In this work we study the orbital stability/instability in the energy space of a specific family of periodic wave solutions of the general $ϕ^{4n}$-model for all $n\in\mathbb{N}$. This family of periodic solutions are orbiting around the origin in the corresponding phase portrait and, in the standing case, are related (in a proper sense) with the aperiodic Kink solution that connect the states $-\tfrac{1}{2}$ with $\tfrac{1}{2}$. In the traveling case, we prove the orbital instability in the whole energy space for all $n\in\mathbb{N}$, while in the standing case we prove that, under some additional parity assumptions, these solutions are orbitally stable for all $n\in\mathbb{N}$.
In the present paper, we establish the existence and orbital instability results of cnoidal periodic waves for the quintic Klein-Gordon and nonlinear Schrödinger equations. The spectral analysis for the corresponding linearized operator is established by using the Floquet theory. The orbital instability is determined by applying an abstract result due to Shatah and Strauss.
We explain how spectrally stable vortices of the Nonlinear Schrödinger Equation in the plane can be orbitally unstable. This relates to the nonlinear Fermi golden rule, a mechanism which exploits the nonlinear interaction between discrete and continuous modes of the NLS.
The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital (nonlinear) instability in appropriate periodic Sobolev spaces. The analysis is based on the well-posedness theory, the smoothness of the data-solution map, and an abstract result of instability of equilibria under nonlinear iterations. The resulting instability criterion is applied to two families of periodic waves. The first family consists of small amplitude …
In this work we find explicit periodic wave solutions for the classical $ϕ^4$-model, and study their corresponding orbital stability/instability in the energy space. In particular, for this model we find at least four different branches of spatially-periodic wave solutions, which can be written in terms of Jacobi elliptic functions. Two of these branches correspond to superluminal waves, a third-one corresponding to a sub-luminal wave and the remaining one corresponding to a stationary complex-valued wave. In this work we prove the orbital instability of both, real-valued sub-luminal traveling…
We study the long-time behaviour of the focusing cubic NLS on $\R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the ground state to polynomial growth at worst; this is a partial analogue of the $H^1$ orbital stability result of Weinstein. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the ``$I$-method'' from our earlier papers, which pushes down from the energy norm, as well as an ``upside-down $I$-method'' …
We consider the quadratic and cubic KP - I and NLS models in $1+2$ dimensions with periodic boundary conditions. We show that the spatially periodic travelling waves (with period $K$) in the form $u(t,x,y)=\vp(x-c t)$ are spectrally and linearly unstable, when the perturbations are taken to be with the same period. This strong instability implies other instabilities considered recently - for example with respect to perturbations with periods $nK, n=2, 3, ...$ or bounded perturbations.
In this paper, we investigate the instability of one-dimensionally stable periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Zakharov-Kuznetsov equation in two space dimensions. By deriving appropriate asymptotic expansions of the periodic Evans function, we derive an index which yields sufficient conditions for transverse instabilities to occur. This index is geometric in nature, and applies to any periodic traveling wave profile under some minor smoothness assumptions on the nonlinearity. We also desc…
We study the existence and orbital stability/instability of periodic standing wave solutions for the Klein-Gordon-Schrödinger system with Yukawa and cubic interactions. We prove the existence of periodic waves depending on the Jacobian elliptic functions. For one hand, the approach used to obtain the stability results is the classical Grillakis, Shatah and Strauss theory in the periodic context. On the other hand, to show the instability results we employ a general criterium introduced by Grillakis, which get orbital instability from linear instability.
We give a new method to prove the existence, non-existence, multiplicity, orbital stability/instability of standing waves for NLS with partial confinement without the subcritical hypothesis, even in the reduction equation. Using this method, we give an affirmative answer for an open problem proposed by [7, Remark 1.10] where the authors conjectured the existence of more than a normalized solution. We also establish uniqueness results of the ground state solutions depending on the bifurcation parameters. We explain that when the effect of partial confinement is strong, a dimension reduction app…
In this paper, we establish a new criterion for the orbital stability of periodic waves related to a general class of regularized dispersive equations. More specifically, we present sufficient conditions for the stability without knowing the positiveness of the associated hessian matrix. As application of our method, we show the orbital stability for the fifth-order model. The orbital stability of periodic waves resulting from a minimization of a convenient functional is also proved.
Using the Fermi Golden Rule analysis developed in several results by the first author, we prove asymptotic stability of asymmetric nonlinear bound states bifurcating from linear bound states for a quintic nonlinear Schrödinger operator with symmetric potential. This goes in the direction of proving that the approximate periodic solutions of the NLS shadowed in a recent work of the second author with Michael Weinstein do not persist for the full NLS.
