For the 5D energy-critical wave equation, we construct excited $N$-solitons with collinear speeds, i.e. solutions $u$ of the equation such that \begin{ equation*} \lim_{ t\to+\infty}\bigg\|\nabla_{ t,x}u(t)-\nabla_{ t,x}\bigg(\sum_{ n=1}^{ N}Q_{ n}(t)\bigg)\bigg\|_{ L^{ 2}}=0, \end{ equation*} where for $n=1,\ldots,N$, $Q_n(t,x)$ is the Lorentz transform of a non-degenerate and sufficiently decaying excited state, each with different but collinear speeds. The existence proof follows the ideas of Martel-Merle and C\^ote-Martel developed for the energy-critical wave and nonlinear Klein-Gordon equations. In particular, we rely on an energy method and on a general coercivity property for the linearized operator.
Categories: math.AP