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Asymptotic Stability and classification of multi-solitons for Klein-Gordon equations
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Abstract
Focusing on multi-solitons for the Klein-Gordon equations, in first part of this paper, we establish their conditional asymptotic stability. In the second part of this paper, we classify pure multi-solitons which are solutions converging to multi-solitons in the energy space as . Using Strichartz estimates developed in our earlier work \cite{CJ2} and the modulation techniques, we show that if a solution stays close to the multi-soliton family, then it scatters to the multi-soliton family in the sense that the solution will converge in large time to a superposition of Lorentz-transformed solitons (with slightly modified velocities), and a radiation term which is at main order a free wave. Moreover, we construct a finite-codimension centre-stable manifold around the well-separated multi-soliton family. Finally, given different Lorentz parameters and arbitrary centers, we show that all the pure multi-solitons form a finite-dimension manifold.
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We are interested in solutions of the nonlinear Klein–Gordon equation (NLKG) in R1+d, d≥1, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schrödinger equations, we obtain an N-parameter family of solutions of (NLKG) which converges exponentially fast to a sum of N given (unstable) solitons. For N=1N=1, this family completely describes the set of solutions converging to the soliton considered; for N≥2, we prove uniqueness in a class with explicit algebraic rate of convergence.
We construct multisoliton solutions to the defocusing energy critical wave equation with potentials in based on regular and reversed Strichartz estimates developed in \cite{GC3} for wave equations with charge transfer Hamiltonians. We also show the asymptotic stability of multisoliton solutions. The multisoliton structures with both stable and unstable solitons are covered. Since each soliton decays slowly with rate , the interactions among the solitons are strong. Some reversed Strichartz estimates and local decay estimates for the charge transfer model are established to handle strong interactions.
In this paper, we study the some reversed Strichartz estimates along general time-like trajectories for wave equations in R3. Some applications of the reversed Strichartz estimates and the structure of wave operators to the wave equation with one potential are also discussed. These techniques are useful to analyze the stability problem of traveling solitons.
We study *infinite soliton trains* solutions of nonlinear Schr\"odinger
equations (NLS), i.e. solutions behaving at large time as the sum of infinitely
many solitary waves. Assuming the composing solitons have sufficiently large
relative speeds, we prove the existence and uniqueness of such a soliton train.
We also give a new construction of multi-solitons (i.e. finite trains) and
prove uniqueness in an exponentially small neighborhood, and we consider the
case of solutions composed of several solitons and kinks (i.e. solutions with a
non-zero background at infinity).
In this paper we consider the existence of multi-soliton structures for the
nonlinear Klein-Gordon equation (NLKG) in R^{1+d}. We prove that, independently
of the unstable character of (NLKG) solitons, it is possible to construct a
N-soliton family of solutions to (NLKG), of dimension 2N, globally well-defined
in the energy space H^1 \times L^2 for all large positive times. The method of
proof involves the generalization of previous works on supercritical NLS and
gKdV equations by Martel, Merle and the first author to the wave case, where we
replace the unstable mode associated to the linear NLKG operator by two
generalized directions that are controlled without appealing to modulation
theory. As a byproduct, we generalize the linear theory described in
Grillakis-Shatah-Strauss and Duyckaerts-Merle to the case of boosted solitons,
and provide new solutions to be studied using the recent Nakanishi- Schlag
theory.
if they start at timet = 0 near the manifold of ground states, under some restrictive spectral conditions, as t ! +1 look like a sum of a solitary wave plus radiation, with the latter dispersing at the same rate as the solutions to the linear constant coecients Schrodinger equation. More specically our result is an extension to n 3 of the result stated in (BP) for n = 1. It is important to emphasize that here we do not try to show that our hypotheses are satised by any particular example. To introduce our result, we recall that under certain conditions on (juj 2 )w hich we review later, and if !> 0, there exists a unique positive, radially symmetric and exponentially decreasing time independent solution !(x )o f
We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig and Merle for the H1 critical wave and Schrödinger equations. Our result includes the H1 critical case, where the threshold is given by the ground state for the massless equation, and the 2D square-exponential case, where the mass for the ground state may be modified, depending on the constant in the sharp Trudinger-Moser inequality. The main difficulty is the lack of scaling invariance in both the linear and the nonlinear terms.
We consider a class of nonlinear Schrödinger equations (conservative and dispersive systems) with localized and dispersive solutions. We obtain a class of initial conditions, for which the asymptotic behavior (t→±∞) of solutions is given by a linear combination of nonlinear bound state (time periodic and spatially localized solution) of the equation and a purely dispersive part (decaying to zero with time at the free dispersion rate). We also obtain a result ofasymptotic stability type: given data near a nonlinear bound state of the system, there is a nonlinear bound state of nearby energy and phase, such that the difference between the solution (adjusted by a phase) and the latter disperses to zero. It turns out that in general, the time-period (and energy) of the localized part is different fort→+∞ from that fort→−∞. Moreover the solution acquires an extra constant asymptotic phasee
iy
±.
