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Description and Classification of 2-Solitary Waves for Nonlinear Damped Klein–Gordon Equations

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Abstract

We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein–Gordon equation ∂ttu+2α∂tu-Δu+u-|u|p-1u=0on RN, for 1⩽N⩽5 and energy subcritical exponents p>2. The description is twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we construct and classify the full family of 2-solitary waves in the case of opposite signs. Close to the sum of two remote solitary waves, it turns out that only the components of the initial data in the unstable direction of each ground state are relevant in the large time asymptotic behavior of the solution. In particular, we show that 2-solitary waves have a universal behavior: the distance between the solitary waves is asymptotic to logt as t→∞. This behavior is due to damping of the initial data combined with strong interactions between the solitary waves.
Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-021-04241-5
Commun. Math. Phys. 388, 1557–1601 (2021) Communications in
Mathematical
Physics
Description and Classification of 2-Solitary Waves for
Nonlinear Damped Klein–Gordon Equations
Raphaël Côte1, Yvan Martel2, Xu Yuan2,3, Lifeng Zhao4
1IRMA UMR 7501, Université de Strasbourg, CNRS, 67000 Strasbourg, France.
E-mail: cote@math.unistra.fr
2CMLS, École polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France.
E-mail: yvan.martel@polytechnique.edu
3Present address: Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong
Kong.
E-mail: xu.yuan@cuhk.edu.hk
4School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui,
China. E-mail: zhaolf@ustc.edu.cn
Received: 24 March 2021 / Accepted: 30 September 2021
Published online: 22 October 2021 – © The Author(s), under exclusive licence to Springer-Verlag GmbH
Germany, part of Springer Nature 2021
Abstract: We describe completely 2-solitary waves related to the ground state of the
nonlinear damped Klein–Gordon equation
ttu+2α∂tuu+u−|u|p1u=0
on RN,for1N5 and energy subcritical exponents p>2. The description is
twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we
construct and classify the full family of 2-solitary waves in the case of opposite signs.
Close to the sum of two remote solitary waves, it turns out that only the components
of the initial data in the unstable direction of each ground state are relevant in the large
time asymptotic behavior of the solution. In particular, we show that 2-solitary waves
have a universal behavior: the distance between the solitary waves is asymptotic to logt
as t→∞. This behavior is due to damping of the initial data combined with strong
interactions between the solitary waves.
1. Introduction
1.1. Setting of the problem. We consider the nonlinear focusing damped Klein–Gordon
equation
ttu+2α∂tuu+uf(u)=0(t,x)R×RN,(1.1)
where f(u)=|u|p1u,α>0, 1 N5, and the exponent pcorresponds to the
energy sub-critical case, i.e.
2<p<for N=1,2 and 2 <p<N+2
N2for N=3,4,5.(1.2)
R. C. was partially supported by the French ANR contract MAToS ANR-14-CE25-0009-01. Y. M. and
X. Y. thank IRMA, Université de Strasbourg, for its hospitality. L. Z. thanks CMLS, École Polytechnique, for
its hospitality. L. Z. was partially supported by the NSFC Grant of China (11771415).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... Both statements generalize and precise results from [13], [14] and are based on the analysis developed in [7,6]. 10] for details and results related to this conjecture for the (undamped) energy critical wave equation. ...
... Our objective is to present additional examples of multi-solitary waves and describe precisely the asymptotics, in the context of configurations with symmetry. We also prove that multi-solitons necessarily have solitons of opposite signs: this generalize a statement of [7] for 2-solitons to the case of any number of solitons. ...
... If more freedom is allowed, one is required to understand a unstable system of ODE under perturbations, which can be untractable. The case of solitons on a line for example in R d , d 2, is not a immediate extension of the case d = 1 (treated in [7]), and will the object of future work. We devoted some effort to abstract the hypothesis needed on the symmetry group G to be amenable to the analysis, keeping in mind we were interested mainly in configuration on regular polytopes as stated in Corollary 1.5. ...
