mathbb{T}_L$ is the torus with the period $2πL$. The orbital and asymptotic stability of the one soliton of Korteweg-de Vries equation on the energy space has been proved by Benjamin, Pego and Weinstein and Martel and Merle. We regard the one soliton of Korteweg-de Vries equation as a line solitary wave of Zakharov-Kuznetsov equation on $\mathbb{R}\times\mat…
In this paper, we consider the Fornberg-Whitham equation and a family of solitary wave solutions is found by using minimization principle, where a related penalization function and the concentration-compactness lemma play a key role in our proof. Besides, we also prove that the family of solitary solutions is orbital stable and decay exponentially when speed wave c is bigger than 1.
In this paper, we consider capillary-gravity waves propagating on the interface separating two fluids of finite depth and constant density. The flow in each layer is assumed to be incompressible and of constant vorticity. We prove the existence of small-amplitude solitary wave solutions to this system in the strong surface tension regime via a spatial dynamics approach. We then use a variant of the classical Grillakis--Shatah--Strauss (GSS) method to study the orbital stability/instability of these waves. We find an explicit function of the parameters (Froude number, Bond number, and the depth…
Considered herein is the integrable two-component Camassa-Holm shallow water system derived in the context of shallow water theory, which admits blow-up solutions and the solitary waves interacting like solitons. Using modulation theory, and combining the almost monotonicity of a local version of energy with the argument on the stability of a single solitary wave, we prove that the train of $N$ solitary waves, which are sufficiently decoupled, is orbitally stable in the energy space $H^1(\R)\times L^2(\R)$.
In this paper, we construct center stable manifolds of unstable line solitary waves for the Zakharov--Kuznetsov equation on $\mathbb{R}\times \mathbb{T}_L$ and show the orbital stability of the unstable line solitary waves on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. The construction is based on the graph transform approach by Nakanishi and Schlag. Applying the bilinear estimate on Fourier restriction spaces by Molinet and Pilod and modifying the mobile distance by Nakanishi a…
This work is devoted to study the dynamics of the supercritical gKDV equations near solitary waves in the energy space $H^1$. We construct smooth local center-stable, center-unstable and center manifolds near the manifold of solitary waves and give a detailed description of the local dynamics near solitary waves. In particular, the instability is characterized as following: any forward flow not starting from the center-stable manifold will leave a neighborhood of the manifold of solitary waves exponentially fast. Moreover, orbital stability is proved on the center manifold, which implies the u…
In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schrödinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist and converge to zero as the velocity tends to $1$. Finally, we solve the Cauchy problem for initial data in $L^{2}_{x}H^{s}_{y}(\mathbb{R}^{2})$, with $s>\frac{1}{2}$.
In this work, we consider the following generalized Boussinesq equation \begin{align*} \partial_{t}^2u-\partial_{x}^2u+\partial_{x}^2(\partial_{x}^2u+|u|^{p}u)=0,\qquad (t,x)\in\mathbb R\times \mathbb R, \end{align*} with $0<p<\infty$. This equation has the traveling wave solutions $ϕ_ω(x-ωt)$, with the frequency $ω\in (-1,1)$ and $ϕ_ω$ satisfying \begin{align*} -\partial_{xx}ϕ_ω+(1-{ω^2})ϕ_ω-ϕ_ω^{p+1}=0. \end{align*} Bona and Sachs (1988) proved that the traveling wave $ϕ_ω(x-ωt)$ is orbitally stable when $0<p<4,$ $\frac p4<ω^2<1$. Liu (1993) proved the orbital instability under the condition…
In this paper we establish a rigorous spectral stability analysis for solitary waves associated to a generalized fractional Benjamin-Bona-Mahony type equation. Besides the well known smooth and positive solitary wave with large wave speed, we present the existence of smooth negative solitary waves having small wave speed. The spectral stability is then determined by analysing the behaviour of the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum at the solitary wave. Since the analytical solution is not known, we generate t…
We consider the KP-I and gKP-I equations in $\mathbb{R}\times (\mathbb{R}/2π\mathbb{Z})$. We prove that the KdV soliton with subcritical speed $0<c<c^*$ is orbitally stable under the global KP-I flow constructed by Ionescu and Kenig \cite{IK}. For supercritical speeds $c>c^*$, in the spirit of the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation.
