8 problems

Let $u$ be a multi-soliton solution of the generalized KdV equation, defined at a given time, together with its spatial derivatives. Pointwise exponential decay conjecture. For eac…

Fix soliton parameters $A$, $\boldsymbol{\lambda}$, $\nu$, and $\sigma$. Let $\phi_{\mathrm{rad}}$ be a smooth solution of the wave equation $\Box\phi_{\mathrm{rad}}=0$ satisfying…

Let $J$ be the number of solitons in a solution of the energy-critical wave equation … on a space of dimension $N\geq 3$, and let … . This conjecture concerns the existence of radi…

General classification conjecture. Every such multi-soliton belongs to the $N$-parameter family $(\varphi_{A_1,\dots,A_N})_{(A_1,\dots,A_N)\in\mathbb{R}^N}$ constructed in Theorem…

Nonexistence conjecture. In the four-dimensional case, there exists no multi-soliton based on the ground state $W$ in this sense, for any $N\ge 2$.

Consider the focusing energy-critical wave equation in space dimension $3$ or $4$, and let a multi-soliton mean a solution satisfying the multi-soliton asymptotic conditions referr…

Let $u(t)$ be the multi-soliton solution constructed in the paper, defined for forward times $t\geq T_0$ and considered globally toward the past as far as its existence permits. Th…

Arbitrary-speed conjecture. The assumption on the speeds $c_1,\ldots,c_N$ is not optimal, and the result should hold for any choice of speeds.