The focusing nonlinear Schrodinger equation possesses special non-dispersive solitary type solutions, solitons. Under certain spectral assumptions we show existence and asymptotic stability of solutions with the asymptoic profile (as time goes to infinity) of a linear combination of N non-colliding solitons.
In this paper, we study some properties of multi-solitons for the non-linear Schrödinger equations in Rd with general non-linearities. Multi-solitons have already been constructed in H1(Rd) in papers by Merle (1990), Martel and Merle (2006), and Côte, Martel and Merle (2011). We show here that multi-solitons are smooth, depending on the regularity of the non-linearity. We obtain also a result of uniqueness in some class, either when the ground states are all stable, or in the mass-critical case.
We give a sufficient condition for the existence of an exponential dichotomy for a general linear dynamical system (not necessarily invertible) in a Banach space, in discrete or continuous time. We provide applications to the backward heat equation with a potential varying in time, and to the heat equation with a finite number of slowly moving potentials. We also consider the Klein–Gordon equation with a finite number of potentials whose centres move at sublight speed with small accelerations.
For the nonlinear Klein-Gordon equation in R 1+d , we prove the existence of multi-solitary waves made of any number N of decoupled bound states. This extends the work of C{\^o}te and Mu{\~n}oz (Forum Math. Sigma 2 (2014)) which was restricted to ground states, as were most previous similar results for other nonlinear dispersive and wave models.
We investigate the real and complex interpolation of L p -spaces and Lorentz spaces over a measure space. In particular, we prove a generalized version of the Marcinkiewicz theorem (the Calderón-Marcinkiewicz theorem). We also investigate the real and the complex interpolation spaces between L p -spaces with different measures, thus extending a theorem by Stein and Weiss. In Section 6, we consider the interpolation of vector-valued L p -spaces of sequences, thus preparing for the interpolation of Besov spaces in the next chapter.
Preface.- 1. Basic analysis.- 2. Littlewood-Paley theory.- 3. Transport and transport-diffusion equations.- 4. Quasilinear symmetric systems.- 5. Incompressible Navier-Stokes system.- 6. Anisotropic viscosity.- 7. Euler system for perfect incompressible fluids.- 8. Strichartz estimates and applications to semilinear dispersive equations.- 9. Smoothing effect in quasilinear wave equations.- 10.- The compressible Navier-Stokes system.- References. - List of notations.- Index.
We prove in this paper the stability and asymptotic stability in H^1 of
a decoupled sum of N solitons for the subcritical generalized KdV
equations (1<p<5). The proof of the
stability result is based on energy arguments and monotonicity of local
L^2 norm. Note that the result is new even for p=2 (the KdV equation).
The asymptotic stability result then follows directly from a rigidity
theorem in [15].
Consider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity
in R^3, i \partial_t \psi + \Delta \psi + |\psi|^2 \psi = 0.
It admits an eight-dimensional manifold of periodic solutions called solitons
e^{i(\Gamma + vx - t|v|^2 + \alpha^2 t)} \phi(x-2tv-D, \alpha), where \phi(x,
\alpha) is a positive ground state solution of the semilinear elliptic equation
-\Delta \phi + \alpha^2\phi = \phi^3.
We prove that in the neighborhood of the soliton manifold there exists a
H^{1/2} real analytic manifold N of asymptotically stable solutions of the
Schroedinger equation, meaning they are the sum of a moving soliton and a
dispersive term. Furthermore, a solution starting on N remains on N for all
positive time and for some finite negative time and N can be identified as the
centre-stable manifold for this equation.
The proof is based on the method of modulation, introduced by Soffer and
Weinstein and adapted by Schlag to the L^2-supercritical case. Novel elements
include a different linearization and a Strichartz estimate for the
time-dependent linear Schroedinger equation. The main result depends on a
spectral assumption concerning the absence of embedded eigenvalues.
We consider the generalized Korteweg–de Vries (gKdV) equations
in the subcritical cases p = 2, 3 or 4. The first objective of this paper is to present a direct, simplified proof of the asymptotic stability of solitons in the energy space H¹, which was proved by the same authors in Martel and Merle (2001 Arch. Ration. Mech. Anal. 157 219–54).
Then, in the case of the KdV equation, we show the optimality of the result by constructing a solution which behaves asymptotically as a soliton located on a curve which is a logarithmic perturbation of a line. This example justifies the apparently weak control on the location of the soliton in the asymptotic stability result.
