arXiv subjects

Vianney Combet

Publications and source records attributed to Vianney Combet.

7 recordsLinked to original sources

arXiv2017

Construction of multi-bubble solutions for the critical gKdV equation

We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation $\partial_t u + \partial_x(\partial_{xx} u + u^5) = 0$ containing an arbitrary number $K\geq 2$ of blow up bubbles, for any choice of sign and scaling parameters: for any $\ell_1>\ell_2>\cdots>\ell_K>0$ and $ε_1,\ldots,ε_K\in\{\pm1\}$, there exists an $H^1$ solution $u$ of the equation such that \[ u(t) - \sum_{k=1}^K \frac {ε_k}{λ_k^\frac12(t)} Q\left( \frac {\cdot - x_k(t)}{λ_k(t)} \right) \longrightarrow 0 \quad\mbox{ in }\ H^1 \mbox{ as }\ t\downarrow 0, \] with $λ_k(t)\sim \ell_k t$ and $x_k(t)\sim -\ell_k^{-2}t^{-1}$ as $t\downarrow 0$. The construction uses and extends techniques developed mainly by Martel, Merle and Raphaël. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved by the authors in arXiv:1602.03519.

math.AP↗

arXiv2016

Sharp asymptotics for the minimal mass blow up solution of critical gKdV equation

Let $S$ be a minimal mass blow up solution of the critical generalized KdV equation as constructed by Martel, Merle and Raphaël in arXiv:1204.4624. We prove both time and space sharp asymptotics for $S$ close to the blow up time. Let $Q$ be the unique ground state of (gKdV), satisfying $Q"+Q^5=Q$. First, we show that there exist universal smooth profiles $Q_k\in\mathcal{S}(\mathbb{R})$ (with $Q_0=Q$) and a constant $c_0\in\mathbb{R}$ such that, fixing the blow up time at $t=0$ and appropriate scaling and translation parameters, $S$ satisfies, for any $m\geqslant 0$, \[ \partial_x^m S(t) - \sum_{k=0}^{[m/2]} \frac 1{t^{\frac 12+m-2k}} Q_k^{(m-k)}\left(\frac{\cdot+ \frac1t}{t}+c_0\right)\to 0\quad \mbox{in}\ L^2 \mbox{as}\ t\downarrow 0. \] Second, we prove that, for $0<t\ll 1$, $x\leqslant -\frac 1t -1$, \[ S(t,x) \sim - \frac 12 \|Q\|_{L^1} |x|^{-3/2}, \] and related bounds for the derivatives of $S(t)$ of any order. We also prove $\int_{\mathbb{R}} S(t,x)\,dx=0$.

math.AP↗

arXiv2015

Classification of minimal mass blow-up solutions for an $L^2$ critical inhomogeneous NLS

We establish the classification of minimal mass blow-up solutions of the $L^2$ critical inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b}|u|^{\frac{4-2b}{N}}u = 0, \] thereby extending the celebrated result of Merle from the classic case $b=0$ to the case $0<b<\min\{2,N\}$, in any dimension $N\ge1$.

math.AP↗

arXiv2013

Local Dynamics Near Unstable Branches of NLS Solitons

Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions, solutions to initial data from a neighbourhood of the branch either converge to a soliton, or exit a larger neighbourhood of the branch transversally. The qualitative dynamic near a branch of unstable solitons is irrespective of whether blowup eventually occurs, which has practical implications for the description of blowup of NLS with supercritical nonlinearity.

math.AP↗

arXiv2010

Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension

For the L2 supercritical generalized Korteweg-de Vries equation, we proved in a previous article the existence and uniqueness of an N-parameter family of N-solitons. Recall that, for any N given solitons, we call N-soliton a solution of the equation which behaves as the sum of these N solitons asymptotically as time goes to infinity. In the present paper, we also construct an N-parameter family of N-solitons for the supercritical nonlinear Schrödinger equation, in dimension 1 for the sake of simplicity. Nevertheless, we do not obtain any classification result; but recall that, even in subcritical and critical cases, no general uniqueness result has been proved yet.

math.AP↗

arXiv2010

Multi-soliton solutions for the supercritical gKdV equations

For the L^2 subcritical and critical (gKdV) equations, Martel 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 time goes to infinity. More recently, for the L^2 supercritical case, Cote, Martel and Merle proved the existence of at least one multi-soliton. In the present paper, as suggested by a previous work concerning the one soliton case, 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).

math.AP↗

arXiv2009

Construction and characterization of solutions converging to solitons for supercritical gKdV equations

We consider the generalized Korteweg-de Vries equation in the supercritical case, and we are interested in solutions which converge to a soliton in large time in H^1. In the subcritical case, such solutions are forced to be exactly solitons by variational characterization, but no such result exists in the supercritical case. In this paper, we first construct a "special solution" in this case by a compactness argument, i.e. a solution which converges to a soliton without being a soliton. Secondly, using a description of the spectrum of the linearized operator around a soliton due to Pego and Weinstein, we construct a one parameter family of special solutions which characterizes all such special solutions.

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