We are interested in the nonlinear damped Klein–Gordon equation ∂ _t^2 u+2α∂ _t u-Δ u+u-|u|^p-1u=0 on ℝ^d for 2⩽ d⩽ 5 and energy sub-critical exponents 2< p < d+2/d-2 . We construct multi-soliton, that is, solutions which behave for large times as a sum of decoupled solitons, in various configurations with symmetry: this includes multi-soliton whose soliton centers lie at the vertices of an expanding regular polygon (with or without a center), of a regular polyhedron (with a center), of a higher dimensional regular polytope, or on a line. We give a precise description of these multi-solitons, in particular the interaction between nearest neighbour solitons is asymptotic to ln t - d-1/2lnln t as t → +∞ . We also prove that in any multi-soliton, the solitons cannot share all the same sign. Both statements generalize and refine the results of [13,14] and are based on the analysis developed in [8,9].

Non radiative solutions of the energy critical non linear wave equation are global solutions $u$ that furthermore have vanishing asymptotic energy outside the lightcone at both $t \to \pm \infty$: \[ \lim_{t \to \pm \infty} \| \nabla_{t,x} u(t) \|_{L^2(|x| \ge |t|+R)} = 0, \] for some $R \gt 0$. They were shown to play an important role in the analysis of long time dynamics of solutions, in particular regarding the soliton resolution: we refer to the seminal works of Duyckaerts, Kenig and Merle, see \cite{DKM:23} and the references therein. We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at $0$ is given by non radiative solutions to the linear equation (described in \cite{CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field.

We consider nonlinear elliptic equations of the form Δ u = f(u,∇ u) for the suitable analytic nonlinearity f, in the vinicity of infinity in ℝ^d , which is on the complement of a compact set. We show that there is a one-to-one correspondence between the nonlinear solution u defined there and the linear solution u_L to the Laplace equation such that, in an adequate space, u - u_L→ 0 as |x|→ +∞ . This is a kind of scattering operator. Our results apply in particular for the energy critical and supercritical pure power elliptic equation and for the 2d (energy critical) harmonic maps and the H-system. Similar results are derived for solutions defined on the neighborhood of a point in ℝ^d . The proofs are based on a conformal change of variables, and studied as an evolution equation (with the radial direction playing the role of time) in spaces with analytic regularity on spheres (the directions orthogonal to the radial direction).

We are concerned with solutions to the linear wave equation. We give an asymptotic formula for large time, valid in the energy space, via an operator related to the Radon transform. This allows us to show that the energy is concentrated near the light cone. This allows to derive further expressions the exterior energy (outside a shifted light cone). We in particular generalize the formulas of [CKS14] obtained in the radial setting. In odd dimension, we study the discrepancy of the exterior energy regarding initial energy, and prove in the general case the results of [KLLS15] (which were restricted to radial data).

We prove a first stability result of self-similar blow-up for the modified Korteweg–de Vries equation on the line. More precisely, given a self-similar solution and a sufficiently small regular profile, there is a unique global solution which behaves at $$t=0$$ as the sum of the self-similar solution and the smooth perturbation.

We consider a ferromagnetic nanowire, with an energy functional $E$ with easy-axis in the direction $e_1$, and which takes into account the Dzyaloshinskii-Moriya interaction. We consider configurations of the magnetization which are perturbations of two well separated domain wall, and study their evolution under the Landau-Lifshitz-Gilbert flow associated to E. Our main result is that, if the two walls have opposite speed, these configurations are asymptotically stable, up to gauges intrinsic to the invariances of the energy $E$. Our analysis builds on the framework developed in [4], taking advantage that it is amenable to space localisation.

We consider multi-breathers of (mKdV). In Semenov (2022), a smooth multi-breather was constructed, and proved to be unique in two cases: first, in the class of super-polynomial convergence to the profile (in the spirit of (Commun. Partial Differ. Equ. 46, 2325–2385, 2021)), and second, under the assumption that all speeds of the breathers involved are positive (without rate of convergence). The goal of this short note is to improve the second result: we show that uniqueness still holds if at most one velocity is negative or zero.

