24 problems

The shallow water equations are quasilinear hyperbolic equations, and their singular solutions may include point singularities of the type of the square root of a quadratic form. M…

Let a spline shallow water moment system be regularized around a linear profile by the transformation described above, yielding a hyperbolic system matrix for systems based on an…

Dispersion conjecture. Localised smooth initial data satisfying the non-cavitation condition do not lead to stable structures localised in both spatial dimension. The solutions exi…

Gradient-catastrophe conjecture. Initial data violating the non-cavitation condition in one point in 1D lead to a gradient catastrophe in finite time.

Discrete Bernoulli instability conjecture. Such Riemann data, although they solve the discrete form of Bernoulli's equation, do not form a stable steady solution. The claim is a nu…

Consider the shallow water equations with discontinuous topography and a Riemann problem whose two states have constant Riemann invariants, so that the stationary contact wave is a…

Consider persistent roll-wave forms emerging either from periodic forcing or naturally from dambreak-type initial data. Persistent-wave stability conjecture. Persistent wave forms…

Consider a canal with a fixed fluid injection rate at one end and the associated homoclinic wave having the same fluid flux. Homoclinic height conjecture. The height of any resulti…

Consider the discontinuous Galerkin methods for the vector-invariant spherical shallow water equations described in the paper, and compare their convergence with the optimal fourth…

Let $q^e=\tilde q-q$ and $\eta^e=\tilde\eta-\eta$ denote the errors in the velocity potential and surface deviation, respectively, and let $\lambda$ be the observer-design paramete…

Let $k=1$, let $u_0\in H^s(\mathbb{R})$ with $s>3/2$, and let $u$ be the corresponding solution of equation (8.0.1), with lifespan $T$. Finite-time wave breaking conjecture. The so…

Let $u$ solve the generalized $0$-Holm-Staley equation … where $m=u-u_{xx}$. Conjecture on further conservation laws. No further conservation laws can be obtained for this equation…

Large-velocity solitary-wave conjecture. For every $c>1$, there exist smooth and rapidly decaying solitary-wave solutions with velocity $c$ such that: (i) the profiles $\zeta_c$ an…

Consider the shallow water equations with an arbitrary bottom topography, together with their local generalized symmetries and local conservation laws. A generalized symmetry is pr…

Consider the class of two-dimensional shallow water equations with variable bottom topography, denoted by $\mathcal L$, and its spaces of zeroth-order conservation laws. Two cases…

Let … , its generalized symmetries are exhausted, up to equivalence of generalized symmetries, by its Lie symmetries. This would show that the Lie-symmetry analysis captures all ge…

Let ProjMesh denote the projected mesh used to discretize the shallow-water equations, and let $L_2$ and $L_{\infty}$ denote the corresponding error norms. For a long-time integrat…

For the rSV system with $obreakvarepsilon0$, let a solution develop a blow-up singularity at some finite time. Post-blow-up singularity conjecture. After the blow-up time, sol…

The shallow-water model under consideration exhibits exponential convergence toward equilibrium with periodic oscillations, and its large-time behavior can be analyzed by computing…

Non-integrability conjecture. In general, the KRRI equation is not integrable by the inverse scattering transform. The equation has obvious integrable special cases when $a=0$ or…

Convergence-loss conjecture. This mismatch between the linearised wave cones and the physical domain of dependence is the origin of the loss of convergence for Thacker's planar sol…

Rigid-lid and fast-mode corrector approximation conjecture. Numerical simulations suggest that

Consider the rod equation studied in the paper, with parameter $amma<1$, and let $alpha_gamma$ be a coefficient replacing the coefficient $beta_gamma$ from the blowup theorem, whos…

Let $a_L$ and $a_R$ be the bottom levels on the left and right sides of a discontinuity, and let $h_L^{\#o}$ and $h_L^{o\#}$ denote the corresponding intermediate left water height…