A one-dimensional, collisionless plasma given by the Vlasov-Poisson system is considered and the stability properties of periodic steady state solutions known as Bernstein-Greene-Kruskal (BGK) waves are investigated. Sufficient conditions are determined under which BGK waves are linearly unstable under perturbations that share the same period as the equilibria. It is also shown that such solutions cannot support a monotonically decreasing particle distribution function.
We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier--Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic "galloping", or "pulsating", solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier--Stokes equations, in the limit as amplitude (hence also he…
The main goal of this paper is to present orbital stability results of periodic standing waves for the one-dimensional logarithmic Klein-Gordon equation. To do so, we first use compactness arguments and a non-standard analysis to obtain the existence and uniqueness of weak solutions for the associated Cauchy problem in the energy space. Second, we prove the orbital stability of standing waves using a stablity analysis of conservative systems.
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…
New results concerning the orbital stability of periodic traveling wave solutions for the "abcd" Boussinesq model will be shown in this manuscript. For the existence of solutions, we use basic tools of ordinary differential equations to show that the corresponding periodic wave depends on the Jacobi elliptic function of cnoidal type. The spectral analysis for the associated linearized operator is determined by using some tools concerning the Floquet theory. The orbital stability is then established by applying the abstract results [2] and [14] which give us sufficient conditions to the orbital…
In this paper we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic dnoidal waves associat…
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…
We study the local dynamics of $L^{2}\left(\mathbb{R}\right)$-perturbations to the zero solution of spatially $2π$-periodic coefficient reaction-diffusion systems. In this case the spectrum of the linearization about the zero solution is purely essential and may be described via the point spectrum of a one-parameter family of Bloch operators. When this essential spectrum is unstable, we characterize a large class of initial perturbations which lead to nonlinear instability of the trivial solution. This is accomplished by using the Bloch transform to construct an appropriate projection to captu…
This paper establishes suficient conditions for the orbital stability of one-parameter spatially periodic traveling-wave solutions for one-dimensional dispersive equations. Our method of proof combines known techniques with some new ideas. As a consequence of our result, we give several applications for well known dispersive equations. Extension of the theory to regularized equations is also established.
We study the energy cascade problematic for some nonlinear Schrödinger equations on the torus $\T^2$ in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schrödinger nonlinearities that are equal or closely related to the standard polynomial one $|u|^2u$. Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the f…
Transient instability and recurrent aggregation in a two-species chemotaxis-competition system under environmental periodicity
Instituto de Matemática Interdisciplinar, Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid（马德里康普顿斯大学交叉数学研究所分析数学与应用数学系）；Departamento de Matemática Aplicada, ICAI, Universidad Pontificia de Comillas（科米利亚斯宗座大学ICAI学院应用数学系）
In this paper, we prove existence and orbital stability results of periodic standing waves for the cubic-quintic nonlinear Schrödinger equation. We use the implicit function theorem to construct a smooth curve of explicit periodic waves with \textit{dnoidal} profile and such construction can be used to prove that the associated period map is strictly increasing in terms of the energy levels. The monotonicity is also useful to obtain the behaviour of the non-positive spectrum for the associated linearized operator around the wave. Concerning the stability, we prove that the dnoidal waves are or…
Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and the Floquet-Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the period…
We study the stability and instability of periodic traveling waves in the vicinity of the origin in the spectral plane, for equations of Benjamin- Bona-Mahony (BBM) and regularized Boussinesq types permitting nonlocal dispersion. We extend recent results for equations of Korteweg-de Vries type and derive modulational instability indices as functions of the wave number of the underlying wave. We show that a sufficiently small, periodic traveling wave of the BBM equation is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value and a sufficiently…
In this work we study the orbital stability of periodic traveling-wave solutions for dispersive models. The study of traveling waves started in the mid-18th century when John S. Russel established that the flow of water waves in a shallow channel has constant evolution. In recent years, the general strategy to obtain orbital stability consists in proving that the traveling wave in question minimizes a conserved functional restricted to a certain manifold. Although our method can be applied to other models, we deal with the regularized Schamel equation, which contains a fractional nonlinear ter…
In this paper, the existence and orbital stability of the periodic standing waves solutions for the nonlinear fractional Schrodinger (fNLS) equation with cubic nonlinearity is studied. The existence is determined by using a minimizing constrained problem in the complex setting and we it is showed that the corresponding real solution is always positive. The orbital stability is proved by combining some tools regarding positive operators, the oscillation theorem for fractional Hill operators and a Vakhitov-Kolokolov condition, well known for Schrodinger equations. We then perform a numerical app…
We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional "action" on a wide variety of constraints. For a general nonlinearity, we extend to the dimension N=2 the classical instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1…
Working in the context of localized modes in periodic potentials, we consider two systems of the massive Dirac equations in two spatial dimensions. The first system, a generalized massive Thirring model, is derived for the periodic stripe potentials. The second one, a generalized massive Gross--Neveu equation, is derived for the hexagonal potentials. In both cases, we prove analytically that the line solitons suffer from instability with respect to periodic transverse perturbations of large periods. The instability is induced by the spatial translation for the massive Thirring model and by the…
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…
The nonlinear Schroedinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degenera…
We study the equations obtained from linearizing the compressible Navier-Stokes equations around a steady-state profile with a heavier fluid lying above a lighter fluid along a planar interface, i.e. a Rayleigh-Taylor instability. We consider the equations with or without surface tension, with the viscosity allowed to depend on the density, and in both periodic and non-periodic settings. In the presence of viscosity there is no natural variational framework for constructing growing mode solutions to the linearized problem. We develop a general method of studying a family of modified variationa…
We study the periodic non-linear Schrodinger equations with odd integer power nonlinearities, for initial data which are assumed to be small in some negative order Sobolev space, but which may have large L^2 mass. These equations are known to be illposed in H^s for all negative s, in the sense that the solution map fails to be uniformly continuous from H^s to itself, even for short times and small norms. Here we show that these equations are even more unstable. For the cubic equation, the solution map is discontinuous from H^s to even the space of distributions. For the quintic and higher orde…
We study a system of nonlinear Schrödinger equations with cubic interactions in one space dimension. The orbital stability and instability of semitrivial standing wave solutions are studied for both non-degenerate and degenerate cases.
