14 problems

Negative-slope instability conjecture. The availability of a section of the stability curve with negative slope is a sufficient condition for instability of the soliton. The paper…

Let $psi$ be the solution under consideration, with modulation parameters and weak convergence statements as in the preceding theorem. For constants…

Consider the perturbed initial-value problem for the KP equation with the types of initial data studied by Kao, including $V$-type initial-value waves. Let $u(x,y,t)$ denote its so…

For a solution in the range of the Bäcklund transform, the codimension-one condition is intrinsic. Intrinsic codimension-one condition conjecture. This condition should correspond…

The Korteweg–de Vries equation is … Its one-soliton solutions are … The Kadomtsev–Petviashvili equations are … where the minus sign gives KPI and the plus sign gives KPII. Kadomtse…

For $\omega<\omega_c$, consider initial data of the form … Here $Q_\omega$ is a cubic-quintic ground state, $M$ denotes mass, and…

Consider a cubic-quintic nonlinear Schrödinger ground state $Q_\omega$ on the unstable branch and a perturbation of the corresponding initial data. Suppose the perturbed solution e…

Let $E$ satisfy $|E|\ll 10$. A small-energy blow-up conjecture asserts that the KdV soliton for small $a$ is stable under NV dynamics, while the KdV soliton for large $a$ is unstab…

Let $E\ll -1$. A large-negative-energy behavior conjecture asserts that the KdV soliton is stable under NV dynamics and that localized initial data will be radiated away to infinit…

Let $E\gg 1$. A large-positive-energy behavior conjecture asserts that the KdV soliton for small $a$ is stable under NV dynamics, whereas the KdV soliton for large $a$ is unstable…

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…

Consider the phase portrait of a driven nonlinear Schrödinger soliton obtained from the collective-coordinate trajectories, and let the resulting closed curve represent the soliton…

Consider unstable solitons and perturbations that preserve a soliton component. Stable-soliton selection conjecture. Soliton-preserving perturbations of unstable solitons dynamical…

Consider solutions of the nonlinear Schrödinger equation that do not disperse as $t\rightarrow\infty$, and let the stable portion of the soliton curve be the region where…