arXiv subjects

Xu Yuan

Publications and source records attributed to Xu Yuan.

58 records · Page 4Linked to original sources

arXiv2020

On multi-solitons for the energy-critical wave equation in dimension 5

In this paper, we construct $K$-solitons of the focusing energy-critical nonlinear wave equation in five-dimensional space, i.e. solutions $u$ of the equation such that \begin{equation*} \|\nabla_{t,x}u(t)-\nabla_{t,x}\big(\sum_{k=1}^{K}W_{k}(t)\big)\|_{L^{2}}\to 0\quad \mathrm{as}\ t\to \infty, \end{equation*} where for any $k\in \{1,\dots,K\}$, $W_{k}$ is Lorentz transform of the explicit standing soliton $W(x)=(1+|x|^{2}/15)^{-3/2}$, with any speed $\boldsymbol{\ell}_{k}\in \mathbb{R}^{5}$ ,$|\boldsymbol{\ell}_{k}|<1$ ($\boldsymbol{\ell}_{k'}\ne \boldsymbol{\ell}_{k}$ for $k'\ne k$) satisfying an explicit smallness condition.

math.AP↗

arXiv2020

Conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity

We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity \begin{equation*} \partial_{t}^{2}u-\partial_{x}^{2}u+u-|u|^{p-1}u+|u|^{q-1}u=0,\quad \mbox{on}\ [0,\infty)\times \mathbb{R}, \end{equation*} with $1<q<p<\infty$. The main result states the stability in the energy space $H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$ of the sums of decoupled solitary waves with different speeds, up to the natural instabilities. The proof is inspired by the techniques developed for the generalized Korteweg-de Vries equation and the nonlinear Schrödinger equation in a similar context by Martel, Merle and Tsai [14,15]. However, the adaptation of this strategy to a wave-type equation requires the introduction of a new energy functional adapted to the Lorentz transform.

math.AP↗

arXiv2020

Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation

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

math.AP↗

arXiv2019

Description and classification of 2-solitary waves for nonlinear damped Klein-Gordon equations

We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein-Gordon equation \begin{equation*} \partial_{tt}u+2α\partial_{t}u-Δu+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.

math.AP↗