We consider the generalized Korteweg-de Vries equations u(t) + (u(xx) + u(P))(x) = 0, t,x is an element of R, in the subcritical and critical cases p = 2,3,4 or 5. Let R-j(t,x) = Qc(j)(x - c(j)t - x(j)), where j is an element of {1,..., N}, be N soliton solutions of this equation, with corresponding speeds 0 < c(1) < c(2) <...< c(N). In this paper, we construct a solution u(t) of the generalized Korteweg-de Vries equation such that N [GRAPHIC] This solution behaves asymptotically as t -> +infinity as the sum of N solitons without loss of mass by dispersion. This is an exceptional behavior, indeed, being given the parameters {cj}1 <= j <= N, {xj} 1 <= j <= N, we prove uniqueness of such a solution. In the integrable cases p = 2 and 3, such solutions are explicitly known and their properties were extensively studied in the literature (they are called N-soliton solutions). Therefore, the existence result is new only for the nonintegrable cases. The uniqueness result is new for all cases.
The wave equation ∂ttψ − Δψ − ψ5 = 0 in ℝ3 is known to exhibit finite time blowup for data of negative energy. Furthermore, it admits the special static solutions φ(x, a) = (3a) ¼ (1 + a|x|2)−½ for all a > 0 which are linearly unstable. We view these functions as a curve in the energy space ˙H1 ×L2. We prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well-defined long time asymptotic behavior as the sum of a bulk term plus a scattering term. Moreover, this family forms a co-dimension one manifold M of small diameter in a suitable topology. Loosely speaking, M acts as a center-stable manifold with the curve Φ(·, a) as an attractor in M.
On presente une nouvelle demonstration de la stabilite orbitale des solitons d'etat fondamental de l'equation de Schrodinger non lineaire pour une grande classe de non-linearites. On demontre la stabilite de l'onde solitaire pour l'equation de Korteweg-de Vries generalisee
We extend our result Nakanishi and Schlag in J. Differ. Equ. 250(5):2299–2333, 2011) to the non-radial case, giving a complete classification of global dynamics of all solutions with energy that is at most
slightly above that of the ground state for the nonlinear Klein–Gordon equation with the focusing cubic nonlinearity in three
space dimensions.
We study the focusing, cubic, nonlinear Klein-Gordon equation in 3D with large radial data in the energy space. This equation admits a unique positive stationary solution, called the ground state. In 1975, Payne and Sattinger showed that solutions with energy strictly below that of the ground state are divided into two classes, depending on a suitable functional: If it is negative, then one has finite time blowup, if it is nonnegative, global existence; moreover, these sets are invariant under the flow. Recently, Ibrahim, Masmoudi and the first author improved this result by establishing scattering to zero in the global existence case by means of a variant of the Kenig-Merle method. In this paper we go slightly beyond the ground state energy and give a complete description of the evolution. For example, in a small neighborhood of the ground states one encounters the following trichotomy: on one side of a center-stable manifold one has finite-time blowup, on the other side scattering to zero, and on the manifold itself one has scattering to the ground state, all for positive time. In total, the class of initial data is divided into nine disjoint nonempty sets, each displaying different asymptotic behavior, which includes solutions blowing up in one time direction and scattering to zero on the other, and also, the analogue of those found by Duyckaerts and Merle for the energy critical wave and Schr\"odinger equations, exactly with the ground state energy. The main technical ingredient is a "one-pass" theorem which excludes the existence of "almost homoclinic" orbits between the ground states.
The nonlinear scattering and stability results of Soffer and Weinstein (Comm. Math. Phys. (1990)) are extended to the case of anisotropic potentials and data. The range of nonlinearities for which the theory is shown to be valid is also extended Considerably.
Standing wave solutions of the one-dimensional nonlinear Schrödinger equation ∂ t ψ+∂ x 2 ψ=-|ψ| 2σ ψ with σ>2 are well known to be unstable. We show that asymptotic stability can be achieved provided the perturbations of these standing waves are small and chosen to belong to a codimension one Lipschitz surface. Thus, we construct codimension one asymptotically stable manifolds for all supercritical NLS in one dimension. The considerably more difficult L 2 -critical case, for which one wishes to understand the conditional stability of the pseudo-conformal blow-up solutions, is studied in the authors’ companion paper [J. Eur. Math. Soc. (JEMS) 11, No. 1, 1–125 (2009; Zbl 1163.35035)].
