Related papers: Refined asymptotics around solitons for gKdV equat…
We consider the generalized Korteweg-de Vries equation \partial_t u + \partial_x (\partial_x^2 u + f(u))=0, \quad (t,x)\in [0,T)\times \mathbb{R}, (1) with general $C^3$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there…
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 $u_t+(u_{xx}+u^p)_x=0$ (1<p<5). The proof of the stability result is based on energy…
We consider the generalized Korteweg-de Vries equation $\partial_t u = -\partial_x(\partial_x^2 u + f(u))$, where $f(u)$ is an odd function of class $C^3$. Under some assumptions on $f$, this equation admits \emph{solitary waves}, that is…
We revisit the phenomenon of instability of solitons in the generalized Korteweg-de Vries equation, $u_t + \partial_x(u_{xx} + u^p) = 0$. It is known that solitons are unstable for nonlinearities $p \geq 5$, with the critical power $p=5$…
We revisit the phenomenon of instability of solitons in the two dimensional generalization of the Korteweg-de Vries equation, the generalized Zakharov-Kuznetsov (ZK) equation, $u_t + \partial_{x_1} (\Delta u + u^p) = 0, (x_1,x_2) \in…
We study the long-time stability of soliton solutions to the Korteweg-deVries equation. We consider solutions $u$ to the KdV with initial data in $H^s$, $0 \leq s < 1$, that are initially close in $H^s$ norm to a soliton. We prove that the…
In this paper, we reconsider the well-known result of Pego-Weinstein \cite{MR1289328} that soliton solutions to the Korteweg-deVries equation are asymptotically stable in exponentially weighted spaces. In this work, we recreate this result…
We continue our study of the collision of two solitons for the subcritical generalized KdV equations. In a previous paper, mainly devoted to the case of the quartic gKdV equation, we have introduced a new framework to understand the…
We consider the two dimensional generalization of the Korteweg-de Vries equation, the generalized Zakharov-Kuznetsov (ZK) equation, $u_t + \partial_{x_1}(\Delta u + u^p) = 0, (x_1,x_2) \in \mathbb{R}^2$. It is known that solitons are stable…
We apply our idea, which previously we used in the analysis of the pure power NLS, consisting in spitting the virial inequality method into a large energy inequality combined with Kato smoothing, to the case of generalized Korteweg--De…
We establish the asymptotic stability of smooth solitons and multi-solitons for the Camassa-Holm (CH) equation in the energy space $H^1(\R)$. We show that solutions initially close to a soliton converge, up to translation, weakly in…
We present a detailed numerical study of solutions to the Zakharov-Kuznetsov equation in three spatial dimensions. The equation is a three-dimensional generalization of the Korteweg-de Vries equation, though, not completely integrable. This…
We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and…
We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schr\"{o}dinger equations $$ iu_t+\Delta u=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}^2\times\mathbb{R}$,}$$ in the energy class.
We prove the asymptotic stability in the energy space of non-zero speed solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy. More precisely, we show that any solution corresponding to an initial datum…
In this paper, we prove the asymptotic stability of the family of solitons of the Benjamin-Ono equation in the energy space. The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of…
We construct $N$-soliton solutions for the fractional Korteweg-de Vries (fKdV) equation $$ \partial_t u - \partial_x\left(|D|^{\alpha}u - u^2 \right)=0, $$ in the whole sub-critical range $\alpha \in]\frac12,2[$. More precisely, if $Q_c$…
We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation $\partial_t u = -\partial_x (\partial_x^2 u + 3u^2-bu)$, where $b(x,t) = b_0(hx,ht)$, $h\ll 1$ is a slowly varying, but not small, potential. We option an…
In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the $I$-method and spectral analysis following Pego and Weinstein, we are…
We consider the nonlinear focusing Klein-Gordon equation in $1 + 1$ dimensions and the global space-time dynamics of solutions near the unstable soliton. Our main result is a proof of optimal decay, and local decay, for even perturbations…