We consider the nonlinear damped Klein–Gordon equation ∂ttu+2α∂tu−Δu+u−|u|p−1u=0on[0,∞)×RNwith α>0, 2⩽N⩽5 and energy subcritical exponents p>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 t−1 to the excited state.

We provide a correction to the proof of Proposition 3 by Cote and Martel [Trans. Amer. Math. Soc. 370 (2018), pp. 7461-7487]. We refer to Yuan [Nonlinearity 32 (2019), pp. 5017-5048] for a self-contained proof in the analogue context of the energy critical wave equation. We also refer to Chen and Jendrej [Trans. Amer. Math. Soc. 372 (2019), pp. 7461-7496] for an alternate proof.

We study pointwise spatial decay of multi-solitons of the generalized Korteweg-de Vries equations. We obtain that, uniformly in time, these solutions and their derivatives decay exponentially in space on the left of and in the solitons region, and prove rapid decay on the right of the solitons. We also prove the corresponding result for multi-solitons of the nonlinear Schrödinger equations, that is, exponential decay in the solitons region and rapid decay outside.

We consider the Zakharov-Kuznetsov equation (ZK) in space dimension 2. Solutions u with initial data u(0) is an element of H-s are known to be global if s >= 1. We prove that for any integer s >= 2, parallel to u(t)parallel to(Hs) grows at most polynomially in t for large times t. This result is related to wave turbulence and how a solution of (ZK) can move energy to high frequencies. It is inspired by analoguous results by Staffilani [21] on the non linear Schrodinger and Korteweg-de-Vries equation. The main ingredients are adequate bilinear estimates in the context of Bourgain's spaces and a careful study of the variation of the H-s norm.

We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein-Gordon equation \begin{equation*} \partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \end{equation*} on $\bf R^N$, for $1\leq N\leq 5$ 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 $\log t$ as $t\to \infty$. This behavior is due to damping of the initial data combined with strong interactions between the solitary waves.

Abstract In this paper, we study some properties of multi-solitons for the non-linear Schrödinger equations in with general non-linearities. Multi-solitons have already been constructed in 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.

For the one-dimensional nonlinear damped Klein–Gordon equation $$\begin{aligned} \partial _{t}^{2}u+2\alpha \partial _{t}u-\partial _{x}^{2}u+u-|u|^{p-1}u=0 \quad \text{ on } \mathbb {R}\times \mathbb {R}, \end{aligned}$$ with $$\alpha >0$$ and $$p>2$$ , we prove that any global finite energy solution either converges to 0 or behaves asymptotically as $$t\rightarrow \infty $$ as the sum of $$K\ge 1$$ decoupled solitary waves. In the multi-soliton case $$K\ge 2$$ , the solitary waves have alternate signs and their distances are of order $$\log t$$ .

We prove a local well-posedness result for the modified Korteweg–de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an asymptotic description of small solutions as $t \to +\infty $.

We consider the nonlinear damped Klein-Gordon equation \[ \partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \quad \text{on} \ \ [0,\infty)\times \mathbb{R}^N \] with $\alpha>0$, $2 \le N\le 5$ and energy subcritical exponents $p>2$. 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 $t^{-1}$ to the excited state.

We give the asymptotics of the Fourier transform of self-similar solutions for the modified Korteweg-de Vries equation. In the defocussingcase, the self-similar profiles are solutions to the Painleve II equation; although they were extensively studied in physical space, no result to our knowledge describe their behavior in Fourier space. These Fourier asymptotics are crucial in the study of stability properties of the self-similar solutions for the modified Korteweg-de Vries flow. Our result is obtained through a fixed point argument in a weighted W-1,W-infinity space around a carefully chosen, two term ansatz, and we are able to relate the constants involved in the description in Fourier space with those of the description in physical space. (C) 2020 Elsevier Masson SAS. All rights reserved.

Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1 + 4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded on the maximal interval of existence. We prove that along a sequence of times tending to the maximal forward time of existence, the solution decomposes into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume that the critical norm of the evolution localized to the light cone (the forward light cone in the case of global solutions and the backwards cone in the case of finite time blow-up) is less than 2 times the critical norm of the ground state solution W, then the decomposition holds without a restriction to a subsequence.