16 problems

Nonuniform nonautonomous Markus–Yamabe conjecture. Under these conditions, the trivial solution of the nonlinear system is globally nonuniformly asymptotically stable.

Conditional asymptotic stability conjecture. The orbit $\mathcal O_H$ is conditionally asymptotically stable in the energy space: roughly, for initial data of the form…

Discreteness conjecture. As in the case $n=3$, $(p_3,p_4)\setminus F_4$ is a discrete subset of $(p_3,p_4)$.

Asymptotic stability conjecture. Asymptotic stability of solitary waves holds for this semilinear model.

Consider the nonlinear Schrödinger equation … where $u:{\mathbb R}^{1+3}\to \mathbb{C}$, $g\in C^\infty({\mathbb R},{\mathbb R})$, $g(0)=0$, and assume a family of positive radial…

Let $\rho(t)$ be the spatial shift appearing in the asymptotic description of a sine-Gordon kink solution with initial data in the energy space. Convergence conjecture. Under some…

Let $\hat{z}$ be the impedance introduced earlier, and let the Cauchy problem be the multidimensional wave equation with the associated impedance boundary condition. Assume that…

Consider an embedding heteroclinic network with transverse eigenvalues associated to its heteroclinic connections. Ashwin–Post conjecture. A sufficient, but not necessary, conditio…

Consider the stationary kink $H(x)=\tanh(x/\sqrt{2})$ of the constant-speed $\phi^4$ model, with perturbations $(\varphi_1,\varphi_2)\in H^1\times L^2$ in the energy space. Kowalcz…

Let the variable-speed $\phi^4$ model be considered near its stationary kink solution $H(x)=\tanh(x/\sqrt{2})$, and let perturbations lie in the energy space $H^1\times L^2$. Soffe…

Consider the discrete nonlinear Schrödinger evolution and its soliton family $e^{-i(-\mu t+\nu)}\alpha_\mu$, parametrized by $(\mu,\nu)$. The associated soliton manifold is the set…

Let $\mathcal{G}$ be the graph realised by a heteroclinic network $X$, let $\tilde{X}$ be a larger network containing $X$, and let $\zeta>0$ denote the noise amplitude. Cylinder-co…

Let $X$ be the constructed heteroclinic network and let $\tilde{X}$ be a larger network containing $X$. The parameter values determine the stability properties of these networks. A…

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

Let $S$ be a denumerable set, let $P\in PM_{ae}(S)$, let ${\cal M}\in G(S,P)$, and let ${\bf P}_{\cal M}$ be the transition probability function induced by ${\cal M}$. Suppose that…

Let $y_m$ be a stationary streamline, so that $U'(y_m)=0$, and let $\psi(y,t)$ be the stream function with velocity components $v_x(y,t)$ and $v_y(y,t)=ik\psi(y,t)$. Here $U$ is th…