8 problems

Let $k\in(0,1)$ be the elliptic modulus and let $c_d=c_d(\lambda)$ denote the velocity of a dark breather parametrized by $\lambda\in(\lambda_2(k),\lambda_3(k))$. Dark breather vel…

Let $k$ be the elliptic modulus and let $c_b=c_b(\lambda)$ denote the velocity of a bright breather parametrized by $\lambda\in(-\infty,\lambda_1(k))$. Bright breather velocity con…

Caballero–Wayne conjecture. Perturbations by damping are formally related to perturbations by noise.

Birnir–McKean–Weinstein conjecture. For the general nonlinear wave equation in $1+1$ dimensions, breathing takes place only for isolated nonlinearities.

Higher-order mKdV breather energy conjecture. For every $n\in\mathbb{N}$, the $(2n+1)$-st order energy of $B$ is

Consider the nonlinear wave equation … A breather is a solution in $H^1\times L^2$ that is periodic in time, up to the symmetries of the equation. Soffer–Weinstein's conjecture. No…

For the modified Korteweg–de Vries equation, consider breathers, namely spatially localized, time-periodic solutions; a breather has positive velocity when its propagation velocity…

Non-resonance conjecture. In this setting, the non-resonance condition should be replaced by