We extend our previous result on the nonlinear Klein-Gordon equation to the
nonlinear Schrodinger equation with the focusing cubic nonlinearity in three
dimensions, for radial data of energy at most slightly above that of the ground
state. We prove that the initial data set splits into nine nonempty, pairwise
disjoint regions which are characterized by the distinct behaviors of the
solution for large time: blow-up, scattering to 0, or scattering to the family
of ground states generated by the phase and scaling freedom. Solutions of this
latter type form a smooth center-stable manifold, which contains the ground
states and separates the phase space locally into two connected regions
exhibiting blow-up and scattering to 0, respectively. The special solutions
found by Duyckaerts and Roudenko appear here as the unique one-dimensional
unstable/stable manifolds emanating from the ground states. In analogy with the
Klein-Gordon case, the proof combines the hyperbolic dynamics near the ground
states with the variational structure away from them. The main technical
ingredient in the proof is a "one-pass" theorem which precludes "almost
homoclinic orbits", i.e., those solutions starting in, then moving away from,
and finally returning to, a small neighborhood of the ground states. The main
new difficulty compared with the Klein-Gordon case is the lack of finite
propagation speed. We need the radial Sobolev inequality for the error estimate
in the virial argument. Another major difference from the Klein-Gordon case is
the need to control two modulation parameters.
For the L ² subcritical and critical (gKdV) equations, Martel [11 Martel , Y. ( 2005 ). Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations . Amer. J. Math. 127 ( 5 ): 1103 – 1140 . [Google Scholar]] proved the existence and uniqueness of multi-solitons. Recall that for any N given solitons, we call multi-soliton a solution of (gKdV) which behaves as the sum of these N solitons asymptotically as t → + ∞. More recently, for the L ² supercritical case, C
te et al. [4 Côte , R. , Martel , Y. , Merle , F. Construction of multi-soliton solutions for the L ²-supercritical gKdV and NLS equations . To appear in Revista Matematica Iberoamericana . ] proved the existence of at least one multi-soliton. In the present paper, as suggested by a previous work concerning the one soliton case [3 Combet , V. ( 2010 ). Construction and characterization of solutions converging to solitons for supercritical gKdV equations . Diff. Int. Eqs. 23 : 513 – 568 . [Google Scholar]], we first construct an N-parameter family of multi-solitons for the supercritical (gKdV) equation, for N arbitrarily given solitons, and then prove that any multi-soliton belongs to this family. In other words, we obtain a complete classification of multi-solitons for (gKdV).
We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, In particular, we study the case wheref(u)=u
p+1/(p+1),p=1, 2, 3 (and 3<p<4, foru>0, withf∈C
4). The same asymptotic stability result for KdV is also proved for the casep=2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4.) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(t) (bound state part), and an infinite dimensional perturbation (radiating part). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. Asp→4−, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations.
Let H be a selfadjoint operator and A a closed operator on a Hilbert
space . If A is H-(super)smooth in the sense of Kato-Yajima,
we prove that is -(super)smooth. This allows to
include wave and Klein-Gordon equations in the abstract theory at the same
level of generality as Schr\"{o}dinger equations.
We give a few applications and in particular, based on the resolvent
estimates of Erdogan, Goldberg and Schlag \cite{ErdoganGoldbergSchlag09-a}, we
prove Strichartz estimates for wave equations perturbed with large magnetic
potentials on , .
We prove dispersive estimates for the time-dependent Schrödinger equation with a charge transfer Hamiltonian. As a by-product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by K. Yajima [Commun. Math. Phys. 75, 153–178 (1980; Zbl 0437.47008)] and G. M. Graf [Helv. Phys. Acta 63, No. 1-2, 107–138 (1990; Zbl 0741.35050)]. We also consider a more general matrix non-self-adjoint charge transfer problem. This model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear Schrödinger equation.
We consider the Cauchy problem for the nonlinear Schroedinger eqiation with initial data close to a sum of N decoupled solitons. Under some suitable assumptions on the spectral structure of the one soliton linearizations we prove that for large time the asymptotics of the solution is given by a sum of solitons with slightly modified parameters and a small dispersive term.
The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, \emph{solitary wave} or \emph{soliton} solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions is relatively straightforward, but the fact that they are \emph{stable} requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the \emph{critical} case in which the nonlinearity is at exactly the right power to potentially allow for a self-similar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and Grillakis-Shatah-Strauss, to the more recent asymptotic stability and blowup analysis of Martel-Merle and Merle-Raphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations.
- G Chen
Chen, G. Strichartz estimates for wave equations with charge transfer Hamiltonians, arXiv:1610.05226
(2016). Mem. Amer. Math. Soc. 273 (2021), no. 1339, v + 84 pp.
Strichrtz estimates for Klein-Gordon equations with moving potentials
- G Chen
- J Jendrej
Chen, G.; Jendrej, J. Strichrtz estimates for Klein-Gordon equations with moving potentials. Preprint 2022.
Nonlinear dispersive equations. Local and global analysis
- T Tao
Tao, T. Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in
Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by
the American Mathematical Society, Providence, RI, 2006. xvi+373 pp.