72 problems

For a general dispersive equation, let a generic solution mean a solution in the class under consideration, and let a multisolitary wave be a sum of solitary waves. Soliton resolut…

The Skyrme–Faddeev model is a theory in $(3+1)$-dimensional Minkowski space-time whose field is a unit vector ${\vec n}$ taking values on $S^2$, with a quadratic kinetic term and a…

Coercivity conjecture. The condition above holds for every $g>1$.

Let $Q$ be the unique positive radial solution of … Consider solutions $v$ of the $L^2$-critical nonlinear Schrödinger equation with critical mass $\|v\|_{L^2}=\|Q\|_{L^2}$, where…

Let $n$-soliton solutions be the multi-soliton solutions constructed on noncommutative Euclidean spaces by noncommutative Darboux and Bäcklund transformations. An isolated localize…

A soliton solution is a solution constructed using Bäcklund transformations, and an attached Akhmediev breather is an Akhmediev breather attached to such a soliton solution. Instab…

Let the nonlinear Klein–Gordon (NLKG) equation be the relativistic analogue of the nonlinear Schrödinger equation, and consider its corresponding stationary soliton solutions. Chen…

Consider a discrete nonlinear Schrödinger equation on a lattice, with localized modes or discrete breathers understood as periodic-in-time and spatially localized solutions. A latt…

Let a soliton equation have a quasi-periodic background, and consider a short-range initial perturbation whose soliton phase shifts add up to zero. Soliton-only asymptotics conject…

Generic-dynamics conjecture. Generic perturbations of an excited soliton should lead either to blowup, or to collapse to a less energetic soliton (or to the vacuum state), plus rad…

Let $K(m,n)$ denote the generalized Korteweg–de Vries equation admitting compacton and anticompacton solutions. Based on numerical experiments with soliton-antisoliton collisions,…

A biopolymer is modeled as a curved and twisted polymer whose curvature and torsion interact with an electron, producing a localized electronic state and a moving soliton solution.…

Soliton–surface correspondence conjecture. The relation above is true; in particular, the soliton describes the "unoriented" pretzel with $g=2$.

In the Gross–Neveu model, consider static soliton multiplets other than the DHN multiplets with $K=1$, $q=0$, the CCGZ kink multiplet with $K=1$, $|q|=1$, and the heavier topologic…

In an integrable system such as KdV, scattering is elastic and factorised: soliton spectral parameters are preserved and multi-soliton scattering decomposes into two-soliton events…

For a nonlinear Schrödinger equation with a general nonlinearity, consider pure multi-solitons, namely exact solutions that decouple asymptotically into sums of travelling waves. C…

The massive Thirring model has a Lax spectral problem whose continuous spectrum contains the imaginary axis. Multiple embedded-eigenvalue conjecture. Multiple embedded eigenvalues…

Let $\mathrm{NLL}(\beta=2n)$ denote the space of soliton states at coupling $\beta=2n$, where the more general formula is obtained from polynomials with distinct roots. Jackiw–Pi d…

Consider a Gaussian initial pulse for the KdV equation, and let the predicted soliton amplitudes be obtained from relations between the pulse parameter and soliton amplitudes using…

Let $J$ be a positive integer, and consider the stable manifolds associated with $2J$-chains and $(2J+1)$-chains of alternating kinks and anti-kinks. The stable-manifold codimensio…

Equivariant wave maps from the $2+1$-dimensional wormhole to the $2$-sphere admit explicit harmonic map solutions which, in suitable coordinates, are sine-Gordon kinks and anti-kin…

Consider the two-soliton configuration in the four-dimensional Wess–Zumino–Witten model, whose Wess–Zumino action is denoted by $S_{\mathrm{WZ}}$. Vanishing Wess–Zumino action conj…

Let $alpha$ be a one-form on $X$ with a single Morse zero, and let $tilde{X}$ be the minimal cover on which the superpotential $tilde{W}$ is defined. Let…

Let $alpha$ be a closed holomorphic one-form on $X$ with a unique Morse vacuum, defining a cyclic holomorphic cover $tilde{X}$ with deck group $Gamma=\mathbb{Z}langlegammarangle$.…

The model has an interior amplitude, obtained from the Landau–Ginzburg thimble and soliton data described above. Finite-triangle conjecture. The interior amplitude of the model is…