Rogue waves for a semilinear wave equation

We consider the semilinear wave equation

𝑉(𝑥) ∂²_𝑡 𝑢 + d(𝑡) 𝑀(𝑥, ∂_𝑥) 𝑢 = 𝑉(𝑥) d̃(𝑡) |𝑢|^(𝑝−1) 𝑢,  (𝑥,𝑡) ∈ ℝᴺ × ℝ

where 𝑀 is in general an elliptic operator and 𝑝 ∈ (1, 𝑝*). Under suitable conditions on 𝑀 and on the coefficients 𝑉, d and d̃, we show the existence of solutions which are localized in space and in time, so‑called rogue waves, by means of variational methods. We discuss our approach in detail and provide examples of 𝑀, 𝑉, d, d̃ which satisfy our assumptions.