14 problems

The generalized Darboux transformation produces rogue-wave solutions of the nonlinear Schrödinger equation and rational solutions of the Painlevé-II equation, both expressible as G…

Let $\mathcal{PT}$-symmetric vector rational rogue waves be solutions of the multi-component nonlinear Schrödinger equations, and let their asymptotic structures refer to the patte…

Let $\{K_n\subset\mathbb{R}^2\}_{n=1}^\infty$ be a sequence of compact sets satisfying … Let $\psi_{2n}$ and $\psi_{2n-1}$ denote the even- and odd-order finite rogue waves, and le…

Lump-circle radius conjecture. For both cases, the radius satisfies

Ankiewicz–Akhmediev conjecture. For any NLS rogue wave, the number of components is a triangular number, and specifically

Let $\Theta_n^{\pm}(x,t)$ be the polynomials defined by … where $P_n(x,t)$ and $Q_n(x,t)$ are the polynomials in the generalised rational solution of the Boussinesq equation. Bilin…

The complex short pulse equation admits an $N$-breather solution given by the determinant formula above, with $M$ and $G$ as defined there. N-breather law conjecture. The $N$-breat…

The nonlinear Schrödinger equation (NLS) is used to model certain water-wave phenomena, and a Peregrine wave solution is a localized solution of the NLS associated with rogue-wave…

Consider the $k$-th order rogue wave of the derivative nonlinear Schrödinger equation of the second type (DNLSII), with $k$ at least two, and a pattern consisting of a lower-order…

Consider the $k$-th order rogue wave of the derivative nonlinear Schrödinger equation of the second type (DNLSII), and regard a first-order rogue wave as one elementary rogue-wave…

Consider the polynomials occurring in the $k$-th order rogue wave of the derivative nonlinear Schrödinger equation of the second type (DNLSII). Polynomial-degree conjecture. Their…

Let $c$ be the asymptotical height of a rogue wave, and let the $k$-th order rogue wave of the derivative nonlinear Schrödinger equation of the second type (DNLSII) be considered.…

Let an order-$n$ rogue wave of the nonlinear Schrödinger equation be considered, and let a pattern mean a spatial-temporal configuration of its peaks. Uniform-peak pattern conjectu…

Let an order-$n$ rogue wave under the fundamental pattern of the nonlinear Schrödinger equation be given, and consider its maxima in space-time. Matveev's peak-count conjecture. Th…