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 \cite{st}. 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 ent…
This paper considers the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is vacuum, and that the waves are acted upon by gravity with surface tension effects on the air--water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted $δ$-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of $0$. Our mai…
We consider the logarithmic Korteweg-de Vries (log-KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in $H^1(\mathbb{R})$ with conserved $L^2$ norm and energy, we construct a weak global solution of the log-KdV equation in a subset of $H^1(\mathbb{R})$. This construction yields conditional orbital stability of Gaussian solitary waves of the log-KdV equation, provided uniqueness and continuous dependence of the constructed solution holds. Furthermore…
J. Angulo and J. F. Montenegro (J. Differential Equations 174 (2001), no. 1, 181-199) published a paper about nonlinear stability of solitary waves for an interaction system between a long internal wave and a short surface wave in a two layer fluid considering that the fluid depth of the lower layer is sufficiently large in comparison with the wavelength of the internal wave. In this note, we show that in a critical step during the proof of Lemma 2.4 in the above mentioned paper, there is a claim used by the authors which fails to be true. Lemma 2.4 is crucial for the proof of Lemma 2.7, and f…
We consider a derivative nonlinear Schrödinger 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, their velocity. We illustrate these results with numerical simulations.
The aim of this paper is to provide a proof of the (conditional) orbital stability of solitary waves solutions to the fractional Korteweg- de Vries equation (fKdV) and to the fractional Benjamin-Bona-Mahony (fBBM) equation in the $L^2$ subcritical case. We also discuss instability and its possible scenarios.
We derive the precise stability criterion for smooth solitary waves in the b-family of Camassa-Holm equations. The smooth solitary waves exist on the constant background. In the integrable cases b = 2 and b = 3, we show analytically that the stability criterion is satisfied and smooth solitary waves are orbitally stable with respect to perturbations in $H^3(\mathbb{R})$. In the non-integrable cases, we show numerically and asymptotically that the stability criterion is satisfied for every b > 1. The orbital stability theory relies on a different Hamiltonian formulation compared to the Hamilton…
We study here the existence of solitary wave solutions of a generalized two-component Camassa-Holm system. In addition to those smooth solitary-wave solutions, we show that there are solitary waves with singularities: peaked and cusped solitary waves. We also demonstrate that all smooth solitary waves are orbitally stable in the energy space. We finally give a sufficient condition for global strong solutions to the equation without certain parameters.
In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schrödinger-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 Schrödinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.
We study the existence and stability of standing waves solutions of a three-coupled nonlinear Schrödinger system related to the Raman amplification in a plasma. By means of the concentration-compacteness method, we provide a characterization of the standing waves solutions as minimizers of an energy functional subject to three independent $L^{2}$ mass constraints. As a consequence, we establish existence and orbital stability of solitary waves.
We study asymptotic stability of solitary wave solutions in the one-dimensional Benney-Luke equation, a formally valid approximation for describing two-way water wave propagation. For this equation, as for the full water wave problem, the classic variational method for proving orbital stability of solitary waves fails dramatically due to the fact that the second variation of the energy-momentum functional is infinitely indefinite. We establish nonlinear stability in energy norm under the spectral stability hypothesis that the linearization admits no non-zero eigenvalues of non-negative real pa…
In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work \cite{carles}, in which the authors established the well-posedness and the linear stability of Gaussian solitary waves. By using the approach put forward recently in \cite{natali1} to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator, we apply the abstract theories in \cite{grillakis1} and \cite{weinstein1} to deduce the orbital stability of the periodic t…
This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small-amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite-dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss, but with hypotheses that are relaxed in a number of ways …
The Camassa-Holm equation possesses well-known peaked solitary waves that are called peakons. Their orbital stability has been established by Constantin and Strauss (2000). We prove here the stability of ordered trains of peakons. We also establish a result on the stability of multipeakons.