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.
A hyperbolic model for diffusion, nonlinear transport (or advection) and production of a scalar quantity, is considered. The model is based on a constitutive law of Cattaneo-Maxwell type expressing non-Fickian diffusion by means of a relaxation time relation. The production or source term is assumed to be of logistic type. This paper studies the existence and spectral stability properties of spatially periodic traveling wave solutions to this system. It is shown that a family of subcharacteristic periodic waves emerges from a local Hopf bifurcation around a critical value of the wave speed. Th…
We study the nonlinear Schrödinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $α\in \mathbb{R}$. We investigate an orbital instability of the standing waves $e^{iωt}\mathbfΦ(x)$ of NLS-$δ$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile $Φ(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory, avoiding the variational techniques…
A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of periodic solutions are proved. The natural phase space is a particular second order Sobolev space that can be orthogonally split into two subspaces containing, respectively, the longitudinal and the torsional movements of the plate. Sufficient conditions for the stability of periodic solutions and of solutions having only a longitudinal component are given. …
We study strong instability (instability by blowup) of standing wave solutions for a nonlinear Schrödinger equation with an attractive delta potential and $L^2$-supercritical power nonlinearity in one space dimension. We also compare our sufficient condition on strong instability with some known results on orbital instability.
In this paper, we investigate the spectral instability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable struc…
It is shown how to compute the instability rates for the double-periodic solutions to the cubic NLS (nonlinear Schrodinger) equation by using the Lax linear equations. The wave function modulus of the double-periodic solutions is periodic both in space and time coordinates; such solutions generalize the standing waves which have the time-independent and space-periodic wave function modulus. Similar to other waves in the NLS equation, the double-periodic solutions are spectrally unstable and this instability is related to the bands of the Lax spectrum outside the imaginary axis. A simple numeri…
We show that the peaked periodic traveling wave of the reduced Ostrovsky equations with quadratic and cubic nonlinearity is spectrally unstable in the space of square integrable periodic functions with zero mean and the same period. The main novelty is that we discover a new instability phenomenon: the instability of the peaked periodic waves is induced by spectrum of a linearized operator which completely covers a closed vertical strip of the complex plane.
We consider a fifth-order Kadomtsev-Petviashvili equation which arises as a two-dimensional model in the classical water-wave problem. This equation possesses a family of generalized line solitary waves which decay exponentially to periodic waves at infinity. We prove that these solitary waves are transversely spectrally unstable and that this instability is induced by the transverse instability of the periodic tails. We rely upon a detailed spectral analysis of some suitably chosen linear operators.
We study transverse stability and instability of one-dimensional small-amplitude periodic traveling waves of a generalized Kadomtsev-Petviashvili equation with respect to two-dimensional perturbations, which are either periodic or square-integrable in the direction of the propagation of the underlying one-dimensional wave and periodic in the transverse direction. We obtain transverse instability results in KP-fKdV, KP-ILW, and KP-Whitham equations. Moreover, assuming the spectral stability of one-dimensional wave with respect to one-dimensional square-integrable periodic perturbations, we obta…
We consider the periodic standing waves in the derivative nonlinear Schrodinger (DNLS) equation arising in plasma physics. By using a newly developed algebraic method with two eigenvalues, we classify all periodic standing waves in terms of eight eigenvalues of the Kaup-Newell spectral problem located at the end points of the spectral bands outside the real line. The analytical work is complemented with the numerical approximation of the spectral bands, this enables us to fully characterize the modulational instability of the periodic standing waves in the DNLS equation.