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We are interested in the nonlinear damped Klein-Gordon equation t2u+2αtuΔu+uup1u=0 \partial_t^2 u+2\alpha \partial_t u-\Delta u+u-|u|^{p-1}u=0 on Rd\mathbb{R}^d for 2d52\le d\le 5 and energy sub-critical exponents 2<p<d+2d22 < p < \frac{d+2}{d-2}. We construct multi-solitons, that is, solutions which behave for large times as a sum of decoupled solitons, in various configurations with symmetry: this includes multi-solitons whose soliton centers lie at the vertices of an expanding regular polygon (with or without a center), of a regular polyhedron (with a center), or of a higher dimensional regular polytope. We give a precise description of these multi-solitons: in particular the interaction between nearest neighbour solitons is asymptotic to ln(t)\ln (t) as t+t \to +\infty. We also prove that in any multi-soliton, the solitons can not all share the same sign. Both statements generalize and precise results from \cite{F98}, \cite{Nak} and are based on the analysis developed in \cite{CMYZ,CMY}.
... More recently, Burq, Raugel and Schlag [3] proved the soliton resolution for (1.1) in the full limit t → ∞ for all radial solutions. On the other hand, Côte, Martel, Yuan and Zhao [8] constructed a Lipschitz manifold in the energy space with codimension 2 of those solutions asymptotic to a sum of two ground states ...
... The goal of this paper is to obtain a similar classification around the 2-solitons (1.11), exploiting the damping, and investigating all the solutions starting close to but not on the manifold in [8]. The main interest is in the transition from 2-solitons to 1-solitons, both by time evolution and by initial perturbation. ...
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Global behavior of solutions is studied for the nonlinear Klein-Gordon equation with a focusing power nonlinearity and a damping term in the energy space on the Euclidean space. We give a complete classification of solutions into 5 types of global behavior for all initial data in a small neighborhood of each superposition of two ground states (2-solitons) with the opposite signs and sufficient spatial distance. The neighborhood contains, for each sign of the ground state, the manifold with codimension one in the energy space, consisting of solutions that converge to the ground state at time infinity. The two manifolds are joined at their boundary by the manifold with codimension two of solutions that are asymptotic to 2-solitons moving away from each other. The connected union of these three manifolds separates the rest of the neighborhood into the open set of global decaying solutions and that of blow-up.
... More recently, Burq, Raugel and Schlag [2] proved the soliton resolution for (1.1) in the full limit t → ∞ for all radial solutions. On the other hand, Côte, Martel, Yuan and Zhao [6] constructed a Lipschitz manifold in the energy space with codimension 2 of those solutions asymptotic to a sum of two ground states ...
... However, one may still expect some qualitative as well as quantitative relations and classifications among different types of solutions. For the undamped equation, Schlag and the second author [19] obtained such a classification into 9 sets in terms of the invariant manifolds of the ground states for E( u) < E( Q) + ε with small ε > 0. The goal of this paper is to obtain a similar classification around the 2-solitons (1.11), exploiting the damping, and investigating all the solutions starting close to but not on the manifold in [6]. The main interest is in the transition from 2-solitons to 1-solitons, both by time evolution and by initial perturbation. ...
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... In [10], it is given a complete description of 2-soliton solutions (that is, solutions which, on at least a sequence of time, behave as the sum of two decoupled ground states), in dimension N 5. Building on the tools developed there, [9] gave a complete description of global solutions in dimension N = 1, that is, the soliton resolution in that case. We aim at considering the behavior of solution without conditions on symmetry (like radiality). ...
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We consider the nonlinear damped Klein-Gordon equation ttu+2αtuΔu+uup1u=0on  [0,)×RN \partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \quad \text{on} \ \ [0,\infty)\times \mathbb{R}^N with α>0\alpha>0, 2N52 \le N\le 5 and energy subcritical exponents p>2p>2. We study the behavior of solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times. We first prove that the nonlinear object is necessarily a bound state. Next, we show that when the nonlinear object is a non-degenerate state or a degenerate excited state satisfying a simplicity condition, the convergence holds for all positive times, with an exponential or algebraic rate respectively. Last, we provide an example where the solution converges exactly at the rate t1t^{-1} to the excited state.
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